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Academic literature on the topic 'Polynomial chao'

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Published: 14 March 2023

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Journal articles on the topic "Polynomial chao"

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Schmid, Christine, and Kyle J. DeMars. "Angular Correlation Using Rogers-Szegő-Chaos." Mathematics 8, no. 2 (February 1, 2020): 171. http://dx.doi.org/10.3390/math8020171.

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Polynomial chaos expresses a probability density function (pdf) as a linear combination of basis polynomials. If the density and basis polynomials are over the same field, any set of basis polynomials can describe the pdf; however, the most logical choice of polynomials is the family that is orthogonal with respect to the pdf. This problem is well-studied over the field of real numbers and has been shown to be valid for the complex unit circle in one dimension. The current framework for circular polynomial chaos is extended to multiple angular dimensions with the inclusion of correlation terms. Uncertainty propagation of heading angle and angular velocity is investigated using polynomial chaos and compared against Monte Carlo simulation.
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Chen, Hong, Ling Wu, Shu Bin Gu, and Qun Ding. "Comparison of the Image Encryption Effects Based on Different Unary Polynomial Transformation Chaos." Advanced Materials Research 846-847 (November 2013): 948–51. http://dx.doi.org/10.4028/www.scientific.net/amr.846-847.948.

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When the chaos is degenerating, the unary polynomial transformation can make the chaotic signal more complicated. Therefore, the transformed chaotic signal is more suitable for image encryption. In this paper, the image is respectively encrypted by the chaotic signal transformed by different unary polynomials and the chaotic signal without transformation. And then their encryption effects are compared and analyzed by various criteria. Experiments and research results indicate that, after unary polynomial transformation, the effect of chaos-based image encryption is improved. Furthermore, different unary polynomials have different effects on the cipher image. It can make the effect of image encryption better by changing the form of unary polynomial.
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Ghanem, R. "The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables." Journal of Applied Mechanics 66, no. 4 (December 1, 1999): 964–73. http://dx.doi.org/10.1115/1.2791806.

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A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.
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Fan, Chengmei, M. Mobeen Munir, Zafar Hussain, Muhammad Athar, and Jia-Bao Liu. "Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks." Complexity 2021 (February 9, 2021): 1–10. http://dx.doi.org/10.1155/2021/6657298.

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Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α , β -Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.
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SEPAHVAND, K., S. MARBURG, and H. J. HARDTKE. "UNCERTAINTY QUANTIFICATION IN STOCHASTIC SYSTEMS USING POLYNOMIAL CHAOS EXPANSION." International Journal of Applied Mechanics 02, no. 02 (June 2010): 305–53. http://dx.doi.org/10.1142/s1758825110000524.

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In recent years, extensive research has been reported about a method which is called the generalized polynomial chaos expansion. In contrast to the sampling methods, e.g., Monte Carlo simulations, polynomial chaos expansion is a nonsampling method which represents the uncertain quantities as an expansion including the decomposition of deterministic coefficients and random orthogonal bases. The generalized polynomial chaos expansion uses more orthogonal polynomials as the expansion bases in various random spaces which are not necessarily Gaussian. A general review of uncertainty quantification methods, the theory, the construction method, and various convergence criteria of the polynomial chaos expansion are presented. We apply it to identify the uncertain parameters with predefined probability density functions. The new concepts of optimal and nonoptimal expansions are defined and it demonstrated how we can develop these expansions for random variables belonging to the various random spaces. The calculation of the polynomial coefficients for uncertain parameters by using various procedures, e.g., Galerkin projection, collocation method, and moment method is presented. A comprehensive error and accuracy analysis of the polynomial chaos method is discussed for various random variables and random processes and results are compared with the exact solution or/and Monte Carlo simulations. The method is employed for the basic stochastic differential equation and, as practical application, to solve the stochastic modal analysis of the microsensor quartz fork. We emphasize the accuracy in results and time efficiency of this nonsampling procedure for uncertainty quantification of stochastic systems in comparison with sampling techniques, e.g., Monte Carlo simulation.
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Xiu, Dongbin, Didier Lucor, C. H. Su, and George Em Karniadakis. "Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos." Journal of Fluids Engineering 124, no. 1 (October 29, 2001): 51–59. http://dx.doi.org/10.1115/1.1436089.

