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Miller, Mark Andrew. "Multiscale techniques for imaging problems." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613033.
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Holst, Henrik. "Multiscale Methods for Wave Propagation Problems." Doctoral thesis, KTH, Numerisk analys, NA, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-48072.
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Simulations of wave propagation in heterogeneous media and at high frequencies are important in many applications such as seismic-, {electro-magnetic-,} acoustic-, fluid flow problems and others. These are classical multiscale problems and often too computationally expensive for direct numerical simulation. The smallest scales must be well resolved over a computational domain represented by the largest scale and this results in a very high computational cost. We develop and analyze numerical techniques based on the heterogeneous multiscale method (HMM) framework for such wave equations with highly oscillatory solutions $u^{\varepsilon}$ where $\varepsilon$ represents the size of the smallest scale. In these techniques the oscillatory microscale is approximated on small local microproblems of size $\varepsilon$ in spatial and time directions. The solution of the microproblems are then coupled to a global macroscale model in divergence form $u_{tt} = \nabla \cdot F$ where the flux $F$ is obtained from the microproblems. The oscillations can either originate from fluctuations in the velocity coefficients or from high frequency initial and boundary conditions. We have developed algorithms that couple micro and macroscales for both these cases. The choice of macroscale variables is inspired by the analytic theories of homogenization and geometrical optics respectively. In the first case local averages $u \approx u^{\varepsilon}$ are used on the macroscale. In the second case, phase $\phi$ and energy are natural macroscopic variables. There are two major goals of this research. One goal is to develop and analyze algorithms for simulating multiscale wave propagation with low computational complexity, and even independent of $\varepsilon$ for finite time problems. This is seen in many examples in one, two and three dimensions. The other goal is to use wave propagation as a model to better understand the HMM framework. An example in this direction is simulation with oscillatory wave field over long time. The dispersive effects that then occur is well approximated by a HMM method that was originally formulated for finite time where added accuracy is required but no explicit adjustment to include dispersion, an evidence of the robustness of the method.Simulering av högfrekventa vågor i heterogena material är viktigt i många tillämpningar, till exempel seismologi, elektromagnetism, akustik och strömningsmekanik. Dessa tillämpningar är exempel på klassiska multiskalproblem och har typiskt en för hög beräkningskostnad, i form av datortid och minne, för en direkt numerisk simulering. De minsta skalorna i problemet måste vara upplösta över ett område som representeras av dom största skalorna och detta innebär en hög beräkningskostnad. Vi har utvecklat och analyserat numeriska metoder för vågekvationer med snabbt oscillerande lösningar $u^{\varepsilon}$ där $\varepsilon$ representerar storleken på den minsta skalan. Metoderna är baserade på ramverket \emph{heterogena multiskalmetoden} (HMM). I dessa metoder approximeras den hastigt oscillerande mikroskalan med små lokala mikroproblem av storleksordning $\varepsilon$ i tids- och rumsriktning. Lösningen till mikroproblemen är kopplade till en global modell på makroskalan i divergensform $u_{tt} = \nabla \cdot F$, där flödet $F$ ges av mikroproblemen. De hastiga oscillationerna kan härröras från snabba variationer i hastighetsfältet, begynnelsevillkor eller randvillkor. Vi har utvecklat algoritmer som kopplar mikro- och makroskalor i bägge fallen. Valet av makroskalvariabler inspireras av de analytiska metoderna homogenisering och geometrisk optik. I det första fallet används lokala medelvärden $u \approx u^{\varepsilon}$ på makroskalnivån. I det andra fallet är fas $\phi$ och energi bra val av makroskalvariabler. Det finns två huvudmål med vår forskning. Ett mål är att utveckla och analysera algoritmer för simulering av vågproblem med multipla skalor med låg beräkningskostnad (om möjligt, oberoende av $\varepsilon$) för problem över begränsad tid. Vi visar numeriska resultat från multiskalproblem i en, två och tre dimensioner. Det andra målet är att att använda vågutbredning som en modell för att bättre förstå HMM ramverket. Ett exempel på detta är simulering med oscillerande hastighetsfält över lång tid. Efter lång tid så uppträder dispersion. Vi har demonstrerat att vår HMM-metod, som ursprungligen var formulerad för begränsad tid, även kan appliceras på detta fall. För att få den rätta dispersionen krävs högre noggrannhetsordning, men metoden ändrar inte form. Detta visar på metodens robusthet.
QC 20111117
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Söderlund, Robert. "Finite element methods for multiscale/multiphysics problems." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-42713.
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In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.
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Elfverson, Daniel. "Discontinuous Galerkin Multiscale Methods for Elliptic Problems." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-138960.
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In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.
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Savchuk, Tatyana. "The multiscale finite element method for elliptic problems." Ann Arbor, Mich. : ProQuest, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3245025.
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Thesis (Ph. D. in Applied Mathematics)--Southern Methodist University, 2007.Title from PDF title page (viewed Mar. 18, 2008). Source: Dissertation Abstracts International, Volume: 67-12, Section: B, page: 7120. Adviser: Zhangxin (John) Chen. Includes bibliographical references.
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Kudreyko, Aleksey. "Multiscale wavelet analysis for integral and differential problems." Doctoral thesis, Universita degli studi di Salerno, 2011. http://hdl.handle.net/10556/176.
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2009 - 2010The object of the present research is wavelet analysis of integral and differential problems by means of harmonic and circular wavelets. It is shown that circular wavelets constitute a complete basis for L2[0; 1] functions, and form multiresolution analysis. Multiresolution analysis can be briefly considered as a decomposition of L2[0; 1] into a complete set of scale depending subspaces of wavelets. Thus, integral operators, differential operators, and L2(R) functions were investigated as scale depending functions through their projection onto these subspaces of wavelets. In particular: - conditions when a certain wavelet can be applied for solution of integral or differential problem are given; - it is shown that the accuracy of this approach exponentially grows when increasing the number of vanishing moments and scaling parameter; - wavelet solutions of low-dimensional nonlinear partial differential equations are compared with other methods; - wavelet-based approach is applied to low-dimensional Fredholm integral equations and the Galerkin method for two-dimensional Fredholm integral equations.[edited by author]. Oggetto della seguente ricerca `e l’analisi di problemi differenziali e integrali, utilizzando wavelet armoniche e wavelet armoniche periodiche. Si dimostra che le wavelet periodiche costituiscono una base completa per le funzioni L2[0; 1] e formano un’analisi multiscala. L’analisi multirisoluzione pu`o essere brevemente considerata come la decomposizione di L2[0; 1] in un insieme completo di sottospazi di wavelet dipendenti da un fattore di scala. Pertanto gli operatori integrali e differenziali e le funzioni L2(R) vengono studiati come funzioni di scala mediante le corrispondenti proiezioni in questi sottospazi di wavelet. In particolare, vengono sviluppati quattro principali argomenti: - sono state individuate le condizioni per applicare una data famiglia di wavelets alla soluzione di un data problema differenziale o integrale; - si `e dimostrato che la precisione di questo approccio cresce esponenzialmente quando decresce il numero dei momenti nulli e del parametro di scala; - soluzioni wavelet di equazioni differenziali a derivate parziali nonlineari di dimensione bassa sono state confrontate con altri metodi di soluzioni; - l’approccio basato sull’uso delle wavelet `e stato applicato anche per ricerca di soluzioni di alcune equazioni integrali di Fredholm e insieme al metodo di Galerkin per risolvere equazioni integrali Fredholm di dimensioni due.[a cura dell'autore]
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Hellman, Fredrik. "Multiscale and multilevel methods for porous media flow problems." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-262276.
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We consider two problems encountered in simulation of fluid flow through porous media. In macroscopic models based on Darcy's law, the permeability field appears as data. The first problem is that the permeability field generally is not entirely known. We consider forward propagation of uncertainty from the permeability field to a quantity of interest. We focus on computing p-quantiles and failure probabilities of the quantity of interest. We propose and analyze improved standard and multilevel Monte Carlo methods that use computable error bounds for the quantity of interest. We show that substantial reductions in computational costs are possible by the proposed approaches. The second problem is fine scale variations of the permeability field. The permeability often varies on a scale much smaller than that of the computational domain. For standard discretization methods, these fine scale variations need to be resolved by the mesh for the methods to yield accurate solutions. We analyze and prove convergence of a multiscale method based on the Raviart–Thomas finite element. In this approach, a low-dimensional multiscale space based on a coarse mesh is constructed from a set of independent fine scale patch problems. The low-dimensional space can be used to yield accurate solutions without resolving the fine scale.
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Parno, Matthew David. "A multiscale framework for Bayesian inference in elliptic problems." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/65322.
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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2011.Page 118 blank. Cataloged from PDF version of thesis.
Includes bibliographical references (p. 112-117).
The Bayesian approach to inference problems provides a systematic way of updating prior knowledge with data. A likelihood function involving a forward model of the problem is used to incorporate data into a posterior distribution. The standard method of sampling this distribution is Markov chain Monte Carlo which can become inefficient in high dimensions, wasting many evaluations of the likelihood function. In many applications the likelihood function involves the solution of a partial differential equation so the large number of evaluations required by Markov chain Monte Carlo can quickly become computationally intractable. This work aims to reduce the computational cost of sampling the posterior by introducing a multiscale framework for inference problems involving elliptic forward problems. Through the construction of a low dimensional prior on a coarse scale and the use of iterative conditioning technique the scales are decouples and efficient inference can proceed. This work considers nonlinear mappings from a fine scale to a coarse scale based on the Multiscale Finite Element Method. Permeability characterization is the primary focus but a discussion of other applications is also provided. After some theoretical justification, several test problems are shown that demonstrate the efficiency of the multiscale framework.
by Matthew David Parno.
S.M.
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Biezemans, Rutger. "Multiscale methods : non-intrusive implementation, advection-dominated problems and related topics." Electronic Thesis or Diss., Marne-la-vallée, ENPC, 2023. http://www.theses.fr/2023ENPC0029.
