Relevant bibliographies by topics / Differential equations, Partial Data processing

Academic literature on the topic 'Differential equations, Partial Data processing'

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Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "Differential equations, Partial Data processing"

1

VAN GENNIP, YVES, and CAROLA-BIBIANE SCHÖNLIEB. "Introduction: Big data and partial differential equations." European Journal of Applied Mathematics 28, no. 6 (November 7, 2017): 877–85. http://dx.doi.org/10.1017/s0956792517000304.

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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].
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Dejnožková, Eva, and Petr Dokládal. "A PARALLEL ARCHITECTURE FOR CURVE-EVOLUTION PARTIAL DIFFERENTIAL EQUATIONS." Image Analysis & Stereology 22, no. 2 (May 3, 2011): 121. http://dx.doi.org/10.5566/ias.v22.p121-132.

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The computation of the distance function is a crucial and limiting element in many applications of image processing. This is particularly true for the PDE-based methods, where the distance is used to compute various geometric properties of the travelling curve. Massive Marchinga is a parallel algorithm computing the distance function by propagating the solution from the sources and permitting simultaneous spreading of component labels in the infiuence zones. Its hardware implementation is conceivable as no sorted data structures are used. The feasibility is demonstrated here on a set of parallely-operating Processing Units arranged in a linear array. The text concludes by a study of the accuracy and the implementation cost.
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Boudellioua, M. S. "Controllable and Observable Polynomial Description for 2D Noncausal Systems." Journal of Control Science and Engineering 2007 (2007): 1–5. http://dx.doi.org/10.1155/2007/87171.

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Two-dimensional state-space systems arise in applications such as image processing, iterative circuits, seismic data processing, or more generally systems described by partial differential equations. In this paper, a new direct method is presented for the polynomial realization of a class of noncausal 2D transfer functions. It is shown that the resulting realization is both controllable and observable.
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Chevallier, Julien, María José Cáceres, Marie Doumic, and Patricia Reynaud-Bouret. "Microscopic approach of a time elapsed neural model." Mathematical Models and Methods in Applied Sciences 25, no. 14 (October 14, 2015): 2669–719. http://dx.doi.org/10.1142/s021820251550058x.

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The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present paper is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.
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Jafarian, Ahmad, and Dumitru Baleanu. "Application of ANNs approach for wave-like and heat-like equations." Open Physics 15, no. 1 (December 29, 2017): 1086–94. http://dx.doi.org/10.1515/phys-2017-0135.

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Abstract Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.
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Miller, Andrew, Jan Petrich, and Shashi Phoha. "Advanced Image Analysis for Learning Underlying Partial Differential Equations for Anomaly Identification." Journal of Imaging Science and Technology 64, no. 2 (March 1, 2020): 20510–1. http://dx.doi.org/10.2352/j.imagingsci.technol.2020.64.2.020510.

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Abstract In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.
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Engl, H. W., O. Scherzer, and M. Yamamoto. "Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data." Inverse Problems 10, no. 6 (December 1, 1994): 1253–76. http://dx.doi.org/10.1088/0266-5611/10/6/006.

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8

Gillespie, Mark, Nicholas Sharp, and Keenan Crane. "Integer coordinates for intrinsic geometry processing." ACM Transactions on Graphics 40, no. 6 (December 2021): 1–13. http://dx.doi.org/10.1145/3478513.3480522.

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This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Many applications demand a correspondence between the intrinsic triangulation and the input surface, but existing data structures either rely on floating point values to encode correspondence, or do not support remeshing operations beyond basic edge flips. We instead provide an integer-based data structure that guarantees valid correspondence, even for meshes with near-degenerate elements. Our starting point is the framework of normal coordinates from geometric topology, which we extend to the broader set of operations needed for mesh processing (vertex insertion, edge splits, etc. ). The resulting data structure can be used as a drop-in replacement for earlier schemes, automatically improving reliability across a wide variety of applications. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset. In turn, we can compute reliable and highly accurate solutions to partial differential equations even on extremely low-quality meshes.
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Kumar, Dr Amresh, and Dr Ram Kishore Singh. "A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”." International Journal of Scientific Research and Management 8, no. 05 (May 15, 2020): 386–96. http://dx.doi.org/10.18535/ijsrm/v8i05.ec02.