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We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flow-structure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener (1938) and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to second-order oscillators to demonstrate convergence, and subsequently is coupled to incompressible Navier-Stokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortex-induced vibrations.
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Gao, Rugao, Keping Zhou, and Yun Lin. "A Flexible Polynomial Expansion Method for Response Analysis with Random Parameters." Complexity 2018 (December 3, 2018): 1–14. http://dx.doi.org/10.1155/2018/7471460.

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The generalized Polynomial Chaos Expansion Method (gPCEM), which is a random uncertainty analysis method by employing the orthogonal polynomial bases from the Askey scheme to represent the random space, has been widely used in engineering applications due to its good performance in both computational efficiency and accuracy. But in gPCEM, a nonlinear transformation of random variables should always be used to adapt the generalized Polynomial Chaos theory for the analysis of random problems with complicated probability distributions, which may introduce nonlinearity in the procedure of random uncertainty propagation as well as leading to approximation errors on the probability distribution function (PDF) of random variables. This paper aims to develop a flexible polynomial expansion method for response analysis of the finite element system with bounded random variables following arbitrary probability distributions. Based on the large family of Jacobi polynomials, an Improved Jacobi Chaos Expansion Method (IJCEM) is proposed. In IJCEM, the response of random system is approximated by the Jacobi expansion with the Jacobi polynomial basis whose weight function is the closest to the probability density distribution (PDF) of the random variable. Subsequently, the moments of the response can be efficiently calculated though the Jacobi expansion. As the IJCEM avoids the necessity that the PDF should be represented in terms of the weight function of polynomial basis by using the variant transformation, neither the nonlinearity nor the errors on random models will be introduced in IJCEM. Numerical examples on two random problems show that compared with gPCEM, the IJCEM can achieve better efficiency and accuracy for random problems with complex probability distributions.
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Franco-Medrano, Fermin, and Francisco J. Solis. "Stability of Real Parametric Polynomial Discrete Dynamical Systems." Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/680970.

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We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept ofcanonical polynomial mapswhich are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termedProduct Position Functionfor a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termedstability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.
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Abbasi, Mostafa, and Ali Gholami. "Polynomial chaos expansion for nonlinear geophysical inverse problems." GEOPHYSICS 82, no. 4 (July 1, 2017): R259—R268. http://dx.doi.org/10.1190/geo2016-0716.1.

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There are lots of geophysical problems that include computationally expensive functions (forward models). Polynomial chaos (PC) expansion aims to approximate such an expensive equation or system with a polynomial expansion on the basis of orthogonal polynomials. Evaluation of this expansion is extremely fast because it is a polynomial function. This property of the PC expansion is of great importance for stochastic problems, in which an expensive function needs to be evaluated thousands of times. We have developed PC expansion as a novel technique to solve nonlinear geophysical problems. To better evaluate the methodology, we use PC expansion for automating the velocity analysis. For this purpose, we define the optimally picked velocity model as an optimizer of a variational integral in a semblance field. However, because computation of a variational integral with respect to a given velocity model is rather expensive, it is impossible to use stochastic methods to search for the optimal velocity model. Thus, we replace the variational integral with its PC expansion, in which computation of the new function is extremely faster than the original one. This makes it possible to perturb thousands of velocity models in a matter of seconds. We use particle swarm optimization as the stochastic optimization method to find the optimum velocity model. The methodology is tested on synthetic and field data, and in both cases, reasonable results are achieved in a rather short time.
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Li, Ning, Bo Meng, Xinlong Feng, and Dongwei Gui. "A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos." East Asian Journal on Applied Mathematics 5, no. 2 (May 2015): 192–208. http://dx.doi.org/10.4208/eajam.250714.020515a.

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AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.
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Dissertations / Theses on the topic "Polynomial chao"

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Xiaochen, Liu. "Statistical Analysis of Integrated Circuits Using Decoupled Polynomial Chaos." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34836.

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One of the major tasks in electronic circuit design is the ability to predict the performance of general circuits in the presence of uncertainty in key design parameters. In the mathematical literature, such a task is referred to as uncertainty quantification. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. To achieve the uncertainty quantification, this thesis presents a new approach based on the concept of generalized Polynomial Chaos (gPC) to perform variability analysis of general nonlinear circuits. The proposed approach is built upon a decoupling formulation of the Galerkin projection (GP) technique, where the large matrix is transformed into a block-diagonal whose diagonal blocks can be factorized independently. The proposed methodology provides a general framework for decoupling the GP formulation based on a general system of orthogonal polynomials. Moreover, it provides a new insight into the error level that is caused by the decoupling procedure, enabling an assessment of the performance of a wide variety of orthogonal polynomials. For example, it is shown that, for the same order, the Chebyshev polynomials outperforms other commonly used gPC polynomials.
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Yorke, Rory. "Chaos control using local polynomial approximation." Master's thesis, University of Cape Town, 2001. http://hdl.handle.net/11427/5075.