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Cette thèse porte sur les méthodes numériques pour les équations aux dérivées partielles (EDP) multi-échelles, et en particulier sur la méthode dite des éléments finis multi-échelles (MsFEM). Celle-ci est une méthode de type éléments finis qui consiste en une approximation de Galerkin de l'EDP sur une base problème-dépendante. Trois difficultés particulières liées à cette méthode sont abordées dans cette thèse. Premièrement, puisque la MsFEM utilise une base problème-dépendante, la méthode ne peut être facilement implémentée dans des codes industriels génériques. Cela freine la diffusion de la MsFEM au-delà des environnements académiques. Une méthodologie générique est proposée pour convertir la MsFEM en un problème effectif qui peut être résolu par des codes génériques. Il est démontré par des résultats théoriques ainsi que des expériences numériques que la nouvelle méthodologie est aussi précise que la MsFEM originale. Deuxièmement, les MsFEM adaptées aux problèmes advection-dominés sont étudiées. Ce régime spécifique rend instables les discrétisations naïves. Une explication est trouvée pour l'instabilité de certaines méthodes proposées précédemment. Des expériences numériques montrent la stabilité d'une MsFEM avec des conditions aux limites de type Crouzeix-Raviart enrichie par des fonctions bulles. Troisièmement, une nouvelle analyse de convergence pour la MsFEM est présentée, permettant pour la première fois d'établir la convergence sous des hypothèses de régularité minimales. Cette démarche est importante pour réduire l'écart entre la théorie pour la MsFEM et son application en pratique, où les hypothèses de régularité habituelles sont rarement satisfaitesThis thesis is concerned with computational methods for multiscale partial differential equations (PDEs), and in particular the multiscale finite element method (MsFEM). This is a finite element type method that performs a Galerkin approximation of the PDE on a problem-dependent basis. Three particular difficulties related to the method are addressed in this thesis. First, the intrusiveness of the MsFEM is considered. Since the MsFEM uses a problem-dependent basis, it cannot easily be implemented in generic industrial codes and this hinders its adoption beyond academic environments. A generic methodology is proposed that translates the MsFEM into an effective problem that can be solved by generic codes. It is shown by theoretical convergence estimates and numerical experiments that the new methodology is as accurate as the original MsFEM. Second, MsFEMs for advection-dominated problems are studied. These problems cause additional instabilities for naive discretizations. An explanation is found for the instability of previously proposed methods. Numerical experiments show the stability of an MsFEM with Crouzeix-Raviart type boundary conditions enriched with bubble functions. Third, a new convergence analysis for the MsFEM is presented that, for the first time, establishes convergence under minimal regularity hypotheses. This bridges an important gap between the theoretical understanding of the method and its field of application, where the usual regularity hypotheses are rarely satisfied
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Litvinenko, Alexander [Verfasser]. "Application of hierarchical matrices for solving multiscale problems / Alexander Litvinenko." Aachen : Universitätsbibliothek der RWTH Aachen, 2019. http://d-nb.info/1193181313/34.
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Alebrand, Sven [Verfasser], and Bernard [Akademischer Betreuer] Haasdonk. "Efficient schemes for parameterized multiscale problems / Sven Alebrand. Betreuer: Bernard Haasdonk." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2015. http://d-nb.info/106850594X/34.
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Johnsen, Pernilla. "Homogenization of Partial Differential Equations using Multiscale Convergence Methods." Licentiate thesis, Mittuniversitetet, Institutionen för matematik och ämnesdidaktik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-42036.
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The focus of this thesis is the theory of periodic homogenization of partial differential equations and some applicable concepts of convergence. More precisely, we study parabolic problems exhibiting both spatial and temporal microscopic oscillations and a vanishing volumetric heat capacity type of coefficient. We also consider a hyperbolic-parabolic problem with two spatial microscopic scales. The tools used are evolution settings of multiscale and very weak multiscale convergence, which are extensions of, or closely related to, the classical method of two-scale convergence. The novelty of the research in the thesis is the homogenization results and, for the studied parabolic problems, adapted compactness results of multiscale convergence type.
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Schöps, Sebastian [Verfasser]. "Multiscale Modeling and Multirate Time-Integration of Field/Circuit Coupled Problems / Sebastian Schöps." Wuppertal : Universitätsbibliothek Wuppertal, 2011. http://d-nb.info/1013799690/34.
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Beck, Andrea [Verfasser]. "High Order Discontinuous Galerkin Methods for the Simulation of Multiscale Problems / Andrea Beck." München : Verlag Dr. Hut, 2015. http://d-nb.info/1074063082/34.
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Miller, Eric Lawrence. "The application of multiscale and stochastic techniques to the solution of inverse problems." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/11616.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (leaves 295-306).
by Eric Lawrence Miller.
Ph.D.
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Fieguth, Paul Werner 1968. "Application of multiscale estimation to large scale multidimensional imaging and remote sensing problems." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/11409.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Vita.
Includes bibliographical references (p. 287-298).
by Paul Werner Fieguth.
Ph.D.
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Irving, William W. (William Wood). "Multiscale stochastic realization and model identification with applications to large-scale estimation problems." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/11094.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (p. 187-191).
by William W. Irving.
Ph.D.
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Mishra, Sudib Kumar. "DEVELOPMENT OF A MULTISCALE AND MULTIPHYSICS SIMULATION FRAMEWORK FOR REACTION-DIFFUSION-CONVECTION PROBLEMS." Diss., The University of Arizona, 2009. http://hdl.handle.net/10150/194088.
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Reaction-diffusion-convection (R-D-C) problems are governed by wide spectrum of spatio-temporal scales associated with ranges of physical and chemical processes. Such Problems are called multiscale, multiphysics problems. The challenge associated with R-D-C problems is to bridge these scales and processes as seamlessly as possible. For this purpose, we develop a wavelet-based multiscale simulation framework that couples diverse scales and physics.In a first stage we focus on R-D models. We treat the `fine' reaction-scales stochastically, with kinetic Monte Carlo (kMC). The transport via diffusion possesses larger spatio-temporal scales which are bridged to the kMC with the Compound Wavelet Matrix (CWM) formalism. Since R-D-C problems are dynamical we extend the CWM method via the dynamic-coupling of the kMC and diffusion models. The process is approximated by sequential increments, where the CWM on each increment is used as the starting point for the next, providing better exploration of phase-space. The CWM is extended to two-dimensional diffusion with a reactive line-boundary to show that the computational gain and error depends on the scale-overlap and wavelet-filtering. We improve the homogenization by a wavelet-based scheme for the exchange of information between a reactive and diffusive field by passing information along fine to coarse (up-scaling) and coarse to fine (down-scaling) scales by retaining the fine-scale statistics (higher-order moments, correlations). Critical to the success of the scheme is the identification of dominant scales. The efficiency of the scheme is compared to the homogenization and benchmark model with scale-disparity.To incorporate transport by convection, we then couple the Lattice Boltzmann Model (LBM) and kMC operating at diverse scales for flows around reactive block. Such model explores markedly different physics due to strong interplay between these time-scales. `Small' reaction induced temperature variations are considered for multiscale coupling of the reactions with the flow, showing the discrepancies in the evolutions and yield comparing to the conventional model. The same framework is used to study the reactions induced by hydrodynamic bubble collapse which shows the similar features of the kinetics and yield comparing to conventional models.We culminate to some problems that could be solved using the developed framework and preliminary results are presented as "proof of concept."
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Marras, Simone. "Variational multiscale stabilization of finite and spectral elements for dry and moist atmospheric problems." Doctoral thesis, Universitat Politècnica de Catalunya, 2012. http://hdl.handle.net/10803/112755.
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In this thesis the finite and spectral element methods (FEM and SEM, respectively) applied to
problems in atmospheric simulations are explored through the common thread of Variational
Multiscale Stabilization (VMS). This effort is justified by three main reasons. (i) the recognized
need for new solvers that can efficiently execute on massively parallel architectures ¿a spreading
framework in most fields of computational physics in which numerical weather prediction
(NWP) occupies a prominent position. Element-based methods (e.g. FEM, SEM, discontinuous
Galerkin) have important advantages in parallel code development; (ii) the inherent flexibility of
these methods with respect to the geometry of the grid makes them a great candidate for dynamically
adaptive atmospheric codes; and (iii) the localized diffusion provided by VMS represents
an improvement in the accurate solution of multi-physics problems where artificial diffusion may
fail. Its application to atmospheric simulations is a novel approach within a field of research
that is still open. First, FEM and VMS are described and derived for the solution of stratified
low Mach number flows in the context of dry atmospheric dynamics. The validity of the method
to simulate stratified flows is assessed using standard two- and three-dimensional benchmarks
accepted by NWP practitioners. The problems include thermal and gravity driven simulations.
It will be shown that stability is retained in the regimes of interest and a numerical comparison
against results from the the literature will be discussed. Second, the ability of VMS to stabilize
the FEM solution of advection-dominated problems (i.e. Euler and transport equations) is taken
further by the implementation of VMS as a stabilizing tool for high-order spectral elements with
advection-diffusion problems. To the author¿s knowledge, this is an original contribution to the
literature of high order spectral elements involved with transport in the atmosphere. The problem
of monotonicity-preserving high order methods is addressed by combining VMS-stabilized
SEM with a discontinuity capturing technique. This is an alternative to classical filters to treat
the Gibbs oscillations that characterize high-order schemes. To conclude, a microphysics scheme
is implemented within the finite element Euler solver, as a first step toward realistic atmospheric
simulations. Kessler microphysics is used to simulate the formation of warm, precipitating clouds.
This last part combines the solution of the Euler equations for stratified flows with the solution
of a system of transport equations for three classes of water: water vapor, cloud water, and rain.