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Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.
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Rvachov, Volodimir Olexijovych, Tatiana Volodimirivna Rvachova, and Evgenia Pavlovna Tomilova. "TOMIC FUNCTIONS AND LACUNARY INTERPOLATION SERIES IN BOUNDARY VALUE PROBLEMS FOR PARTIAL DERIVATIVES EQUATIONS AND IMAGE PROCESSING." RADIOELECTRONIC AND COMPUTER SYSTEMS, no. 1 (January 28, 2020): 58–69. http://dx.doi.org/10.32620/reks.2020.1.06.

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In the paper we consider and solve the problem of construction of the so called tomic functions – the systems of infinitely differentiable functions which while retaining many important properties of the shifts of atomic function up(x) such as locality and representation of algebraic polynomials and being based on the atomic functions nevertheless have nonuniform character and therefore allow to take into account the inhomogeneous and changing character of the data encountered in real world problems in particular in boundary value problems for partial differential equations with variable coefficients and complex geometry of domains in which these boundary value problems must be solved. The same class of tomic functions can be applied to processing,denoising and sparse storage of signals and images by lacunary interpolation. The lacunary or Birkhoff interpolation of functions in which the function is being restored by the values of derivatives of orderin points in which values of function and derivatives of order k<r are unknown is of great importance in many real world problems such as remote sensing. The lacunary interpolation methods using the tomic functions possesss important advantages over currently widely applied lacunary spline interpolation in view of infinite smoothness of tomic functions.The tomic functions can also be applied to connect (to stitch) atomic expansions with different steps on different intervals preserving smoothness and optimal approximation properties. The equations for of construction oftomic functions tofuj(x) –analogues of the basic functions of the generalized atomic Taylor expansions are obtained – which are needed for lacunary (Birkhoff) interpolation. For the applications in variational and collocation methods for solving bondary value problems for partial derivative and integral equations the tomic functions ftupr,j(x) are obtained that are analogues of B-splines and atomic functions fupn(x). Using similar methods, the tomic functions based on other atomic functions such as Ξn(x) can be obtained.
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Dissertations / Theses on the topic "Differential equations, Partial Data processing"

1

He, Chuan. "Numerical solutions of differential equations on FPGA-enhanced computers." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-1248.

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Karasev, Peter A. "Feedback augmentation of pde-based image segmentation algorithms using application-specific exogenous data." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50257.

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This thesis is divided into five chapters. The scope of problems considered is defined in chapter I. Next, chapter II provides background material on image processing with partial differential equations and a review of prior work in the field. Chapter III covers the medical imaging portion of the research; the key contribution is a control-based algorithm for interactive image segmentation. Applications of the feedback-augmented level set method to fracture reconstruction and surgical planning are shown. Problems in vision-based control are considered in Chapters IV and V. A method of improving performance in closed-loop target tracking using level set segmentation is developed, with unmanned aerial vehicle or next-generation missile guidance being the primary applications of interest. Throughout this thesis, the two application types are connected into a unified viewpoint of open-loop systems that are augmented by exogenous data.
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Lazcano, Vanel. "Some problems in depth enhanced video processing." Doctoral thesis, Universitat Pompeu Fabra, 2016. http://hdl.handle.net/10803/373917.