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Chaotic systems may be defined as those whose behaviour is sensitively dependent on initial conditions. Such systems may be made periodic using small input perturbations, as proposed in [OGY90]; this is called Ott-Grebogi-Yorke (OGY) chaos control. The original method used a linear model for controller design; a later development of chaos control was [CCdF99], in which a polynomial model is used. This dissertation proposes using local Taylor polynomial models as a basis for chaos control.
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Templeton, Brian Andrew. "A Polynomial Chaos Approach to Control Design." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/28840.

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A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired. From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented. The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method.
Ph. D.
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Whittle, Lisa. "Stochastic Optimal Trajectory Generation via Multivariate Polynomial Chaos." Thesis, Luleå tekniska universitet, Institutionen för system- och rymdteknik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-65746.

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This thesis presents a framework that has been developed in order to compute stochastic optimal trajectories. This is achieved by transforming the initial set of stochastic ordinary differential equations into their deterministic equivalent by application of Multivariate Polynomial Chaos. Via Galerkin projection, it is possible to include stochastic information in the optimal-trajectory generation process, and to solve the corresponding optimal-control problem using pseudospectral methods. The resultant trajectory is therefore less sensitive to the uncertainties included in the analysis, e.g., those present in system parameters, initial conditions or path constraints. The accurate, yet computationally efficient manner in which solutions are obtained is presented and a comparison with deterministic results show the benefits of the proposed approach for a variety of numerical examples.
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Szepietowska, Katarzyna. "POLYNOMIAL CHAOS EXPANSION IN BIO- AND STRUCTURAL MECHANICS." Thesis, Bourges, INSA Centre Val de Loire, 2018. http://www.theses.fr/2018ISAB0004/document.

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Cette thèse présente une approche probabiliste de la modélisation de la mécanique des matériaux et des structures. Le dimensionnement est influencé par l'incertitude des paramètres d'entrée. Le travail est interdisciplinaire et les méthodes décrites sont appliquées à des exemples de biomécanique et de génie civil. La motivation de ce travail était le besoin d'approches basées sur la mécanique dans la modélisation et la simulation des implants utilisés dans la réparation des hernies ventrales. De nombreuses incertitudes apparaissent dans la modélisation du système implant-paroi abdominale. L'approche probabiliste proposée dans cette thèse permet de propager ces incertitudes et d’étudier leurs influences respectives. La méthode du chaos polynomial basée sur la régression est utilisée dans ce travail. L'exactitude de ce type de méthodes non intrusives dépend du nombre et de l'emplacement des points de calcul choisis. Trouver une méthode universelle pour atteindre un bon équilibre entre l'exactitude et le coût de calcul est encore une question ouverte. Différentes approches sont étudiées dans cette thèse afin de choisir une méthode efficace et adaptée au cas d’étude. L'analyse de sensibilité globale est utilisée pour étudier les influences des incertitudes d'entrée sur les variations des sorties de différents modèles. Les incertitudes sont propagées aux modèles implant-paroi abdominale. Elle permet de tirer des conclusions importantes pour les pratiques chirurgicales. À l'aide de l'expertise acquise à partir de ces modèles biomécaniques, la méthodologie développée est utilisée pour la modélisation de joints de bois historiques et la simulation de leur comportement mécanique. Ce type d’étude facilite en effet la planification efficace des réparations et de la rénovation des bâtiments ayant une valeur historique
This thesis presents a probabilistic approach to modelling the mechanics of materials and structures where the modelled performance is influenced by uncertainty in the input parameters. The work is interdisciplinary and the methods described are applied to medical and civil engineering problems. The motivation for this work was the necessity of mechanics-based approaches in the modelling and simulation of implants used in the repair of ventral hernias. Many uncertainties appear in the modelling of the implant-abdominal wall system. The probabilistic approach proposed in this thesis enables these uncertainties to be propagated to the output of the model and the investigation of their respective influences. The regression-based polynomial chaos expansion method is used here. However, the accuracy of such non-intrusive methods depends on the number and location of sampling points. Finding a universal method to achieve a good balance between accuracy and computational cost is still an open question so different approaches are investigated in this thesis in order to choose an efficient method. Global sensitivity analysis is used to investigate the respective influences of input uncertainties on the variation of the outputs of different models. The uncertainties are propagated to the implant-abdominal wall models in order to draw some conclusions important for further research. Using the expertise acquired from biomechanical models, modelling of historic timber joints and simulations of their mechanical behaviour is undertaken. Such an investigation is important owing to the need for efficient planning of repairs and renovation of buildings of historical value
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Nydestedt, Robin. "Application of Polynomial Chaos Expansion for Climate Economy Assessment." Thesis, KTH, Optimeringslära och systemteori, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-223985.