The method is verified using idealized two- and three-dimensional storm simulations.En esta tesis los métodos de elementos finitos y espectrales (FEM - finite element method y SEM- spectral element method, respectivamente), aplicados a los problemas de simulaciones atmosféricas, se exploran a través del método de estabilización conocidocomo Variational Multiscale Stabilization (VMS). Tres razones fundamentales justifican este esfuerzo: (i) la necesidad de tener nuevos métodos de solución de las ecuaciones diferenciales a las derivadas parciales usando máquinas paralelas de gran escala –un entorno en expansión en muchos campos de la mecánica computacional, dentro de la cual la predicción numérica de la dinámica atmosférica (NWP-numerical weather prediction)representa una aplicación importante. Métodos del tipo basado en elementos(por ejemplo, FEM, SEM, Galerkin discontinuo) presentan grandes ventajas en el desarrollo de códigos paralelos; (ii) la flexibilidad intrínseca de tales métodos respecto a lageometría de la malla computacional hace que esos métodos sean los candidatos ideales para códigos atmosféricos con mallas adaptativas; y (iii) la difusión localizada que VMSintroduce representa una mejora en las soluciones de problemas con física compleja en los cuales la difusión artificial clásica no funcionaría. La aplicación de FEM o SEM con VMS a problemas de simulaciones atmosféricas es una estrategia innovadora en un campo de investigación abierto. En primera instancia, FEM y VMS vienen descritos y derivados para la solución de flujos estratificados a bajo número de Mach en el contexto de la dinámica atmosférica. La validez del método para simular flujos estratificados es verificada por medio de test estándar aceptado por la comunidad dentro del campo deNWP. Los test incluyen simulaciones de flujos térmicos con efectos de gravedad. Se demostrará que la estabilidad del método numérico se preserva dentro de los regímenesde interés y se discutirá una comparación numérica de los resultados frente a aquellos hallados en la literatura. En segunda instancia, la capacidad de VMS para estabilizarmétodos FEM en problemas de advección dominante (i.e. ecuaciones de Euler y ecuaciones de transporte) se implementa además en la solución a elementos espectrales de alto orden en problemas de advección-difusión. Hasta donde el autor sabe, esta es una contribución original a la literatura de métodos basados en elementos espectrales en problemas de transporte atmosférico. El problema de monotonicidad con métodos de alto orden es tratado mediante la combinación de SEM+VMS con una técnica de shockcapturing para un mejor tratamiento de las discontinuidades. Esta es una alternativa a los filtros que normalmente se aplican a SEM para eilminar las oscilaciones de Gibbsque caracterizan las soluciones de alto orden. Como último punto, se implementa un esquema de humedad acoplado con el núcleo en elementos finitos; este es un primer paso hacia simulaciones atmosféricas más realistas. La microfísica de Kessler se emplea para simular la formación de nubes y tormentas cálidas (warm clouds: no permite la formación de hielo). Esta última parte combina la solución de las ecuaciones de Eulerpara atmósferas estratificadas con la solución de un sistema de ecuaciones de transporte de tres estados de agua: vapor, nubes y lluvia. La calidad del método es verificadautilizando simulaciones de tormenta en dos y tres dimensiones.
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Millward, Raymond. "A new adaptive multiscale finite element method with applications to high contrast interface problems." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.544505.
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In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent. The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.
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Wei, Xiupeng. "Multiscale modeling and simulation of material phase change problems: ice melting and copper crystallization." Thesis, University of Iowa, 2010. https://ir.uiowa.edu/etd/904.
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The primary objective of this work is to propose a state-of-the-art physics based multiscale modeling framework for simulating material phase change problems. Both ice melting and copper crystallization problems are selected to demonstrate this multiscale modeling and simulation. The computational methods employed in this thesis include: classical molecular dynamics, finite element method, phase-field method, and multiscale (nano/micro coupling) methods.
Classical molecular dynamics (MD) is a well-known method to study material behaviors at atomic level. Due to the limit of MD, it is not realistic to provide a complete molecular model for simulations at large length and time scales. Continuum methods, including finite element methods, should be employed in this case.
In this thesis, MD is employed to study phase change problems at the nanoscale. In order to study material phase change problems at the microscale, a thermal wave method one-way coupling with the MD and a phase-field method one-way coupling with MD are proposed. The thermal wave method is more accurate than classical thermal diffusion for the study of heat transfer problems especially in crystal based structures. The second model is based on the well-known phase-field method. It is modified to respond to the thermal propagation in the crystal matrix by the thermal wave method, as well as modified to respond to temperature gradients and heat fluxes by employing the Dual-Phase-Lag method. Both methods are coupled with MD to obtain realistic results.
It should be noted that MD simulations can be conducted to obtain material/thermal properties for microscopic and/or macroscopic simulations for the purpose of hierarchical/sequential multiscale modeling. These material parameters include thermal conductivity, specific heat, latent heat, and relaxation time. Other type of interfacial parameters that occur during the phase change process, such as nucleus shape, interfacial energy, interfacial thickness, etc., are also obtained by MD simulation since these have so far been too difficult to measure experimentally.
I consider two common phase change phenomena, ice melting and copper crystallization, in this thesis. For the case of ice melting, MD is first employed to study its phase change process and obtain thermal properties of ice and water. Several potential models are used. I conduct simulations of both bulk ice and ice/water contacting cases. It is found that various potential models result in similar melting phenomena, especially melting speed. Size effects are also studied and it is found that the melting time is longer for larger bulk ice segments but that the average melting speed is size dependent. There is no size effect for the melting speed at ice/water interface at the nanoscale if the same temperature gradient is applied. The melting speed of ice should depend on the temperature gradient. To study ice melting at the microscale, the thermal wave model is employed with parameters obtained from MD simulations. It is found that ice melting speed is scale, for both length scale and time scale, dependent.
For the case of copper crystallization, an EAM potential is first employed to conduct MD simulations for studying the copper crystallization process at the nanoscale. I obtain thermal properties and interfacial parameters, including thermal diffusion coefficient, latent heat, relaxation time, interfacial thickness, interfacial energy and the anisotropy coefficients, and nucleus shape etc. A central symmetry parameter is used to identify an atom in solid state or liquid state. And then an initial nucleus shape is obtained and used as the input for microscale simulation, in which the phase-field method is used to study copper crystallization at the microscale.
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Persson, Jens. "Selected Topics in Homogenization." Doctoral thesis, Mittuniversitetet, Institutionen för teknik och hållbar utveckling, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-16230.
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The main focus of the present thesis is on the homogenization of some selected elliptic and parabolic problems. More precisely, we homogenize: non-periodic linear elliptic problems in two dimensions exhibiting a homothetic scaling property; two types of evolution-multiscale linear parabolic problems, one having two spatial and two temporal microscopic scales where the latter ones are given in terms of a two-parameter family, and one having two spatial and three temporal microscopic scales that are fixed power functions; and, finally, evolution-multiscale monotone parabolic problems with one spatial and an arbitrary number of temporal microscopic scales that are not restricted to be given in terms of power functions. In order to achieve homogenization results for these problems we study and enrich the theory of two-scale convergence and its kins. In particular the concept of very weak two-scale convergence and generalizations is developed, and we study an application of this convergence mode where it is employed to detect scales of heterogeneity.Huvudsakligt fokus i avhandlingen ligger på homogeniseringen av vissa elliptiska och paraboliska problem. Mer precist så homogeniserar vi: ickeperiodiska linjära elliptiska problem i två dimensioner med homotetisk skalning; två typer av evolutionsmultiskaliga linjära paraboliska problem, en med två mikroskopiska skalor i både rum och tid där de senare ges i form av en tvåparameterfamilj, och en med två mikroskopiska skalor i rum och tre i tid som ges i form av fixa potensfunktioner; samt, slutligen, evolutionsmultiskaliga monotona paraboliska problem med en mikroskopisk skala i rum och ett godtyckligt antal i tid som inte är begränsade till att vara givna i form av potensfunktioner. För att kunna uppnå homogeniseringsresultat för dessa problem så studerar och utvecklar vi teorin för tvåskalekonvergens och besläktade begrepp. Speciellt så utvecklar vi begreppet mycket svag tvåskalekonvergens med generaliseringar, och vi studerar en tillämpningav denna konvergenstyp där den används för att detektera förekomsten av heterogenitetsskalor.
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Ferreira, Rita Alexandra Gonçalves. "Spectral and homogenization problems." Doctoral thesis, Faculdade de Ciências e Tecnologia, 2011. http://hdl.handle.net/10362/7856.
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Dissertation for the Degree of Doctor of Philosophy in MathematicsFundação para a Ciência e a Tecnologia through the Carnegie Mellon | Portugal Program under Grant SFRH/BD/35695/2007, the Financiamento Base 20010 ISFL–1–297, PTDC/MAT/109973/2009 and UTA
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Goncalves-Ferreira, Rita Alexandria. "Spectral and Homogenization Problems." Research Showcase @ CMU, 2011. http://repository.cmu.edu/dissertations/83.
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In this dissertation we will address two types of homogenization problems. The first one is a spectral problem in the realm of lower dimensional theories, whose physical motivation is the study of waves propagation in a domain of very small thickness and where it is introduced a very thin net of heterogeneities. Precisely, we consider an elliptic operator with "ε-periodic coefficients and the corresponding Dirichlet spectral problem in a three-dimensional bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is of order smaller than that of δ (δ = ετ , τ < 1), or ε is of order greater than that of δ (δ = ετ , τ > 1). We consider all three cases.
The second problem concerns the study of multiscale homogenization problems with linear growth, aimed at the identification of effective energies for composite materials in the presence of fracture or cracks. Precisely, we characterize (n+1)-scale limit pairs (u,U) of sequences {(uεLN⌊Ω,Duε⌊Ω)}ε>0 ⊂ M(Ω;ℝd) × M(Ω;ℝd×N) whenever {uε}ε>0 is a bounded sequence in BV (Ω;ℝd). Using this characterization, we study the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space BV of functions of bounded variation and described by n ∈ ℕ microscales
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Kronsbein, Cornelia [Verfasser], and Oleg [Akademischer Betreuer] Iliev. "On selected efficient numerical methods for multiscale problems with stochastic coefficients / Cornelia Kronsbein. Betreuer: Oleg Iliev." Kaiserslautern : Technische Universität Kaiserslautern, 2013. http://d-nb.info/1030521409/34.
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Beck, Andrea [Verfasser], and Claus-Dieter [Akademischer Betreuer] Munz. "High order discontinuous Galerkin methods for the simulation of multiscale problems / Andrea Beck. Betreuer: Claus-Dieter Munz." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2015. http://d-nb.info/1075191025/34.
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Persson, Jens. "Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence." Licentiate thesis, Mid Sweden University, Department of Engineering and Sustainable Development, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-11991.
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The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.
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Álamo, Miguel del [Verfasser], Axel [Akademischer Betreuer] Munk, Axel [Gutachter] Munk, and Thorsten [Gutachter] Hohage. "Multiscale Total Variation Estimators for Regression and Inverse Problems / Miguel del Álamo ; Gutachter: Axel Munk, Thorsten Hohage ; Betreuer: Axel Munk." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2019. http://d-nb.info/1189904624/34.
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Wang, Xiaojun. "Well-posedness results for a class of complex flow problems in the high Weissenberg number limit." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/27669.
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For simple fluids, or Newtonian fluids, the study of the Navier-Stokes equations in the high Reynolds number limit brings about two fundamental research subjects, the Euler equations and the Prandtl's system. The consideration of infinite Reynolds number reduces the Navier-Stokes equations to the Euler equations, both of which are dealing with the entire flow region. Prandtl's system consists of the governing equations of the boundary layer, a thin layer formed at the wall boundary where viscosity cannot be neglected.