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In this thesis we tackle two problems, namely, the data interpolation prob- lem in the context of depth computation both for images and for videos, and the problem of the estimation of the apparent movement of objects in image sequences. The rst problem deals with completion of depth data in a region of an image or video where data are missing due to occlusions, unreliable data, damage or lost of data during acquisition. In this thesis we tackle it in two ways. First, we propose a non-local gradient-based energy which is able to complete planes locally. We consider this model as an extension of the bilateral lter to the gradient domain. We have successfully evaluated our model to complete synthetic depth images and also incomplete depth maps provided by a Kinect sensor. The second approach to tackle the problem is an experimental study of the Biased Absolutely Minimizing Lipschitz Extension (biased AMLE in short) for anisotropic interpolation of depth data to big empty regions without informa- tion. The AMLE operator is a cone interpolator, but the biased AMLE is an exponential cone interpolator which makes it more addapted to depth maps of real scenes that usually present soft convex or concave surfaces. Moreover, the biased AMLE operator is able to expand depth data to huge regions. By con- sidering the image domain endowed with an anisotropic metric, the proposed method is able to take into account the underlying geometric information in order not to interpolate across the boundary of objects at di erent depths. We have proposed a numerical model to compute the solution of the biased AMLE which is based on the eikonal operators. Additionally, we have extended the proposed numerical model to video sequences. The second problem deals with the motion estimation of the objects in a video sequence. This problem is known as the optical ow computation. The Optical ow problem is one of the most challenging problems in computer vision. Traditional models to estimate it fail in presence of occlusions and non-uniform illumination. To tackle these problems we proposed a variational model to jointly estimate optical ow and occlusion. Moreover, the proposed model is able to deal with the usual drawback of variational methods in dealing with fast displacements of objects in the scene which are larger than the object it- self. The addition of a term that balance gradient and intensities increases the robustness to illumination changes of the proposed model. The inclusions of a supplementary matches given by exhaustive search in speci cs locations helps to follow large displacements.
En esta tesis se abordan dos problemas: interpolación de datos en el contexto del cálculo de disparidades tanto para imágenes como para video, y el problema de la estimación del movimiento aparente de objetos en una secuencia de imágenes. El primer problema trata de la completación de datos de profundidad en una región de la imagen o video dónde los datos se han perdido debido a oclusiones, datos no confiables, datos dañados o pérdida de datos durante la adquisición. En esta tesis estos problemas se abordan de dos maneras. Primero, se propone una energía basada en gradientes no-locales, energía que puede (localmente) completar planos. Se considera este modelo como una extensión del filtro bilateral al dominio del gradiente. Se ha evaluado en forma exitosa el modelo para completar datos sintéticos y también mapas de profundidad incompletos de un sensor Kinect. El segundo enfoque, para abordar el problema, es un estudio experimental del biased AMLE (Biased Absolutely Minimizing Lipschitz Extension) para interpolación anisotrópica de datos de profundidad en grandes regiones sin información. El operador AMLE es un interpolador de conos, pero el operador biased AMLE es un interpolador de conos exponenciales lo que lo hace estar más adaptado a mapas de profundidad de escenas reales (las que comunmente presentan superficies convexas, concavas y suaves). Además, el operador biased AMLE puede expandir datos de profundidad a regiones grandes. Considerando al dominio de la imagen dotado de una métrica anisotrópica, el método propuesto puede tomar en cuenta información geométrica subyacente para no interpolar a través de los límites de los objetos a diferentes profundidades. Se ha propuesto un modelo numérico, basado en el operador eikonal, para calcular la solución del biased AMLE. Adicionalmente, se ha extendido el modelo numérico a sequencias de video. El cálculo del flujo óptico es uno de los problemas más desafiantes para la visión por computador. Los modelos tradicionales fallan al estimar el flujo óptico en presencia de oclusiones o iluminación no uniforme. Para abordar este problema se propone un modelo variacional para conjuntamente estimar flujo óptico y oclusiones. Además, el modelo propuesto puede tolerar, una limitación tradicional de los métodos variacionales, desplazamientos rápidos de objetos que son más grandes que el tamaño objeto en la escena. La adición de un término para el balance de gradientes e intensidades aumenta la robustez del modelo propuesto ante cambios de iluminación. La inclusión de correspondencias adicionales (obtenidas usando búsqueda exhaustiva en ubicaciones específicas) ayuda a estimar grandes desplazamientos.
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Michel, Thomas. "Analyse mathématique et calibration de modèles de croissance tumorale." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0222/document.