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In climate economics integrated assessment models (IAMs) are used to predict economic impacts resulting from climate change. These IAMs attempt to model complex interactions between human and geophysical systems to provide quantifications of economic impact, typically using the Social Cost of Carbon (SCC) which represents the economic cost of a one ton increase in carbon dioxide. Another difficulty that arises in modeling a climate economics system is that both the geophysical and economic submodules are inherently stochastic. Even in frequently cited IAMs, such as DICE and PAGE, there exists a lot of variation in the predictions of the SCC. These differences stem both from the models of the climate and economic modules used, as well as from the choice of probability distributions used for the random variables. Seeing as IAMs often take the form of optimization problems these nondeterministic elements potentially result in heavy computational costs. In this thesis a new IAM, FAIR/DICE, is introduced. FAIR/DICE is a discrete time hybrid of DICE and FAIR providing a potential improvement to DICE as the climate and carbon modules in FAIR take into account feedback coming from the climate module to the carbon module. Additionally uncertainty propagation in FAIR/DICE is analyzed using Polynomial Chaos Expansions (PCEs) which is an alternative to Monte Carlo sampling where the stochastic variables are projected onto stochastic polynomial spaces. PCEs provide better computational efficiency compared to Monte Carlo sampling at the expense of storage requirements as a lot of computations can be stored from the first simulation of the system, and conveniently statistics can be computed from the PCE coefficients without the need for sampling. A PCE overloading of FAIR/DICE is investigated where the equilibrium climate sensitivity, modeled as a four parameter Beta distribution, introduces an uncertainty to the dynamical system. Finally, results in the mean and variance obtained from the PCEs are compared to a Monte Carlo reference and avenues into future work are suggested.
Inom klimatekonomi används integrated assessment models (IAMs) för att förutspå hur klimatförändringar påverkar ekonomin. Dessa IAMs modellerar komplexa interaktioner mellan geofysiska och mänskliga system för att kunna kvantifiera till exempel kostnaden för den ökade koldioxidhalten på planeten, i.e. Social Cost of Carbon (SCC). Detta representerar den ekonomiska kostnaden som motsvaras av utsläppet av ett ton koldioxid. Faktumet att både de geofysiska och ekonomiska submodulerna är stokastiska gör att SCC-uppskattningar varierar mycket även inom väletablerade IAMs som PAGE och DICE. Variationen grundar sig i skillnader inom modellerna men också från att val av sannolikhetsfördelningar för de stokastiska variablerna skiljer sig. Eftersom IAMs ofta är formulerade som optimeringsproblem leder dessutom osäkerheterna till höga beräkningskostnader. I denna uppsats introduceras en ny IAM, FAIR/DICE, som är en diskret tids hybrid av DICE och FAIR. Den utgör en potentiell förbättring av DICE eftersom klimat- och kolmodulerna i FAIR även behandlar återkoppling från klimatmodulen till kolmodulen. FAIR/DICE är analyserad med hjälp av Polynomial Chaos Expansions (PCEs), ett alternativ till Monte Carlo-metoder. Med hjälp av PCEs kan de osäkerheter projiceras på stokastiska polynomrum vilket har fördelen att beräkningskostnader reduceras men nackdelen att lagringskraven ökar. Detta eftersom många av beräkningarna kan sparas från första simuleringen av systemet, dessutom kan statistik extraheras direkt från PCE koefficienterna utan behov av sampling. FAIR/DICE systemet projiceras med hjälp av PCEs där en osäkerhet är introducerad via equilibrium climate sensitivity (ECS), vilket i sig är ett värde på hur känsligt klimatet är för koldioxidförändringar. ECS modelleras med hjälp av en fyra-parameters Beta sannolikhetsfördelning. Avslutningsvis jämförs resultat i medelvärde och varians mellan PCE implementationen av FAIR/DICE och en Monte Carlo-baserad referens, därefter ges förslag på framtida utvecklingsområden.
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Perez, Rafael A. "Uncertainty Analysis of Computational Fluid Dynamics Via Polynomial Chaos." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/28984.

