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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

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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Анотація:

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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2

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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Анотація:

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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3

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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Анотація:

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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4

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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Анотація:

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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5

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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Анотація:

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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6

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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Анотація:

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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7

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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Анотація:

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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8

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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Анотація:

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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9

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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Анотація:

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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10

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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11

Wei, Xiao, Haichao Chang, Baiwei Feng, and Zuyuan Liu. "Sensitivity Analysis Based on Polynomial Chaos Expansions and Its Application in Ship Uncertainty-Based Design Optimization." Mathematical Problems in Engineering 2019 (January 23, 2019): 1–19. http://dx.doi.org/10.1155/2019/7498526.

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Анотація:

In order to truly reflect the ship performance under the influence of uncertainties, uncertainty-based design optimization (UDO) for ships that fully considers various uncertainties in the early stage of design has gradually received more and more attention. Meanwhile, it also brings high dimensionality problems, which may result in inefficient and impractical optimization. Sensitivity analysis (SA) is a feasible way to alleviate this problem, which can qualitatively or quantitatively evaluate the influence of the model input uncertainty on the model output, so that uninfluential uncertain variables can be determined for the descending dimension to achieve dimension reduction. In this paper, polynomial chaos expansions (PCE) with less computational cost are chosen to directly obtain Sobol' global sensitivity indices by its polynomial coefficients; that is, once the polynomial of the output variable is established, the analysis of the sensitivity index is only the postprocessing of polynomial coefficients. Besides, in order to further reduce the computational cost, for solving the polynomial coefficients of PCE, according to the properties of orthogonal polynomials, an improved probabilistic collocation method (IPCM) based on the linear independence principle is proposed to reduce sample points. Finally, the proposed method is applied to UDO of a bulk carrier preliminary design to ensure the robustness and reliability of the ship.

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12

Guo, Ling, Yongle Liu та Liang Yan. "Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions". Numerical Mathematics: Theory, Methods and Applications 10, № 4 (12 вересня 2017): 775–97. http://dx.doi.org/10.4208/nmtma.2017.0001.

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AbstractIn this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

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13

Ghanem, Roger, and P. D. Spanos. "Polynomial Chaos in Stochastic Finite Elements." Journal of Applied Mechanics 57, no. 1 (March 1, 1990): 197–202. http://dx.doi.org/10.1115/1.2888303.

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A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials. The proposed method involves a mathematical formulation which is a natural extension of the deterministic finite element concept to the space of random functions. A beam problem and a plate problem are investigated using the new method. The corresponding results are found in good agreement with those obtained through a Monte-Carlo simulation solution of the problems.

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14

Ben Said, Mohamed, Lahcen Azrar, and Driss Sarsri. "Numerical Procedures for Random Differential Equations." Journal of Applied Mathematics 2018 (2018): 1–23. http://dx.doi.org/10.1155/2018/7403745.

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Анотація:

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

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15

Chi, Yaodan, Bin Li, Xiaotian Yang, Tianhao Wang, Kaiyu Yang, and Yinhan Gao. "Research on the Statistical Characteristics of Crosstalk in Naval Ships Wiring Harness Based on Polynomial Chaos Expansion Method." Polish Maritime Research 24, s2 (August 28, 2017): 205–14. http://dx.doi.org/10.1515/pomr-2017-0084.

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Abstract Crosstalk in wiring harness has been studied extensively for its importance in the naval ships electromagnetic compatibility field. An effective and high-efficiency method is proposed in this paper for analyzing Statistical Characteristics of crosstalk in wiring harness with random variation of position based on Polynomial Chaos Expansion (PCE). A typical 14-cable wiring harness was simulated as the object of research. Distance among interfering cable, affected cable and GND is synthesized and analyzed in both frequency domain and time domain. The model of naval ships wiring harness distribution parameter was established by utilizing Legendre orthogonal polynomials as basis functions along with prediction model of statistical characters. Detailed mean value, mean square error, probability density function and reasonable varying range of crosstalk in naval ships wiring harness are described in both time domain and frequency domain. Numerical experiment proves that the method proposed in this paper, not only has good consistency with the MC method can be applied in the naval ships EMC research field to provide theoretical support for guaranteeing safety, but also has better time-efficiency than the MC method. Therefore, the Polynomial Chaos Expansion method.