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Cette thèse présente des travaux sur l’étude et la calibration de modèles d’équations aux dérivées partielles pour la croissance tumorale. La première partie porte sur l’analyse d’un modèle de croissance tumorale pour le cas de métastases au foie de tumeurs gastro-intestinales (GIST). Le modèle est un système d’équations aux dérivées partielles couplées et prend en compte plusieurs traitements dont un traitement anti-angiogénique. Le modèle permet de reproduire des données cliniques. La première partie de ce travail concerne la preuve d’existence/unicité de la solution du modèle. La seconde partie du travail porte sur l’étude du comportement asymptotique de la solution du modèle lorsqu’un paramètre du modèle, décrivant la capacité de la tumeur à évacuer la nécrose, converge vers 0. La seconde partie de la thèse concerne le développement d’un modèle de croissance pour des sphéroïdes tumoraux ainsi que sur la calibration de ce modèle à partir de données expérimentales in vitro. L’objectif est de développer un modèle permettant de reproduire quantitativement la distribution des cellules proliférantes à l’intérieur d’un sphéroïde en fonction de la concentration en nutriments. Le travail de modélisation et de calibration du modèle a été effectué à partir de données expérimentales permettant d’obtenir la répartition spatiale de cellules proliférantes dans un sphéroïde tumoral
In this thesis, we present several works on the study and the calibration of partial differential equations models for tumor growth. The first part is devoted to the mathematical study of a model for tumor drug resistance in the case of gastro-intestinal tumor (GIST) metastases to the liver. The model we study consists in a coupled partial differential equations system and takes several treatments into account, such as a anti-angiogenic treatment. This model is able to reproduce clinical data. In a first part, we present the proof of the existence/uniqueness of the solution to this model. Then, in a second part, we study the asymptotic behavior of the solution when a parameter of this model, describing the capacity of the tumor to evacuate the necrosis, goes to 0. In the second part of this thesis, we present the development of model for tumor spheroids growth. We also present the model calibration thanks to in vitro experimental data. The main objective of this work is to reproduce quantitatively the proliferative cell distribution in a spheroid, as a function of the concentration of nutrients. The modeling and calibration of this model have been done thanks to experimental data consisting of proliferative cells distribution in a spheroid
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Sum, Kwok-wing Anthony. "Partial differential equation based methods in medical image processing." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/hkuto/record/B38958624.

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Ozmen, Neslihan. "Image Segmentation And Smoothing Via Partial Differential Equations." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12610395/index.pdf.

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In image processing, partial differential equation (PDE) based approaches have been extensively used in segmentation and smoothing applications. The Perona-Malik nonlinear diffusion model is the first PDE based method used in the image smoothing tasks. Afterwards the classical Mumford-Shah model was developed to solve both image segmentation and smoothing problems and it is based on the minimization of an energy functional. It has numerous application areas such as edge detection, motion analysis, medical imagery, object tracking etc. The model is a way of finding a partition of an image by using a piecewise smooth representation of the image. Unfortunately numerical procedures for minimizing the Mumford-Shah functional have some difficulties because the problem is non convex and it has numerous local minima, so approximate approaches have been proposed. Two such methods are the Ambrosio-Tortorelli approximation and the Chan-Vese active contour method. Ambrosio and Tortorelli have developed a practical numerical implementation of the Mumford-Shah model which based on an elliptic approximation of the original functional. The Chan-Vese model is a piecewise constant generalization of the Mumford-Shah functional and it is based on level set formulation. Another widely used image segmentation technique is the &ldquo
Active Contours (Snakes)&rdquo
model and it is correlated with the Chan-Vese model. In this study, all these approaches have been examined in detail. Mathematical and numerical analysis of these models are studied and some experiments are performed to compare their performance.
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Sum, Kwok-wing Anthony, and 岑國榮. "Partial differential equation based methods in medical image processing." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B38958624.

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Kadhum, Nashat Ibrahim. "The spline approach to the numerical solution of parabolic partial differential equations." Thesis, Loughborough University, 1988. https://dspace.lboro.ac.uk/2134/6725.