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The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs.
Ph. D.
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8

Ishak, Hassoun. "Étude stochastique de l'impact des défauts de porosités et de plissements dans les matériaux composites." Thesis, Nantes, 2017. http://www.theses.fr/2017NANT4090/document.

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Les matériaux composites à matrice organique sont de plus en plus utilisés dans divers domaines tels que l'aérospatiale ou les énergies marines renouvelables en raison de leurs excellentes propriétés spécifiques. Cependant, les procédés de fabrication des structures composites sont complexes et peuvent conduire à l'apparition de défauts, en particulier de plissement des plis et de porosité, qui affectent les propriétés mécaniques de la structure. Les pièces composites sont ainsi systématiquement soumises à des contrôles CND long et coûteux. En cas de résultats négatifs par rapport à des critères conservatifs, celles-ci peuvent être rejetées, avec des conséquences économiques non négligeables. L'objectif de cette étude est de quantifier l'impact des défauts observés et des incertitudes associées sur le comportement de pièce composite. Dans ce travail, nous adoptons une vision paramétrique des incertitudes consistant à représenter le contenu probabiliste à travers d’un ensemble fini de variables aléatoires. Nous nous concentrons sur la propagation des incertitudes basée sur des méthodes stochastiques spectrales. L'étude portant sur le défaut de porosités se fait à l’échelle microscopique puis macroscopique. Les paramètres aléatoires d'entrée sont liés à la géométrie des porosités et à leur taux. L'étude du défaut plissements à l'échelle mésoscopique est basée sur une représentation paramétrique de la géométrie du plissement. Les paramètres aléatoires d'entrée représentent alors la forme et la taille de ces défauts. Il est donc possible d'analyser l'impact de ces défauts à l'échelle structurelle par des grandeurs mécaniques classiques et des critères de rupture
Composite materials are increasingly used in various fields such as aerospace or renewable marine energies due to their excellent specific properties. However, the manufacturing processes of the composite structures are complex, which can lead to the appearance of defects, particularly wrinkles and porosities, which affect the mechanical properties of the structure. Based on conservative criteria, a system of non-destructive testing of composite parts thus makes it possible to judge their conformity. In case of non-conformity, those components are rejected, with non-negligible economic consequences. The objective of this study is to quantify the impact of the defects and associated uncertainties on the behavior of composite parts. In this work, we adopt a parametric vision of the uncertainties consisting in representing the probabilistic content through a finite set of random variables. We focus on the propagation of uncertainties based on spectral stochastic methods. The study involving porosity is done at the micro-scale and then at the macro-scale. The random input parameters are related to the geometry of the porosities and their rates. The study of the wrinkle defect, done at the mesoscopic scale, is based on a parametric representation of the geometry of the wrinkle. The random input parameters then represent the shape and size of this defect. It is therefore possible to analyze the impact of these two manufacturing defects at a structural scale through classical mechanical quantities and check the failure of the structure with failure criteria
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Koehring, Andrew. "The application of polynomial response surface and polynomial chaos expansion metamodels within an augmented reality conceptual design environment." [Ames, Iowa : Iowa State University], 2008.

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Fisher, James Robert. "Stability analysis and control of stochastic dynamic systems using polynomial chaos." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-2853.

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Books on the topic "Polynomial chao"

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Pettersson, Mass Per, Gianluca Iaccarino, and Jan Nordström. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-10714-1.

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Ozen, Hasan Cagan. Long Time Propagation of Stochasticity by Dynamical Polynomial Chaos Expansions. [New York, N.Y.?]: [publisher not identified], 2017.

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S, Taqqu Murad, ed. Wiener chaos: Moments, cumulants and diagrams : a survey with computer implementation. Milan: Springer, 2011.

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Wiener Chaos : Moments, Cumulants and Diagrams: A Survey with Computer Implementation. Springer Milan, 2011.

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Nordström, Jan, Mass Per Pettersson, and Gianluca Iaccarino. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer, 2015.