In this dissertation, we investigate the upper convected Maxwell(UCM) model for complex fluids, or non-Newtonian fluids, in the high Weissenberg number limit. This is analogous to the Newtonian fluids in the high Reynolds number limit. We present two well-posedness results.
The first result is on an initial-boundary value problem for incompressible hypoelastic materials which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume the stress tensor is rank-one and develop energy estimates to show the problem is locally well-posed. Then we show the more general case can be handled in the same spirit. This problem is closely related to the incompressible ideal magneto-hydrodynamics (MHD) system.
The second result addresses the formulation of a time-dependent elastic boundary layer through scaling analysis. We show the well-posedness of this boundary layer by transforming to Lagrangian coordinates. In contrast to the possible ill-posedness of Prandtl's system in Newtonian fluids, we prove that in non-Newtonian fluids the stress boundary layer problem is well-posed.Ph. D.
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Hellman, Fredrik. "Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-318589.
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We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.
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Madiot, François. "Méthodes éléments finis de type MsFEM pour des problèmes d'advection-diffusion." Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1052/document.
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Ce travail a porté principalement sur le développement et l'étude de méthodes numériques de type éléments finis multi-échelles pour un problème d'advection diffusion multi-échelles dominé par l'advection. Deux types d'approches sont envisagées: prendre en compte l'advection dans la construction de l'espace d'approximation, ou appliquer une méthode de stabilisation. On commence par l'étude d'un problème d'advection diffusion, dominé par l'advection, dans un milieu hétérogène. On poursuit sur des problèmes d'advection-diffusion, sous le régime où l'advection domine, posés dans un domaine perforé. On se focalise ici sur la condition aux bords de type Crouzeix Raviart pour la construction des éléments finis multi-échelles. On considère deux situations différentes selon la condition prescrite au bord des perforations: la condition de Dirichlet homogène ou la condition de Neumann homogène. Cette étude repose sur une hypothèse de coercivité.Pour finir, on se place dans un cadre général où l'opérateur d'advection-diffusion est non coercif, possiblement dominé par l'advection. On propose une approche éléments finis basée sur une mesure invariante associée à l'opérateur adjoint. Cette approche est bien posée inconditionnellement en la taille du maillage. On la compare numériquement à une méthode standard de stabilisationThis work essentially deals with the development and the study of multiscale finite element methods for multiscale advection-diffusion problems in the advection-dominated regime. Two types of approaches are investigated: Take into account the advection in the construction of the approximation space, or apply a stabilization method. We begin with advection-dominated advection-diffusion problems in heterogeneous media. We carry on with advection-dominated advection-diffusion problems posed in perforated domains.Here, we focus on the Crouzeix-Raviart type boundary condition for the construction of the multiscale finite elements. We consider two different situations depending on the condition prescribed on the boundary of the perforations: the homogeneous Dirichlet condition or the homogeneous Neumann condition. This study relies on a coercivity assumption.Lastly, we consider a general framework where the advection-diffusion operator is not coercive, possibly in the advection-dominated regime. We propose a Finite Element approach based on the use of an invariant measure associated to the adjoint operator. This approach is unconditionally well-posed in the mesh size. We compare it numerically to a standard stabilization method
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Waldspurger, Irène. "Wavelet transform modulus : phase retrieval and scattering." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0036/document.
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Les tâches qui consistent à comprendre automatiquement le contenu d’un signal naturel, comme une image ou un son, sont en général difficiles. En effet, dans leur représentation naïve, les signaux sont des objets compliqués, appartenant à des espaces de grande dimension. Représentés différemment, ils peuvent en revanche être plus faciles à interpréter. Cette thèse s’intéresse à une représentation fréquemment utilisée dans ce genre de situations, notamment pour analyser des signaux audio : le module de la transformée en ondelettes. Pour mieux comprendre son comportement, nous considérons, d’un point de vue théorique et algorithmique, le problème inverse correspondant : la reconstruction d’un signal à partir du module de sa transformée en ondelettes. Ce problème appartient à une classe plus générale de problèmes inverses : les problèmes de reconstruction de phase. Dans un premier chapitre, nous décrivons un nouvel algorithme, PhaseCut, qui résout numériquement un problème de reconstruction de phase générique. Comme l’algorithme similaire PhaseLift, PhaseCut utilise une relaxation convexe, qui se trouve en l’occurence être de la même forme que les relaxations du problème abondamment étudié MaxCut. Nous comparons les performances de PhaseCut et PhaseLift, en termes de précision et de rapidité. Dans les deux chapitres suivants, nous étudions le cas particulier de la reconstruction de phase pour la transformée en ondelettes. Nous montrons que toute fonction sans fréquence négative est uniquement déterminée (à une phase globale près) par le module de sa transformée en ondelettes, mais que la reconstruction à partir du module n’est pas stable au bruit, pour une définition forte de la stabilité. On démontre en revanche une propriété de stabilité locale. Nous présentons également un nouvel algorithme de reconstruction de phase, non-convexe, qui est spécifique à la transformée en ondelettes, et étudions numériquement ses performances. Enfin, dans les deux derniers chapitres, nous étudions une représentation plus sophistiquée, construite à partir du module de transformée en ondelettes : la transformée de scattering. Notre but est de comprendre quelles propriétés d’un signal sont caractérisées par sa transformée de scattering. On commence par démontrer un théorème majorant l’énergie des coefficients de scattering d’un signal, à un ordre donné, en fonction de l’énergie du signal initial, convolé par un filtre passe-haut qui dépend de l’ordre. On étudie ensuite une généralisation de la transformée de scattering, qui s’applique à des processus stationnaires. On montre qu’en dimension finie, cette transformée généralisée préserve la norme. En dimension un, on montre également que les coefficients de scattering généralisés d’un processus caractérisent la queue de distribution du processusAutomatically understanding the content of a natural signal, like a sound or an image, is in general a difficult task. In their naive representation, signals are indeed complicated objects, belonging to high-dimensional spaces. With a different representation, they can however be easier to interpret. This thesis considers a representation commonly used in these cases, in particular for theanalysis of audio signals: the modulus of the wavelet transform. To better understand the behaviour of this operator, we study, from a theoretical as well as algorithmic point of view, the corresponding inverse problem: the reconstruction of a signal from the modulus of its wavelet transform. This problem belongs to a wider class of inverse problems: phase retrieval problems. In a first chapter, we describe a new algorithm, PhaseCut, which numerically solves a generic phase retrieval problem. Like the similar algorithm PhaseLift, PhaseCut relies on a convex relaxation of the phase retrieval problem, which happens to be of the same form as relaxations of the widely studied problem MaxCut. We compare the performances of PhaseCut and PhaseLift, in terms of precision and complexity. In the next two chapters, we study the specific case of phase retrieval for the wavelet transform. We show that any function with no negative frequencies is uniquely determined (up to a global phase) by the modulus of its wavelet transform, but that the reconstruction from the modulus is not stable to noise, for a strong notion of stability. However, we prove a local stability property. We also present a new non-convex phase retrieval algorithm, which is specific to the case of the wavelet transform, and we numerically study its performances. Finally, in the last two chapters, we study a more sophisticated representation, built from the modulus of the wavelet transform: the scattering transform. Our goal is to understand which properties of a signal are characterized by its scattering transform. We first prove that the energy of scattering coefficients of a signal, at a given order, is upper bounded by the energy of the signal itself, convolved with a high-pass filter that depends on the order. We then study a generalization of the scattering transform, for stationary processes. We show that, in finite dimension, this generalized transform preserves the norm. In dimension one, we also show that the generalized scattering coefficients of a process characterize the tail of its distribution
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Heinlein, Alexander Verfasser], Axel [Gutachter] Klawonn, and Oliver [Gutachter] [Rheinbach. "Parallel Overlapping Schwarz Preconditioners and Multiscale Discretizations with Applications to Fluid-Structure Interaction and Highly Heterogeneous Problems / Alexander Heinlein. Gutachter: Axel Klawonn ; Oliver Rheinbach." Köln : Universitäts- und Stadtbibliothek Köln, 2016. http://d-nb.info/1105645061/34.
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Krishnan, Bharath Kumar. "A multiscale approximation algorithm for the cardinality constrained knapsack problem." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/34612.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2006.Includes bibliographical references (leaves 83-86).
I develop a multiscale approximation algorithm for the cardinality constrained knapsack problem. The algorithm consists of three steps: a rounding and reduction step where a hierarchical representation of the problem data ranging from coarse to fine is generated, a solution step where a coarse solution is computed, and a refinement step where the accuracy of the solution is improved by refining the problem representation. I demonstrate that the algorithm is fully polynomial with a runtime complexity that improves upon the previous best known fully polynomial approximation scheme. Through an extensive computational study, I show that the running times of the algorithm is less than or equal to that of a commercial integer programming package with little loss in solution accuracy.
by Bharath Kumar Krishnan.
Ph.D.
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Akhtar, Nahid. "A multiscale harmonic spline interpolation method for the inverse spheroidal gravimetric problem." Aachen Shaker, 2009. http://d-nb.info/1000335380/04.
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Berkel, Paula. "Multiscale methods for the combined inversion of normal mode and gravity variations." Aachen Shaker, 2009. http://d-nb.info/997085304/04.
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Akhtar, Nahid [Verfasser]. "A Multiscale Harmonic Spline Interpolation Method for the Inverse Spheroidal Gravimetric Problem / Nahid Akhtar." Aachen : Shaker, 2010. http://d-nb.info/1124365478/34.
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Nguyen, Manh Tu. "Identification multi-échelle du champ d'élasticité apparent stochastique de microstructures hétérogènes : application à un tissu biologique." Thesis, Paris Est, 2013. http://www.theses.fr/2013PEST1135/document.