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This thesis is concerned with the Numerical Solution of Partial Differential Equations. Initially some definitions and mathematical background are given, accompanied by the basic theories of solving linear systems and other related topics. Also, an introduction to splines, particularly cubic splines and their identities are presented. The methods used to solve parabolic partial differential equations are surveyed and classified into explicit or implicit (direct and iterative) methods. We concentrate on the Alternating Direction Implicit (ADI), the Group Explicit (GE) and the Crank-Nicolson (C-N) methods. A new method, the Splines Group Explicit Iterative Method is derived, and a theoretical analysis is given. An optimum single parameter is found for a special case. Two criteria for the acceleration parameters are considered; they are the Peaceman-Rachford and the Wachspress criteria. The method is tested for different numbers of both parameters. The method is also tested using single parameters, i. e. when used as a direct method. The numerical results and the computational complexity analysis are compared with other methods, and are shown to be competitive. The method is shown to have good stability property and achieves high accuracy in the numerical results. Another direct explicit method is developed from cubic splines; the splines Group Explicit Method which includes a parameter that can be chosen to give optimum results. Some analysis and the computational complexity of the method is given, with some numerical results shown to confirm the efficiency and compatibility of the method. Extensions to two dimensional parabolic problems are given in a further chapter. In this thesis the Dirichlet, the Neumann and the periodic boundary conditions for linear parabolic partial differential equations are considered. The thesis concludes with some conclusions and suggestions for further work.
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Elyan, Eyad, and Hassan Ugail. "Reconstruction of 3D human facial images using partial differential equations." Academy Publisher, 2007. http://hdl.handle.net/10454/2644.

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One of the challenging problems in geometric modeling and computer graphics is the construction of realistic human facial geometry. Such geometry are essential for a wide range of applications, such as 3D face recognition, virtual reality applications, facial expression simulation and computer based plastic surgery application. This paper addresses a method for the construction of 3D geometry of human faces based on the use of Elliptic Partial Differential Equations (PDE). Here the geometry corresponding to a human face is treated as a set of surface patches, whereby each surface patch is represented using four boundary curves in the 3-space that formulate the appropriate boundary conditions for the chosen PDE. These boundary curves are extracted automatically using 3D data of human faces obtained using a 3D scanner. The solution of the PDE generates a continuous single surface patch describing the geometry of the original scanned data. In this study, through a number of experimental verifications we have shown the efficiency of the PDE based method for 3D facial surface reconstruction using scan data. In addition to this, we also show that our approach provides an efficient way of facial representation using a small set of parameters that could be utilized for efficient facial data storage and verification purposes.
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Dong, Bin. "Applications of variational models and partial differential equations in medical image and surface processing." Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1872060431&sid=3&Fmt=2&clientId=1564&RQT=309&VName=PQD.

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Books on the topic "Differential equations, Partial Data processing"

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Partial differential equations with Mathematica. Wokingham, England: Addison-Wesley Pub. Co., 1993.

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1943-, Flaherty J. E., and Workshop on Adaptive Computational Methods for Partial Differential Equations (1988 : Rensselaer Polytechnic Institute), eds. Adaptive methods for partial differential equations. Philadelphia: Society for Industrial and Applied Mathematics, 1989.

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Ruth, Petzold Linda, ed. Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia: Society for Industrial and Applied Mathematics, 1998.

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Yi-Tung, Chen, ed. Computational partial differential equations using MATLAB. Boca Raton, Fla: Chapman & Hall/CRC Press, 2008.

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E, Schiesser W., ed. Ordinary and partial differential equation routines in C, C++, Fortran, Java, Maple, and MATLAB. Boca Raton: Chapman & Hall/CRC, 2004.

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Introduction to numerical ordinary and partial differential equations using MATLAB. Hoboken, N.J: Wiley-Interscience, 2005.

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Reed, Daniel A. Stencils and problem partitionings: Their influence on the performance of multiple processor systems. Urbana, Ill: Dept. of Computer Science, University of Illinois at Urbana-Champaign, 1986.

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Computational partial differential equations: Numerical methods and Diffpack programming. Berlin: Springer, 1999.

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The numerical solution of ordinary and partial differential equations. San Diego, CA: Academic Press, 1988.

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The numerical solution of ordinary and partial differential equations. 2nd ed. Hoboken, N.J: John Wiley, 2005.

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Book chapters on the topic "Differential equations, Partial Data processing"

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Villegas, Rossmary, Oliver Dorn, Miguel Moscoso, and Manuel Kindelan. "Shape Reconstruction from Two-Phase Incompressible Flow Data using Level Sets." In Image Processing Based on Partial Differential Equations, 381–401. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-33267-1_21.