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Nordström, Jan, Mass Per Pettersson, and Gianluca Iaccarino. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer, 2016.

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Nordström, Jan, Mass Per Pettersson, and Gianluca Iaccarino. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer, 2015.

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Peccati, Giovanni, and Murad S. Taqqu. Wiener Chaos : Moments, Cumulants and Diagrams: A survey with Computer Implementation. Springer, 2014.

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Book chapters on the topic "Polynomial chao"

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Pettersson, Mass Per, Gianluca Iaccarino, and Jan Nordström. "Polynomial Chaos Methods." In Polynomial Chaos Methods for Hyperbolic Partial Differential Equations, 23–29. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-10714-1_3.

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Hesthaven, Jan S., and Dongbin Xiu. "Polynomial Chaos Expansions." In Encyclopedia of Applied and Computational Mathematics, 1162–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_331.

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Chu, Liu. "Polynomial Chaos Expansion." In Uncertainty Quantification of Stochastic Defects in Materials, 37–49. Boca Raton: CRC Press, 2021. http://dx.doi.org/10.1201/9781003226628-5.

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Lototsky, Sergey V., and Boris L. Rozovsky. "The Polynomial Chaos Method." In Universitext, 233–380. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58647-2_5.

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Dmitrishin, Dmitriy, Anna Khamitova, and Alexander M. Stokolos. "Fejér Polynomials and Chaos." In Springer Proceedings in Mathematics & Statistics, 49–75. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-10545-1_7.

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Ghanem, Roger, and John Red-Horse. "Polynomial Chaos: Modeling, Estimation, and Approximation." In Handbook of Uncertainty Quantification, 521–51. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-12385-1_13.

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Ghanem, Roger, and John Red-Horse. "Polynomial Chaos: Modeling, Estimation, and Approximation." In Handbook of Uncertainty Quantification, 1–31. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-11259-6_13-1.

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Xiu, Dongbin, Didier Lucor, C. H. Su, and George Em Karniadakis. "Performance Evaluation of Generalized Polynomial Chaos." In Lecture Notes in Computer Science, 346–54. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44864-0_36.

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Russo, Francesco, and Pierre Vallois. "Hermite Polynomials and Wiener Chaos Expansion." In Stochastic Calculus via Regularizations, 309–32. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-09446-0_9.

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Peccati, Giovanni, and Murad S. Taqqu. "Some facts about Charlier polynomials." In Wiener Chaos: Moments, Cumulants and Diagrams, 171–75. Milano: Springer Milan, 2011. http://dx.doi.org/10.1007/978-88-470-1679-8_10.

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Conference papers on the topic "Polynomial chao"

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Gerritsma, Marc, Peter Vos, and Jan‐Bart van der Steen. "Time‐Dependent Polynomial Chaos." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS: International Conference on Numerical Analysis and Applied Mathematics 2008. American Institute of Physics, 2008. http://dx.doi.org/10.1063/1.2990897.

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Xu, Can, Zhao Liu, Wei Tao, and Ping Zhu. "A Novel Hierarchical Framework for Uncertainty Analysis of Multiscale Systems Combined Vine Copula With Sparse PCE." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-97832.

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Abstract Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt transformation is adopted to generalize independent standard variables. Then the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model, can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modelled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.
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Mai, Chu V., and Bruno Sudret. "HIERARCHICAL ADAPTIVE POLYNOMIAL CHAOS EXPANSIONS." In 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering. Athens: Institute of Structural Analysis and Antiseismic Research School of Civil Engineering National Technical University of Athens (NTUA) Greece, 2015. http://dx.doi.org/10.7712/120215.4253.517.

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Bakhtiari-Nejad, Firooz, Naserodin Sepehry, and Mahnaz Shamshirsaz. "Polynomial Chaos Expansion Sensitivity Analysis for Electromechanical Impedance of Plate." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59129.