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Dans le cadre de l'élasticité linéaire 3D des microstructures complexes qui ne peuvent pas être simplement décrites en terme de constituants telles que des tissus biologiques, nous proposons, dans ce travail de recherche, une méthodologie d'identification expérimentale multi-échelle du champ stochastique d'élasticité apparent de la microstructure à l'échelle mésoscopique en utilisant des mesures de champ de déplacements aux échelles macroscopique et mésoscopique. On peut alors utiliser cette méthodologie dans le cadre de changement d'échelle pour obtenir des propriétés mécaniques à l'échelle macroscopique. Dans ce contexte, la question majeure est celle de l'identification expérimentale par résolution d'un problème statistique inverse de la modélisation stochastique introduite pour le champ d'élasticité apparent à l'échelle mésoscopique. Cette identification expérimentale permet non seulement de valider la modélisation mais encore de la rendre utile pour des matériaux existants ayant une microstructure complexe. Le présent travail de recherche est une contribution proposée dans ce cadre pour lequel l'expérimentation et validation expérimentale basée sur des mesures simultanées d'imagerie de champ aux échelles macroscopique et mésoscopique sont faites sur de l'os corticalIn the framework of linear elasticity 3D for complex microstructures that cannot be simply described in terms of components such as biological tissues, we propose, in this research work, a methodology for multiscale experimental identification of the apparent elasticity random field of the microstructure at mesoscopic scale using displacement field measurements at macroscopic scale and mesoscopic scale. We can then use this methodology in the case of changing scale to obtain the mechanical properties at macroscale. In this context, the major issue is the experimental identification by solving a statistical inverse problem of the stochastic modeling introduced for the apparent elasticity random field at mesoscale. This experimental identification allows to validate the modeling and makes it useful for existing materials with complex microstructures. This research work is proposed in this context in which experimentation and experimental validation based on simultaneous measurements of field imaging at macroscale and mesoscale are made on the cortical bonemakes it useful for existing materials with complex microstructures. This research work is proposed in this context in which experimentation and experimental validation based on simultaneous measurements of field imaging at macroscale and mesoscale are made on the cortical bone
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Santesarti, Gianluca. "Simulazioni agli elementi finti per problemi di interazione fluido-struttura con approccio multiscala." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019.
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Il lavoro svolto in questa tesi riguarda le simulazioni agli elementi finiti di problemi d'interazione fluido-struttura FSI. In particolare si sono implementati due tipi di modelli matematici, ovvero quello monoscala, in cui solido e fluido possiedono la stessa dimensione spaziale, e quello multiscala, in cui l'oggetto solido viene modellato con una dimensione spaziale inferiore. Quest'ultimo approccio permette di ottenere una considerevole riduzione dei tempi di calcolo ed una maggiore stabilità numerica dell'algoritmo. Infine i risultati ottenuti dalle simulazioni numeriche eseguite con entrambi i modelli mostrano una buona accuratezza del nuovo metodo.
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Marzi, Emanuela. "Controllo ottimo di problemi di interazione fluido-struttura multiscala attraverso simulazioni agli elementi finiti." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020.
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Durante la fase di progettazione di un sistema fisico è ormai consuetudine costruire un modello matematico per simularne il comportamento. Il passo successivo è l’ottimizzazione del sistema stesso con lo scopo di poterlo controllare, modificando i parametri che vi compaiono, affinché agisca come desiderato. Spesso questo tipo di problemi viene affrontato con un approccio di tipo try and fail. Questo metodo risulta però essere molto dispendioso sia da un punto di vista dei calcoli che delle tempistiche. È piuttosto utile in questi casi passare allo studio del problema inverso. Si parte cioè dall’obiettivo, e si ricavano i valori dei parametri di progetto adeguati per ottenere i risultati desiderati attraverso la minimizzazione di un funzionale di costo, nel quale vengono imposte le condizioni di ottimo. In questa tesi si affrontano problemi inversi di interazione fluido-struttura (FSI) utilizzando la teoria del controllo ottimo. Un problema FSI è un sistema fisico in cui un fluido e un solido interagiscono tra loro, il moto del fluido comporta una deformazione del solido ed essa incide a sua volta sul moto del fluido. Il problema di ottimo è risolto con il metodo dell’equazione aggiunta ed è proposto un controllo ottimo per il problema stazionario, in cui l’obiettivo è ottenere un valore di deformazione del solido desiderato, tramite un controllo sulla pressione del fluido in ingresso. Il sistema di ottimo viene ricavato utilizzando il teorema dei moltiplicatori di Lagrange per la minimizzazione del funzionale, e si risolve iterativamente attraverso il metodo di discesa del gradiente. Si utilizza inoltre un approccio multiscala per modellare il solido, semplificato a un’entità di una dimensione inferiore rispetto al dominio del fluido, diventando così una condizione al contorno per il moto del fluido. La risoluzione del problema viene effettuata attraverso il metodo agli elementi finiti e il problema di controllo ottimo trovato viene implementato nel codice FEMuS.
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Pont, Ribas Arnau. "Numerical simulation of aeroacoustics using the variational multiscale method : application to the problem of human phonation." Doctoral thesis, Universitat Politècnica de Catalunya, 2018. http://hdl.handle.net/10803/461955.
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The solution of the human phonation problem applying computational mechanics is covered by several research branches, such as Computational Fluid Dynamics (CFD), biomechanics or acoustics, among others. In the present thesis, the problem is approached from the Computational Aeroacoustics (CAA) point of view and the first main objective consists in developing numerical methods of general application that can take part in the solution of any scenario related to human phonation with a reasonable cost. In this sense, only the compressible Navier-Stokes equations can describe all flow and acoustic scales without any modeling, which is known as Direct Numerical Simulation (DNS), but its computational cost is usually unaffordable. Even in the case of a Large Eddy Simulation (LES), where the small scales are modeled, the cost can still be a handicap due to the complexity of the problem. This drawback gets worse in the low Mach regime due to the large disparity between flow velocity and sound speed, which leads to an ill-conditioning of the system of equations, specially for conservative schemes. At this point, it makes sense to move towards the incompressible flow approximation, bearing in mind the low velocities expected in human phonation problems.
Incompressible flows do not yield any acoustics, for which a second problem containing the propagation of the sound sources needs to be modeled and solved. These are the so called hybrid methods, which allow a better conditioning of the problem by segregating flow and acoustic scales. Lighthill's analogy has been taken as starting point for the present work, but its restriction to free-field scenarios has motivated the extension of the method to arbitrary geometries and non-uniform flows. The first development in this direction consists in a splitting of Lighthill's analogy into a quadrupolar and dipolar component, which does not change the original problem but allows assessing the contribution of solid boundaries to the generation of sound. The second step consists in the development of a stabilized Finite Element (FEM) formulation for the Acoustic Perturbation Equations (APE) which account for non-uniform flows and perform a complete filtering of the acoustic scales. The final step assumes the compressible approach but omitting the energy equation and thus considering both flow and acoustic propagation as isentropic. In this case the solver is unified and hence a method for applying compatible boundary conditions for flow and acoustics has been developed. Moreover, the whole numerical framework has been extended to dynamic phonation cases, which require using an Arbitrary Lagrangian Eulerian (ALE) reference. Also, a novel remeshing strategy with conservative interpolation between meshes is presented.
In the last chapter a challenging case in human phonation has been chosen for testing the developed computational framework: the fricative phoneme /s/. Unlike vowels, which are voiced sounds defined by a few characteristic frequencies, fricatives cannot be simulated as the propagation of a known analytic solution (glottal pulse) because the sound sources correspond to a wide range of turbulent scales. Therefore, a CFD calculation is mandatory in order to capture all relevant eddies behind the generation of sound. This problem is solved with an LES together with the Variational Multiscale (VMS) stabilization method as turbulence model, which is supplemented with several acoustic formulations when using incompressible flow. The analysis of the results focuses on the numerical representation of turbulence and the acoustic signal at the far-field, which has been compared to experimental recordings. Finally, the role of the upper incisors in the generation of the fricative sound has been evaluated. All simulations have been run with the parallel multiphysics FEM code FEMUSS, based on FORTRAN Object-Oriented-Programming land the OpenMPI parallel library.La solució del problema de la veu humana des de la mecànica computacional és objecte d'estudi per part de diverses disciplines, com per exemple la Dinàmica de Fluids Computacional (CFD), la biomecànica o l'acústica. En la present tesi s'encara el problema des de l'Aeroacústica Computacional (CAA) i el primer objectiu consisteix en desenvolupar mètodes numèrics d'aplicació general que puguin ser part de la solució, amb un cost computacional raonable, de qualsevol escenari relacionat amb la fonació humana. En aquest sentit, només les equacions de flux compressible de Navier-Stokes aconsegueixen descriure totes les escales alhora, tant les dinàmiques com les acústiques, sense recórrer a cap modelització, conegut com a Simulació Numèrica Directa (DNS), però el seu cost computacional és normalment inassumible. Fins i tot en el cas d'una Large Eddy Simulation (LES), on les escales petites són modelades, el cost pot resultar excessiu a causa de la complexitat del problema. Aquest fet encara és més accentuat en el règim de baix nombre de Mach donada la gran disparitat entre la velocitat del fluid i la del so i el conseqüent mal condicionament del sistema d'equacions, sobretot en esquemes conservatius. Per tant, tenint en compte les baixes velocitats de l'aire al tracte vocal, té sentit recórrer a l'aproximació de flux incompressible. Els fluids incompressibles no inclouen la part acústica, de manera que cal calcular un segon problema que descrigui la propagació de les fonts de so. Aquests són els anomenats mètodes híbrids, que permeten un millor condicionament del problema gràcies a la segregació de les escales acústiques de les dinàmiques. S'ha pres l'analogia de Lighthill com a punt de partida, però la seva restricció a casos en camp obert ha motivat l'extensió del mètode cap a geometries arbitràries i fluxos no uniformes. El primer desenvolupament en aquesta direcció consisteix en la divisió de l'analogia de Lighthill en una component quadrupolar i una altra de dipolar, fet que no altera el problema original però que permet analitzar la contribució de cossos sòlids en la generació de so. El segon pas consisteix en el desenvolupament d'una formulació estabilitzada en elements finits (FEM) de les Acoustic Perturbation Equations (APE), que incorporen la propagació en fluxos no uniformes i que realitzen un filtrat complet de les escales acústiques. El pas final assumeix la compressibilitat del fluid però omet l'equació d'energia, i per tant considera la dinàmica i l'acústica fenòmens isentròpics. En aquest cas el solver és unificat i per tant s'ha desenvolupat un mètode per imposar condicions de contorn compatibles entre ambdues escales del fluid. Finalment, les formulacions numèriques han estat adaptades a casos de fonació dinàmica mitjançant una referència Arbitrària Lagrangiana Euleriana (ALE). A més, es presenta una estratègia de remallat amb interpolació conservativa entre malles. En l'últim capítol es presenta un cas de fonació humana que suposa un repte per la seva complexitat i que ha servit per validar les formulacions numèriques presentades: la fricativa sorda /s/. A diferència de les vocals, que són sons sonors definits per unes poques freqüències característiques, les fricatives no poden ser simulades com la propagació d'una funció analítica coneguda (pols glotal) perquè les fonts de so corresponen a un rang ampli d'escales turbulents. Per tant és necessària una simulació CFD per tal de capturar-les. El problema se soluciona amb un model de turbulència LES amb el mètode d'estabilització Variational Multiscale. L'anàlisi se centra en la representació numèrica de la turbulència i en el senyal acústic al camp llunyà, tot comparant-lo amb dades experimentals. Finalment, s'avalua la contribució dels incisius superiors en la generació del so fricatiu sord /s/. Totes les simulacions han estat realitzades amb el codi FEM multi-físic en paral·lel FEMUSS, basat en programació orientada a objectes en FORTRAN i en OpenMPI.