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Mikula, Karol. "Image processing with partial differential equations." In Modern Methods in Scientific Computing and Applications, 283–321. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-010-0510-4_8.

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Bredies, Kristian, and Dirk Lorenz. "Partial Differential Equations in Image Processing." In Applied and Numerical Harmonic Analysis, 171–250. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-01458-2_5.

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Chen, Li M. "Gradual Variations and Partial Differential Equations." In Digital Functions and Data Reconstruction, 145–59. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5638-4_10.

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Bærentzen, Jakob Andreas, Jens Gravesen, François Anton, and Henrik Aanæs. "Finite Difference Methods for Partial Differential Equations." In Guide to Computational Geometry Processing, 65–79. London: Springer London, 2012. http://dx.doi.org/10.1007/978-1-4471-4075-7_4.

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Choi, Sunhi, and Inwon C. Kim. "Homogenization with oscillatory Neumann boundary data in general domain." In Geometric Partial Differential Equations proceedings, 105–18. Pisa: Scuola Normale Superiore, 2013. http://dx.doi.org/10.1007/978-88-7642-473-1_5.

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Preusser, Tobias, Robert M. Kirby, and Torben Pätz. "Partial Differential Equations and Their Numerics." In Stochastic Partial Differential Equations for Computer Vision with Uncertain Data, 7–26. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-031-02594-5_2.

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Kreinovich, Vladik, Anatoly Lakeyev, Jiří Rohn, and Patrick Kahl. "Solving Differential Equations." In Computational Complexity and Feasibility of Data Processing and Interval Computations, 219–23. Boston, MA: Springer US, 1998. http://dx.doi.org/10.1007/978-1-4757-2793-7_20.

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Nikitin, Alexey. "On an Optimal Control Problem for the Wave Equation in One Space Dimension Controlled by Third Type Boundary Data." In Progress in Partial Differential Equations, 223–38. Heidelberg: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-00125-8_10.

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Burgeth, Bernhard, Joachim Weickert, and Sibel Tari. "Minimally Stochastic Schemes for Singular Diffusion Equations." In Image Processing Based on Partial Differential Equations, 325–39. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-33267-1_18.

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Conference papers on the topic "Differential equations, Partial Data processing"

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Hasan, Ali, Joao M. Pereira, Robert Ravier, Sina Farsiu, and Vahid Tarokh. "Learning Partial Differential Equations From Data Using Neural Networks." In ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2020. http://dx.doi.org/10.1109/icassp40776.2020.9053750.

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Liu, Zeyi, Zhong Liu, Yanghe Feng, Qing Cheng, Xingxing Liang, Rongxiao Wang, Yuling Yang, Naifu Xu, and Yan Li. "Quasi-Spectral Method for Nonlinear Partial Differential KdV Equation in Image Processing." In 2019 5th International Conference on Big Data and Information Analytics (BigDIA). IEEE, 2019. http://dx.doi.org/10.1109/bigdia.2019.8802837.

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Kumar, Ashutosh. "Quantum Computation for End-to-End Seismic Data Processing with Its Computational Advantages and Economic Sustainability." In ADIPEC. SPE, 2022. http://dx.doi.org/10.2118/211843-ms.