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Piezoelectric wafer active sensors (PWAS) have been the widely used in impedance based damage detection applications. A most important matter in impedance method is applied voltage to PWAS and measuring current in PWAS. In this paper, for modeling of impedance based structural health monitoring, a 3D spectral finite element method (SFEM) is developed for plate structure with PWAS. Because of high frequency application of impedance method, high degree of freedom (DOF) is needed for modeling of impedance of PWAS attached on the plate. Uncertainty of plate and PWAS parameters could be effect on the natural frequencies of structure. So, impedance signal of modeling would be different based on uncertainty parameters. Polynomial chaos expansion (PC) is a probabilistic method consisting in the projection of the model output on a basis of orthogonal stochastic polynomials in the random inputs. In this paper, PCE is used for sensitivity analysis of the electromechanical impedance of plate structure with PWAS.
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Tagade, Piyush M., and Han-Lim Choi. "A Polynomial Chaos Based Bayesian Inference Method With Uncertain Hyper-Parameters." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47632.

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This paper proposes stochastic spectral representation for Bayesian calibration of computer simulators with parametric and model structure uncertainty with unknown/poorly known prior hyper-parameters. The methodology is specifically developed for calibration of simulators with spatially/temporally varying parameters. Uncertainty in parameters and model structure is represented using independent stationary Gaussian processes with uncertain hyper-parameters. Gaussian processes are spectrally represented using Karhunnen-Loeve expansion. A methodology based on decomposition of parametric space and orthogonal polynomials defined on the decomposed space is developed for evaluating coefficients of Karhunnen-Loeve expansion of Gaussian process with uncertain hyper-parameters. Galerkin projection method is used to evaluate the resultant stochastic spectral decomposition of predicted system response. Bayesian inference is used to update the prior probability distribution of the polynomial chaos basis. The proposed method is demonstrated for calibration of a simulator of quasi-one dimensional flow through a convergent-divergent nozzle.
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Thapa, Mishal, Sameer B. Mulani, and Robert W. Walters. "Polynomial Chaos Decomposition with Differentiation Operation." In 17th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2016. http://dx.doi.org/10.2514/6.2016-4288.

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Smith, A., A. Monti, and F. Ponci. "Robust Controller Using Polynomial Chaos Theory." In Conference Record of the 2006 IEEE Industry Applications Conference Forty-First IAS Annual Meeting. IEEE, 2006. http://dx.doi.org/10.1109/ias.2006.256892.

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Cheng, Haiyan, and Adrian Sandu. "Collocation least-squares polynomial chaos method." In the 2010 Spring Simulation Multiconference. New York, New York, USA: ACM Press, 2010. http://dx.doi.org/10.1145/1878537.1878621.

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Mulani, Sameer, Rakesh Kapania, and Robert Walters. "Stochastic Eigenvalue Problem with Polynomial Chaos." In 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
14th AIAA/ASME/AHS Adaptive Structures Conference
7th. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2006. http://dx.doi.org/10.2514/6.2006-2068.

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Walters, Robert. "Stochastic Fluid Mechanics via Polynomial Chaos." In 41st Aerospace Sciences Meeting and Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2003. http://dx.doi.org/10.2514/6.2003-413.

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Reports on the topic "Polynomial chao"

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Nance, Douglas V. Stochastic Estimation via Polynomial Chaos. Fort Belvoir, VA: Defense Technical Information Center, October 2015. http://dx.doi.org/10.21236/ada627811.

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Jakeman, John, Fabian Franzelin, Akil Narayan, Michael Eldred, and Dirk Pflueger. Polynomial chaos expansions for dependent random variables. Office of Scientific and Technical Information (OSTI), May 2019. http://dx.doi.org/10.2172/1762354.

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Jardak, M., C. Su, and G. E. Karniadakis. Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation. Fort Belvoir, VA: Defense Technical Information Center, October 2001. http://dx.doi.org/10.21236/ada460601.

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Xiu, Dongbin, and George E. Karniadakis. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, January 2003. http://dx.doi.org/10.21236/ada460654.

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Xiu, Dong, and George E. Karniadakis. Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos. Fort Belvoir, VA: Defense Technical Information Center, October 2002. http://dx.doi.org/10.21236/ada461813.

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Field, Jr, Richard V., .), and Mircea Grigoriu. Convergence properties of polynomial chaos approximations for L2 random variables. Office of Scientific and Technical Information (OSTI), March 2007. http://dx.doi.org/10.2172/903430.

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Xiu, Dongbin, Didier Lucor, C. Su, and George E. Karniadakis. Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos. Fort Belvoir, VA: Defense Technical Information Center, September 2001. http://dx.doi.org/10.21236/ada461832.

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Xiu, Dongbin, and George E. Karniadakis. Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos. Fort Belvoir, VA: Defense Technical Information Center, July 2002. http://dx.doi.org/10.21236/ada460658.

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