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Cai, Xiran. "Multiscale investigation of the elastic properties of human cortical bone measured by resonant ultrasound spectroscopy." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS059/document.
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L’os présente la propriété remarquable de s’adapter à son environnement et s’est forgé au cours de l’évolution des caractéristiques exceptionnelles qui fascinent les scientifiques mais aussi les ingénieurs : léger mais d’une rigidité à toute épreuve, une capacité de résistance à la fracture hors norme tout en gardant une certaine flexibilité. Ces propriétés mécaniques de l’os sont l’œuvre d’une optimisation de sa composition et d’une structure fortement hiérarchisée et organisée en multiples niveaux allant de l'échelle nanométrique à l'échelle macroscopique. L’amélioration de la prise en charge des maladies osseuses, l’optimisation des implants orthopédiques et la conception de nouveaux matériaux bio-inspirés passent par une connaissance approfondie des multiples facteurs qui déterminent les propriétés mécaniques de l’os. Dans ce travail, nous mettons l’accent sur les propriétés élastiques de l'os cortical humain à la fois aux échelles millimétrique et micrométrique. Nous avons caractérisé l’élasticité (à l’échelle mésoscopique), la composition et la microstructure de l’os cortical, à partir d’échantillons de fémur, tibia et radius prélevés sur des donneurs âgés, à l’aide d’une batterie de tests expérimentaux comportant des mesures en résonance ultrasonore spectroscopique, micro-tomographie par rayonnement synchrotron, microscopie infrarouge à transformée de Fourier et analyse biochimique. Ces mesures mettent à jour le rôle prépondérant joué par la porosité et le degré de minéralisation dans la détermination de l’élasticité et suffisent à eux seuls à en expliquer les variations. En particulier, les caractéristiques de la microstructure, comme la forme des pores, leur nombre, taille ou connectivité ne semblent pas avoir d’effets mesurables sur l’élasticité à l’échelle mésoscopique. Dans un second temps, une nouvelle approche d’homogénéisation inverse introduite dans cette thèse a permis l’estimation du tenseur des coefficients élastiques de la matrice osseuse à l’échelle microscopique. Connaissant l’élasticité de la matrice, nous avons évalué la gamme des microdéformations qui se produisent localement en réponse à des contraintes physiologiques. Les microdéformations étant à l’origine des signaux qui déclenchent la réponse des cellules mécanosensibles, ce dernier résultat devrait contribuer à une meilleure compréhension du comportement mécanique osseux au niveau microscopique. En conclusion, ce travail de thèse a permis l’obtention d’une base de données unique sur les caractéristiques élastiques de l’os cortical humain et la caractérisation des relations qui existent entre l’élasticité, la microstructure et la compositionBone as an important organ in human body is an extraordinary material which exhibits highly optimized properties, strong yet light weight, stiff yet flexible. Its distinct mechanical properties which fascinates not only scientists but also engineers are the results of the highly hierarchized and organized structure and the compositional properties spanning over several lengths from the nanoscale to the macroscale. Hence, a deep understanding of the parameters affecting bone mechanical behavior is necessary to better predict and treat bone diseases, improve orthopedic implants design, and engineer bio-inspired materials. In this work, a special focus is placed on human cortical bone elastic properties both at the millimeter and micrometer scales. Based on a multimodal approach (resonant ultrasound spectroscopy, synchrotron radiation micro-computed tomography, Fourier transform infrared microspectroscopy and biochemistry experiments) involving an exhaustive amount of microstructural and compositional properties, our results provide strong evidence that intra-cortical porosity and degree of mineralization are the most important determinants of bone stiffness at millimeter scale in an elderly population. Further, the other microstructure characteristics independent of porosity have non measurable effects on bone stiffness at this level. At the micrometer scale, a novel inverse homogenization approach is introduced in this work which can evaluate bone matrix anisotropic elastic properties with a good accuracy for all the stiffness constants. Based on the determined bone matrix elasticity data, we investigated the possible range of the magnitude of microstrain experienced by bone matrix. This work opens a way to better evaluate and understand bone mechanical behaviour at the micrometer level, such as the microstrain that can be sensed by osteocytes and builds the bridge to comprehensively investigate the connections between bone anisotropic properties at the millimeter and micrometer scale, and between the anisotropic microelastic properties and the characteristics at the nanometer scale
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Rocca, Layza Vladimir Jaime 1987. "Uso de técnicas de recuperação de imagens para o problema de reidentificação de pessoas." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/275548.
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Orientadores: Ricardo da Silva Torres, Hélio PedriniDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Vários sistemas de vigilância baseados no uso de múltiplas câmeras têm sido propostos recentemente. No entanto, a identificação de pessoas em sequências de vídeos obtidas por várias câmeras com vistas não sobrepostas, comumente conhecida como reidentificação de pessoas, é um problema em aberto. As razões para que este problema seja considerado desafiador referem-se principalmente às restrições nas quais o problema deve ser resolvido. Estas restrições são definidas a partir das características do cenário e dos objetos de interesse (as pessoas): primeiro, as características biométricas de pessoas não podem ser utilizadas como características discriminantes; segundo, a aparência das pessoas muda drasticamente em virtude de variações na posição, iluminação e parâmetros de câmera. Tais restrições fazem com que uma mesma pessoa possa ser observada por múltiplas câmeras como uma pessoa diferente para cada uma delas. Nesta pesquisa, busca-se investigar alternativas para a criação de sistemas de vigilância visando à reidentificação de pessoas. Foram empregadas técnicas de recuperação de imagens por conteúdo tais como descritores de imagens tradicionais e propostos recentemente, análise multiescala, e técnicas de rank aggregation. Os experimentos realizados consideram a utilização de quatro bases de dados comumente utilizadas na avaliação de sistemas de reidentificação de pessoas. Os resultados obtidos mostraram que as técnicas de recuperação de imagens por conteúdo são promissoras para a reidentificação de pessoas, obtendo resultados comparáveis aos métodos do estado da arte
Abstract: Several surveillance systems based on the use of multiple cameras have been proposed recently. However, the identification of people in video sequences obtained from several cameras with non-overlapping views, commonly known as the person reidentification problem, is still an open problem. Person reidentification is a challenging problem due to the constraints under which the problem should be solved. These constraints come from the characteristics of the scenario and the objects of interest (people): first, biometric features may not be used as discriminant information; second, appearance is dramatically modified by changes in position, lighting conditions, and camera parameters. Therefore, in these conditions a unique person can be ''seen'' as a distinct person by different cameras. This research is focused on the investigation of alternatives for the creation of surveillance systems aiming at person reidentification. We intend to use content-based image retrieval techniques, such as traditional and recently proposed image descriptors, multiscale analysis, and rank aggregation approaches. Conducted experiments considered the use of four different datasets, commonly used in the evaluation of person reidentification systems. Obtained results show that the content-based image retrieval techniques are promising to reidentify people, producing equivalent results to the state-of-the-art methods
Mestrado
Ciência da Computação
Mestre em Ciência da Computação
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Quintela, Bárbara de Melo. "Implementação computacional paralela da homogeneização por expansão assintótica para análise de problemas mecânicos em 3D." Universidade Federal de Juiz de Fora (UFJF), 2011. https://repositorio.ufjf.br/jspui/handle/ufjf/3536.
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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico
FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais
A Homogeneização por Expansão Assintótica (HEA) é uma técnica multiescala empregada ao cálculo de propriedades efetivas de meios contínuos com estrutura periódica. As principais vantagens desta técnica são a redução do tamanho do problema a resolver e a possibilidade de se empregar uma propriedade homogeneizada que guarda informações da microestrutura heterogênea. Quando associada ao Método dos Elementos Finitos (MEF), a HEA demanda o emprego de malhas que permitam a imposição de condições de contorno periódicas – sendo portanto necessário especificar tal particularidade quando da geração dos modelos em MEF. Tais modelos representam as células periódicas, que são volumes representativos do meio heterogêneo e, em alguns casos, apresentam uma complexidade geométrica e física que torna imprescindível o emprego de malhas com alto grau de refinamento – levando a um custo computacional significativo. Este trabalho tem por objetivo a obtenção de um programa em Elementos Finitos para a aplicação da HEA à Elasticidade em 3D, empregando técnicas de programação paralela. Foram desenvolvidas versões do programa em 2D: uma sequencial em C e duas paralelas empregando OpenMP e CUDA. Foi implementado com sucesso o programa HEA3D em uma versão sequencial, em linguagem FORTRAN e uma paralela, empregando OpenMP. Para validação dos programas, foram analisadas células periódicas bifásicas e os resultados apresentaram boa concordância com valores experimentais e numéricos disponíveis na literatura. A versão paralela obteve expressivos ganhos de desempenho, com acelerações de desempenho de até 5.3 vezes em relação a versão sequencial.
The Asymptotic Expansion Homogenization (AEH) is a multiscale technique applied to estimate the effective properties of heterogeneous media with periodical structure. The main advantages of this technique are the reduction of the problem size to be solved and the ability to employ an homogenized property that keeps information from the heterogeneous microstructure. In association with the Finite Element Method (FEM), the AEH requires the application of periodic boundary conditions, which must be taken into account during the generation of FE meshes. Such models represent periodic cells, which are representative volumes for heterogeneous media and, in some cases, present a geometric and physics complexity that demands refined meshes, leading to a significant computational cost. The aim of this work is to develop a parallel program that applies both FEM and AEH to estimate the elasticity properties of 3D bodies. A sequential version of the 2D program using C, and parallel versions using OpenMP and CUDA were implemented. A sequential version of the program, called HEA3D, was successfully implemented using FORTRAN. Also, a parallel version of the code was implemented using OpenMP. The validation of the codes consisted of comparisons of the numerical results obtained, with numerical and experimental data available in the literature, showing good agreement. Significant speedups were obtained by the parallel version of the code, achieving speedups up to 5.3 times over its sequential version.