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Abstract Mathematical and computational challenges involved in seismic data processing presents an opportunity for early adoption of quantum computation methods for end-to-end seismic data processing. Existing methods of seismic data processing involve processes with exponential complexities that result in approximations as well as conversion of some of the continuous phenomena into a stochastic one. In the classical computation methods, the mentioned approximations and assumptions enable us to obtain acceptable results in commercially viable time. This paper proposes alternatives of the classical computations that exist in the quantum computation ecosystem along with the computational advantages it holds. The paper also presents potential contributions of the petroleum industry towards sustaining the quantum computation technologies. Fundamentally seismic data processing involves solutions for systems of linear equations and its derivatives. Quantum computation ecosystem holds efficient solutions for systems of linear equations. In the frequency domain, Finite-Difference modelling reduces seismic-wave equations to systems of linear equations. In the classical computational setup the seismic acquisition involves treatment of the recorded waves as rays and has limited summation provision for recreating the natural reflection or refraction phenomena that is continuous instead of being a stochastic process. The algorithms in the quantum ecosystem allow us to consider summation of signals from all possible paths between the source and the receiver, by amplitude-probability. In addition to the systems of linear equations and their solution with corresponding methods in the quantum ecosystem the fourier transformation and partial differential equations enable us to decompose the waves and apply the physics equation to obtain the desired objective. Quantum-algorithms facilitate exponential speed-up in seismic data processing. The PDE-constrained optimization inverts subsurface P-wave velocity. While going through the seismic data processing steps it is found that the fourier transformation algorithms are derived as a decomposition of the diagonal matrix. The key difference between the fast fourier transform and the quantum fourier transform is that the quantum fourier transformation is used as the building block of several quantum algorithms. Seismic inversion involves laws of physics and calculation that are guided by the ordinary differential equations. In the quantum computation ecosystem these algorithms for linear ordinary differential equations for linear partial differential equations have the complexity of (1/e), where ‘e’ is the tolerance. The insights brought by successful implementation of end-to-end seismic data processing with algorithms in the quantum computation domain enables us to drill most optimally located wells and hence facilitate cost saving. Even with a reduction of 10% in the total number of wells that we drill, we can possibly fund development of one quantum computer hence ensuring economic sustainability of the technology. The novelty of the presented paper lies in the comparative analysis of the classical methods with its counterparts in the quantum ecosystem. It explains the technological and economical aspects of the technology such that extensive knowledge of quantum technology is not compulsory for grasping its contents.
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Narang, H. N., and Rajiv K. Nekkanti. "Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations." In ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/fedsm2003-45030.

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The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.
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Крысько, Вадим, Vadim Krys'ko, Ирина Папкова, Irina Papkova, Екатерина Крылова, Ekaterina Krylova, Антон Крысько, and Anton Krysko. "Visualization of Transition's Scenarios from Harmonic to Chaotic Flexible Nonlinear-elastic Nano Beam's Oscillations." In 29th International Conference on Computer Graphics, Image Processing and Computer Vision, Visualization Systems and the Virtual Environment GraphiCon'2019. Bryansk State Technical University, 2019. http://dx.doi.org/10.30987/graphicon-2019-2-62-65.

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In this study, a mathematical model of the nonlinear vibrations of a nano-beam under the action of a sign-variable load and an additive white noise was constructed and visualized. The beam is heterogeneous, isotropic, elastic. The physical nonlinearity of the nano-beam was taken into account. The dependence of stress intensity on deformations intensity for aluminum was taken into account. Geometric non-linearity according to Theodore von Karman’s theory was applied. The equations of motion, the boundary and initial conditions of the Hamilton-Ostrogradski principle with regard to the modified couple stress theory were obtained. The system of nonlinear partial differential equations to the Cauchy problem by the method of finite differences was reduced. The Cauchy problem by the finite-difference method in the time coordinate was solved. The Birger variable method was used. Data visualization is carried out from the standpoint of the qualitative theory of differential equations and nonlinear dynamics were carried out. Using a wide range of tools visualization allowed to established that the transition from ordered vibrations to chaos is carried out according to the scenario of Ruelle-Takens-Newhouse. With an increase of the size-dependent parameter, the zone of steady and regular vibrations increases. The transition from regular to chaotic vibrations is accompanied by a tough dynamic loss of stability. The proposed method is universal and can be extended to solve a wide class of various problems of mechanics of shells.
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Carrigan, Travis J., Jacob Watt, and Brian H. Dennis. "Using GPU-Based Computing to Solve Large Sparse Systems of Linear Equations." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-48452.