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Pacquaut, Guillaume. "Couplage Stokes/Darcy dans un cadre Level-set en grandes déformations pour la simulation des procédés d'élaboration par infusion de résine." Phd thesis, Ecole Nationale Supérieure des Mines de Saint-Etienne, 2010. http://tel.archives-ouvertes.fr/tel-00609670.
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Ce travail de recherche propose un modèle numérique pour simuler les procédés par infusion de résine en utilisant la méthode des éléments finis. Ce modèle permet de représenter l'écoulement d'une résine liquide dans des préformes poreuses subissant de grandes déformations. Dans cette étude, une modélisation macroscopique est utilisée. Au niveau du procédé, une zone de résine liquide est déposée sur les préformes. Ces dernières étant considérées comme un milieu poreux. Les équations de Stokes et de Darcy sont utilisées pour modéliser l'écoulement de la résine respectivement dans le drainant et dans les préformes. L'originalité du modèle réside dans le fait qu'un seul maillage est utilisé pour les deux milieux. La discrétisation est réalisée avec des éléments mixtes : dans Stokes, des éléments P1+/P1 sont utilisés et dans Darcy, des éléments P1/P1 stabilisés avec une formulation multi-échelle sont employés. Des fonctions distances signées sont utilisées pour représenter l'interface entre Stokes-Darcy et pour représenter le front de résine. Concernant la déformation des préformes, une formulation Lagrangienne réactualisée est utilisée. Dans cette formulation Lagrangienne, le comportement des préformes humides est représenté à l'aide du modèle de Terzaghi dans lequel les préformes sèches ont un comportement élastique non-linéaire. La perméabilité est reliée à la porosité via la relation de Carman-Kozeny. Celle-ci est déterminée à partir de l'équation de conservation de la masse. Ce modèle a été implémenté dans ZéBuLoN. Plusieurs simulations numériques d'infusion de résine sont présentées à la fin de ce manuscrit.
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Lelièvre, Tony. "Modèles multi-échelles pour les fluides viscoélastiques." Marne-la-vallée, ENPC, 2004. https://hal.science/tel-00006797.
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Ce travail porte principalement sur l'analyse mathématique de modèles multi-échelles pour la simulation de fluides polymériques. Ces modèles couplent, au niveau microscopique, une description moléculaire de l'évolution des chaînes de polymère (sous forme d'une équation différentielle stochastique) avec, au niveau macroscopique, les équations de conservation de la masse et de la quantité de mouvement pour le solvant (sous forme d'équations aux dérivées partielles). Le chapitre 1 introduit les modèles et donne les principaux résultats obtenus. Dans les chapitres 2, 4, 5 et 7 on montre en quel sens les équations sont bien posées pour divers modèles de polymère, en considérant soit des écoulements homogènes, soit des écoulements cisaillés plans. Dans les chapitres 2, 3, 6 et 7, on analyse et on prouve la convergence de méthodes numériques pour ces modèles. Enfin, le chapitre 8 concerne le comportement en temps long du système. Une deuxième partie de ce document est constituée de trois chapitres portant sur un travail en magnétohydrodynamique (MHD), en collaboration avec l'industrie. Le chapitre 9 est une introduction à la problématique ainsi qu'aux méthodes numériques utilisées. Le chapitre 10 décrit un nouveau cas-test en MHD. Enfin, le chapitre 11 donne une analyse de la stabilité du schéma numérique utiliséThe most important part of this work deals with the mathematical analysis of multiscale models of polymeric fluids. These models couple, at the microscopic level, a molecular description of the evolution of the polymer chains (in terms of stochastic differential equations) and, at the macroscopic level, the mass conservation and momentum equations for the solvent (which are partial differential equations). In Chapter 1, we introduce the models and give the main results obtained. In Chapters 2, 4, 5 and 7 we make precise the mathematical meaning and the well-posedness of the equations in either homogeneous flows or plane shear flows for some specific models of polymer chains. In Chapters 2, 3, 6 and 7, we analyse and prove convergence of some numerical schemes. Finally, in Chapter 8, we deal with the longtime behaviour of the coupled system. A second part of this document concerns a magnetohydrodynamic (MHD) problem coming from industry. In Chapter 9, we introduce the problem and the numerical methods used. We present a new test-case in MHD in Chapter 10. Finally, we give a stability analysis of the scheme in Chapter 11
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Ferraz, Paola Cunha 1988. "Implementação de um algoritmo multi-escala para sistemas de equações lineares de grande porte mal condicionados provenientes da discretização de problemas elípticos em dinâmica de fluidos em meios porosos." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307022.
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Orientador: Eduardo Cardoso de AbreuDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O foco deste trabalho é aproximação numérica de problemas envolvendo equações diferenciais parciais (EDPs), de natureza elíptica, no contexto de aplicações em dinâmica de fluidos em meios porosos. Especificamente, a dissertação pretende contribuir com uma implementação de um algoritmo multiescala e multigrid, recentemente introduzido na literatura, para resolução aproximada de sistemas de equações lineares de grande porte e mal condicionados, proveniente dessa classe de EDPs, tipicamente associada a problemas de Poisson de pressão-velocidade com condições de contornos típicas de fluxo em meios porosos. O problema concreto de Poisson discutido neste trabalho será desacoplado do sistema de transporte de EDPs de convecção-difusão, com convecção dominante, e linearizado por meio do emprego de uma técnica de decomposição de operadores. A metodologia para a discretização do problema elíptico de Poisson é elementos finitos mistos híbridos. A resolução numérica do sistema linear resultante deste procedimento será realizado via um método do tipo Gradientes Conjugados com Pré-condicionamento (PCG) multiescala e multigrid. Combinamos as metodologias multi-escala e multigrid de modo a capturar os distintos comprimentos de onda associados aos diferentes comprimentos de onda do operador diferencial auto-adjunto de Poisson, fortemente influenciado pela heterogeneidade das propriedades geológicas do meio poroso, em particular da permeabilidade absoluta, que pode exibir flutuações em várias ordens de grandeza. Experimentos computacionais em aplicações de problemas de dinâmica de fluidos em meios porosos são apresentados e discutidos para verificação dos resultados obtidos
Abstract: The focus of this work is the numerical approximation of differential problems involving partial differential equations (PDE's) of elliptic nature, in the context of modelling and simulation in fluid dynamics in porous media. The dissertation aims to contribute with an implementation of a multiscale multigrid algorithm, recently introduced in the literature, designed for solving ill-conditioned large linear systems of equations derived from those classes of PDE's, typically associated with Poisson problems of pressure-velocity with boundary conditions typical of flow in porous media. The Poisson problem discussed here is identified from the coupled convection-diffusion transport system counterpart of PDE's, with dominated convection, and by a linearization by means the use of an operator splitting approach. The methodology used for the discretization of the Poisson elliptic problem is by mixed hybrid finite elements. The numerical solution of the resulting linear system will be addressed by a multiscale multigrid preconditioned conjugate gradient (PCG) method. We combine both methodologies in order to capture the distinct wavelengths associated with the different wavelengths from the assosiated self-adjoint Poisson operator, strongly influenced by the heterogeneity of the geological properties of the porous media, in particular to the absolute permeability tensor, which in turn might exhibit very large fluctuations of orders of magnitude. Numerical experiments in applications of fluid dynamics problems in porous media are presented and discussed for a verification of the results obtained by direct numerical simulations with the multiscale multigrid algorithm under consideration
Mestrado
Matematica Aplicada
Mestra em Matemática Aplicada
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Nguepedja, Nankep Mac jugal. "Modélisation stochastique de systèmes biologiques multi-échelles et inhomogènes en espace." Thesis, Rennes, École normale supérieure, 2018. http://www.theses.fr/2018ENSR0012/document.
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Les besoins grandissants de prévisions robustes pour des systèmes complexes conduisent à introduire des modèles mathématiques considérant un nombre croissant de paramètres. Au temps s'ajoutent l'espace, l'aléa, les échelles de dynamiques, donnant lieu à des modèles stochastiques multi-échelles avec dépendance spatiale (modèles spatiaux). Cependant, l'explosion du temps de simulation de tels modèles complique leur utilisation. Leur analyse difficile a néanmoins permis, pour les modèles à une échelle, de développer des outils puissants: loi des grands nombres (LGN), théorème central limite (TCL), ..., puis d'en dériver des modèles simplifiés et algorithmes accélérés. Dans le processus de dérivation, des modèles et algorithmes dits hybrides ont vu le jour dans le cas multi-échelle, mais sans analyse rigoureuse préalable, soulevant ainsi la question d'approximation hybride dont la consistance constitue l'une des motivations principales de cette thèse.En 2012, Crudu, Debussche, Muller et Radulescu établissent des critères d'approximation hybride pour des modèles homogènes en espace de réseaux de régulation de gènes. Le but de cette thèse est de compléter leur travail et le généraliser à un cadre spatial.Nous avons développé et simplifié différents modèles, tous des processus de Markov de sauts pures à temps continu. La démarche met en avant, d'une part, des conditions d'approximations déterministes par des solutions d'équations d'évolution (type réaction-advection-diffusion), et, d'autre part, des conditions d'approximations hybrides par des processus stochastiques hybrides. Dans le cadre des réseaux de réactions biochimiques, un TCL est établi. Il correspond à une approximation hybride d'un modèle homogène simplifié à deux échelles de temps (suivant Crudu et al.). Puis, une LGN est obtenue pour un modèle spatial à deux échelles de temps. Ensuite, une approximation hybride est établie pour un modèle spatial à deux échelles de dynamique en temps et en espace. Enfin, des comportements asymptotiques en grandes populations et en temps long sont présentés pour un modèle d'épidémie de choléra, via une LGN suivie d'une borne supérieure pour les sous-ensembles compacts, dans le cadre d'un principe de grande déviation (PGD) correspondant.À l'avenir, il serait intéressant, entre autres, de varier la géométrie spatiale, de généraliser le TCL, de compléter les estimations du PGD, et d'explorer des systèmes complexes issus d'autres domainesThe growing needs of precise predictions for complex systems lead to introducing stronger mathematical models, taking into account an increasing number of parameters added to time: space, stochasticity, scales of dynamics. Combining these parameters gives rise to spatial --or spatially inhomogeneous-- multiscale stochastic models. However, such models are difficult to study and their simulation is extremely time consuming, making their use not easy. Still, their analysis has allowed one to develop powerful tools for one scale models, among which are the law of large numbers (LLN) and the central limit theorem (CLT), and, afterward, to derive simpler models and accelrated algorithms. In that deduction process, the so-called hybrid models and algorithms have arisen in the multiscale case, but without any prior rigorous analysis. The question of hybrid approximation then shows up, and its consistency is a particularly important motivation of this PhD thesis.In 2012, criteria for hybrid approximations of some homogeneous regulation gene network models were established by Crudu, Debussche, Muller and Radulescu. The aim of this PhD thesis is to complete their work and generalize it afterward to a spatial framework.We have developed and simplified different models. They all are time continuous pure jump Markov processes. The approach points out the conditions allowing on the the one hand deterministic approximations by solutions of evolution equations of type reaction-advection-diffusion, and, on the other hand, hybrid approximations by hybrid stochastic processes. In the field of biochemical reaction networks, we establish a CLT. It corresponds to a hybrid approximation of a simplified homogeneous model (due to Crudu et al.). Then a LLN is obtained for a spatial model with two time scales. Afterward, a hybrid approximation is established, for a two time-space scales spatial model. Finally, the asymptotic behaviour in large population and long time are respectively presented for a model of cholera epidemic, through a LLN followed by the upper bound for compact sets, in the context of a corresponding large deviation principle (LDP).Interesting future works would be, among others, to study other spatial geometries, to generalize the CLT, to complete the LDP estimates, and to study complex systems from other fields
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Moreau, Antoine. "Calcul des propriétés homogénéisées de transfert dans les matériaux poreux par des méthodes de réduction de modèle : Application aux matériaux cimentaires." Thesis, La Rochelle, 2022. http://www.theses.fr/2022LAROS024.
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Cette thèse propose de coupler deux outils préexistant pour la modélisation mathématique en mécanique : l’homogénéisation périodique et la réduction de modèle, afin de modéliser la corrosion des structures de béton armé exposées à la pollution atmosphérique et au sel marin. Cette dégradation est en effet difficile à simuler numériquement, eu égard la forte hétérogénéité des matériaux concernés, et la variabilité de leur microstructure. L’homogénéisation périodique fournit un modèle multi-échelle permettant de s’affranchir de la première de ces deux difficultés. Néanmoins, elle repose sur l’existence d’un volume élémentaire représentatif (VER) de la microstructure du matériau poreux modélisé. Afin de prendre en compte la variabilité de cette dernière, on est amenés à résoudre en temps réduit les équations issues du modèle multi-échelle pour un grand nombre VER. Ceci motive l’utilisation de la méthode POD de réduction de modèle. Cette thèse propose de recourir à des transformations géométriques pour transporter ces équations sur la phase fluide d’un VER de référence. La méthode POD ne peut, en effet, pas être utilisée directement sur un domaine spatial variable (ici le réseau de pores du matériau). Dans un deuxième temps, on adapte ce nouvel outil à l’équation de Poisson-Boltzmann, fortement non linéaire, qui régit la diffusion ionique à l’échelle de la longueur de Debye. Enfin, on combine ces nouvelles méthodes à des techniques existant en réduction de modèle (MPS, interpolation ITSGM), pour tenir compte du couplage micro-macroscopique entre les équations issues de l’homogénéisation périodiqueIn this thesis, we manage to combine two existing tools in mechanics: periodic homogenization, and reduced-order modelling, to modelize corrosion of reinforced concrete structures. Indeed, chloride and carbonate diffusion take place their pores and eventually oxydate their steel skeleton. The simulation of this degradation is difficult to afford because of both the material heterogenenity, and its microstructure variability. Periodic homogenization provides a multiscale model which takes care of the first of these issues. Nevertheless, it assumes the existence of a representative elementary volume (REV) of the material at the microscopical scale. I order to afford the microstructure variability, we must solve the equations which arise from periodic homogenization in a reduced time. This motivates the use of model order reduction, and especially the POD. In this work we design geometrical transformations that transport the original homogenization equations on the fluid domain of a unique REV. Indeed, the POD method can’t be directly performed on a variable geometrical space like the material pore network. Secondly, we adapt model order reduction to the Poisson-Boltzmann equation, which is strongly nonlinear, and which rules ionic electro diffusion at the Debye length scale. Finally, we combine these new methods to other existing tools in model order reduction (ITSGM interpolatin, MPS method), in order to couple the micro- and macroscopic components of periodic homogenization
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Zhang, Tianyu. "Problème inverse statistique multi-échelle pour l'identification des champs aléatoires de propriétés élastiques." Thesis, Paris Est, 2019. http://www.theses.fr/2019PESC2068.
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Dans le cadre de la théorie de l'élasticité linéaire, la modélisation et simulation numérique du comportement mécanique des matériaux hétérogènes à microstructure aléatoire complexe soulèvent de nombreux défis scientifiques à différentes échelles. Bien qu'à l'échelle macroscopique, ces matériaux soient souvent modélisés comme des milieux homogènes et déterministes, ils sont non seulement hétérogènes et aléatoires à l'échelle microscopique, mais ils ne peuvent généralement pas non plus être explicitement décrits par les propriétés morphologiques et mécaniques locales de leurs constituants. Par conséquent, une échelle mésoscopique est introduite entre l'échelle macroscopique et l'échelle mésoscopique, pour laquelle les propriétés mécaniques d'un tel milieu élastique linéaire aléatoire sont décrites par un modèle stochastique prior non-gaussien paramétré par un nombre faible ou modéré d'hyperparamètres inconnus. Afin d'identifier ces hyperparamètres, une méthodologie innovante a été récemment proposée en résolvant un problème statistique inverse multi-échelle en utilisant uniquement des données expérimentales partielles et limitées aux deux échelles macroscopique et mésoscopique. Celui-ci a été formulé comme un problème d'optimisation multi-objectif qui consiste à minimiser une fonction-coût multi-objectif (à valeurs vectorielles) définie par trois indicateurs numériques correspondant à des fonctions-coût mono-objectif (à valeurs scalaires) permettant de quantifier et minimiser des distances entre les données expérimentales multi-échelles mesurées simultanément aux deux échelles macroscopique et mésoscopique sur un seul échantillon soumis à un essai statique, et les solutions des modèles numériques déterministe et stochastique utilisés pour simuler la configuration expérimentale multi-échelle sous incertitudes. Ce travail de recherche vise à contribuer à l'amélioration de la méthodologie d'identification inverse statistique multi-échelle en terme de coût de calcul, de précision et de robustesse en introduisant (i) une fonction-coût mono-objectif (indicateur numérique) supplémentaire à l'échelle mésoscopique quantifiant la distance entre la(les) longueur(s) de corrélation spatiale des champs expérimentaux mesurés et celle(s) des champs numériques calculés, afin que chaque hyperparamètre du modèle stochastique prior ait sa propre fonction-coût mono-objectif dédiée, permettant ainsi d'éviter d'avoir recours à l'algorithme d'optimisation global (algorithme génétique) utilisé précédemment et de le remplacer par un algorithme plus performant en terme d'efficacité numérique, tel qu'un algorithme itératif de type point fixe, pour résoudre le problème d'optimisation multi-objectif avec un coût de calcul plus faible, et (ii) une représentation stochastique ad hoc des hyperparamètres impliqués dans le modèle stochastique prior du champ d'élasticité aléatoire à l'échelle mésoscopique en les modélisant comme des variables aléatoires, pour lesquelles les distributions de probabilité peuvent être construites en utilisant le principe du maximum d'entropie sous un ensemble de contraintes définies par les informations objectives et disponibles, et dont les hyperparamètres peuvent être déterminés à l'aide de la méthode d'estimation du maximum de vraisemblance avec les données disponibles, afin d'améliorer à la fois la robustesse et la précision de la méthode d'identification inverse du modèle stochastique prior. En parallèle, nous proposons également de résoudre le problème d'optimisation multi-objectif en utilisant l’apprentissage automatique par des réseaux de neurones artificiels. Finalement, la méthodologie améliorée est tout d'abord validée sur un matériau virtuel fictif dans le cadre de l'élasticité linéaire en 2D contraintes planes et 3D, puis illustrée sur un matériau biologique hétérogène réel (os cortical de bœuf) en élasticité linéaire 2D contraintes planesWithin the framework of linear elasticity theory, the numerical modeling and simulation of the mechanical behavior of heterogeneous materials with complex random microstructure give rise to many scientific challenges at different scales. Despite that at macroscale such materials are usually modeled as homogeneous and deterministic elastic media, they are not only heterogeneous and random at microscale, but they often also cannot be properly described by the local morphological and mechanical properties of their constituents. Consequently, a mesoscale is introduced between macroscale and microscale, for which the mechanical properties of such a random linear elastic medium are represented by a prior non-Gaussian stochastic model parameterized by a small or moderate number of unknown hyperparameters. In order to identify these hyperparameters, an innovative methodology has been recently proposed by solving a multiscale statistical inverse problem using only partial and limited experimental data at both macroscale and mesoscale. It has been formulated as a multi-objective optimization problem which consists in minimizing a (vector-valued) multi-objective cost function defined by three numerical indicators corresponding to (scalar-valued) single-objective cost functions for quantifying and minimizing distances between multiscale experimental data measured simultaneously at both macroscale and mesoscale on a single specimen subjected to a static test, and the numerical solutions of deterministic and stochastic computational models used for simulating the multiscale experimental test configuration under uncertainties. This research work aims at contributing to the improvement of the multiscale statistical inverse identification method in terms of computational efficiency, accuracy and robustness by introducing (i) an additional mesoscopic numerical indicator allowing the distance between the spatial correlation length(s) of the measured experimental fields and the one(s) of the computed numerical fields to be quantified at mesoscale, so that each hyperparameter of the prior stochastic model has its own dedicated single-objective cost-function, thus allowing the time-consuming global optimization algorithm (genetic algorithm) to be avoided and replaced with a more efficient algorithm, such as the fixed-point iterative algorithm, for solving the underlying multi-objective optimization problem with a lower computational cost, and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model of the random elasticity field at mesoscale by modeling them as random variables, for which the probability distributions can be constructed by using the maximum entropy principle under a set of constraints defined by the available and objective information, and whose hyperparameters can be determined using the maximum likelihood estimation method with the available data, in order to enhance both the robustness and accuracy of the statistical inverse identification method of the prior stochastic model. Meanwhile, we propose as well to solve the multi-objective optimization problem by using machine learning based on artificial neural networks. Finally, the improved methodology is first validated on a fictitious virtual material within the framework of 2D plane stress and 3D linear elasticity theory, and then illustrated on a real heterogenous biological material (beef cortical bone) in 2D plane stress linear elasticity