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Often thought of as tools for image rendering or data visualization, graphics processing units (GPU) are becoming increasingly popular in the areas of scientific computing due to their low cost massively parallel architecture. With the introduction of CUDA C by NVIDIA and CUDA enabled GPUs, the ability to perform general purpose computations without the need to utilize shading languages is now possible. One such application that benefits from the capabilities provided by NVIDIA hardware is computational continuum mechanics (CCM). The need to solve sparse linear systems of equations is common in CCM when partial differential equations are discretized. Often these systems are solved iteratively using domain decomposition among distributed processors working in parallel. In this paper we explore the benefits of using GPUs to improve the performance of sparse matrix operations, more specifically, sparse matrix-vector multiplication. Our approach does not require domain decomposition, so it is simpler than corresponding implementation for distributed memory parallel computers. We demonstrate that for matrices produced from finite element discretizations on unstructured meshes, the performance of the matrix-vector multiplication operation is just under 13 times faster than when run serially on an Intel i5 system. Furthermore, we show that when used in conjunction with the biconjugate gradient stabilized method (BiCGSTAB), a gradient based iterative linear solver, the method is over 13 times faster than the serially executed C equivalent. And lastly, we emphasize the application of such method for solving Poisson’s equation using the Galerkin finite element method, and demonstrate over 10.5 times higher performance on the GPU when compared with the Intel i5 system.
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RAUTELA, MAHINDRA, MANISH RAUT, and S. GOPALAKRISHNAN. "SIMULATION OF GUIDED WAVES FOR STRUCTURAL HEALTH MONITORING USING PHYSICS-INFORMED NEURAL NETWORKS." In Structural Health Monitoring 2021. Destech Publications, Inc., 2022. http://dx.doi.org/10.12783/shm2021/36297.

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Guided wave propagation is a valuable and reliable technique for structural health monitoring (SHM) of aerospace structures. Along with its higher sensitivity towards small damages, it offers advantages in traveling long distances with minimum attenuation. Simulation of guided wave propagation is essential to understand wave behavior, and calculating the dispersion relations forms an integral part of the procedure. Application of the current numerical techniques for complex media is highly involved and faces issues related to accuracy, stability, and computational resources. Development in the field of machine learning and graphical processing units (GPUs) leads to the implementation of a faster, automated, and scalable deep neural networks-based learning approach for such problems. Most of the implementation in the field is based on data collection and uses neural networks for nonlinear mapping from input space to target space. However, a large amount of prior information in the form of a governing differential equation is not utilized. In this paper, we have used Physics-Informed Neural Networks (PINNs), in which neural networks are utilized to solve governing partial differential equations. PINNs are implemented to obtain the solution of a one-dimensional wave equation with Dirichlet boundary conditions. The exact solutions and predicted responses match closely with lower mean square error in limited computational time. We have also conducted a detailed comparison of the effect of neural architecture on the mean square error and the training time. This study shows the merit of deep neural networks leveraging the available physical information to simulate the wave phenomenon for SHM efficiently.
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Zhao, Zhenyu, Chenping Hou, Yi Wu, and Yuanyuan Jiao. "Learning partial differential equations for saliency detection." In 2016 IEEE International Conference on Big Data Analysis (ICBDA). IEEE, 2016. http://dx.doi.org/10.1109/icbda.2016.7509838.

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Totounferoush, Amin, Neda Ebrahimi Pour, Sabine Roller, and Miriam Mehl. "Parallel Machine Learning of Partial Differential Equations." In 2021 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW). IEEE, 2021. http://dx.doi.org/10.1109/ipdpsw52791.2021.00106.

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Kamgar-Parsi, Behzad, Behrooz Kamgar-Parsi, and Kian Kamgar-Parsi. "Notes on image processing with partial differential equations." In 2015 IEEE International Conference on Image Processing (ICIP). IEEE, 2015. http://dx.doi.org/10.1109/icip.2015.7351098.

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Reports on the topic "Differential equations, Partial Data processing"

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Osher, Stanley, and Leonid Rudin. Feature-Oriented Signal Processing Under Nonlinear Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, January 1992. http://dx.doi.org/10.21236/ada259951.

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Jones, Richard H. Fitting Stochastic Partial Differential Equations to Spatial Data. Fort Belvoir, VA: Defense Technical Information Center, September 1993. http://dx.doi.org/10.21236/ada279870.

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Webster, Clayton, Raul Tempone, and Fabio Nobile. The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data. Office of Scientific and Technical Information (OSTI), December 2007. http://dx.doi.org/10.2172/934852.

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