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16

Yang, Yongfeng, Tingdong Jiang, Zhong Ren, Junyao Zhao, and Zheng Zhang. "Rationalize the irrational and fractional expressions in nonlinear analysis." Modern Physics Letters B 30, no. 04 (February 10, 2016): 1650068. http://dx.doi.org/10.1142/s0217984916500688.

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Chebyshev polynomial approximation is an effective method to study the stochastic bifurcation and chaos. However, due to irrational and fractional expressions existing in the denominator of some mechanical systems, the integral process is very complicated. The Taylor series expansion is proposed to expand the irrational and fractional expressions into a series of polynomials. Smooth and discontinuous oscillator was taken as an example, and the results show that the Taylor series expansion method is acceptable. The rub-impact force was taken as another example. Numerical results indicate that the method is suitable for the rub-impact rotor system.

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17

Schobi, Roland, Bruno Sudret, and Joe Wiart. "POLYNOMIAL-CHAOS-BASED KRIGING." International Journal for Uncertainty Quantification 5, no. 2 (2015): 171–93. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.2015012467.

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18

SEPAHVAND, K., S. MARBURG, and H. J. HARDTKE. "STOCHASTIC STRUCTURAL MODAL ANALYSIS INVOLVING UNCERTAIN PARAMETERS USING GENERALIZED POLYNOMIAL CHAOS EXPANSION." International Journal of Applied Mechanics 03, no. 03 (September 2011): 587–606. http://dx.doi.org/10.1142/s1758825111001147.

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Анотація:

In this paper, the application of generalized polynomial chaos expansion in stochastic structural modal analysis including uncertain parameters is investigated. We review the theory of polynomial chaos and relating error analysis. A general formulation for the representation of modal problems by the polynomial chaos expansion is derived. It shows how the modal frequencies and modal shapes are influenced by the parameter uncertainties. The key issues that arise in the polynomial chaos simulation of modal analysis are discussed for two examples: a discrete 2-DOF system and continuous model of a microsensor. In both cases, the polynomial chaos expansion is used for the approximation of uncertain parameters, eigenfrequencies and eigenvectors. We emphasize the accuracy and time efficiency of the method in estimation of the stochastic modal responses in comparison with the sampling techniques, such as the Monte Carlo simulation.

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19

WAUBKE, HOLGER. "BOUNDARY ELEMENT METHOD FOR ISOTROPIC MEDIA WITH RANDOM SHEAR MODULI." Journal of Computational Acoustics 13, no. 01 (March 2005): 229–58. http://dx.doi.org/10.1142/s0218396x05002530.

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Анотація:

Green's functions for elastic solids with random properties are usually derived by means of the perturbation method. This paper deals with a new approach that has the potential to deal with a large variability of random shear modulus based on the transformation of a polynomial chaos. The deterministic Green's functions for stresses and displacements and the principal values in the boundary integrals caused by a pressure load on the surface are nonlinear transformations of the random variables. A series of transformations of the polynomial chaos is used to transform significant parts of the equation. The first operation is a projection of the log normal distributed shear modulus to a series of Hermite's polynomials based on a Gaussian variable. The second operation is the determination of an arbitrary potential of the wave velocity. The last operation, similar to the first one, consists in the determination of an exponential function depending on the inverse of the wave velocity. These operations, together with multiplications and summations, transform the complete relation from the random shear modulus to Green's functions and principal values. The inversion of the system matrix is already derived for the random finite element approach. The operations are independent of the specific problem and can be applied to almost all acoustic media and similar nonlinear problems.

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20

HOANG, VIET HA, and CHRISTOPH SCHWAB. "N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS." Mathematical Models and Methods in Applied Sciences 24, no. 04 (January 28, 2014): 797–826. http://dx.doi.org/10.1142/s0218202513500681.

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We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner–Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on ℝℕ. It is shown that the weak solution can be represented as Wiener–Itô Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in ℝ1. We establish sufficient conditions on the random inputs for weighted sequence norms of the Wiener–Itô decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions' Wiener–Itô decomposition are shown to be p-summable for the same value of 0 < p < 1. We prove rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as of Finite Element discretizations of these approximations from a dense, nested family V0 ⊂ V1 ⊂ V2 ⊂ ⋯ V of finite element spaces of continuous, piecewise linear Finite Elements.

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21

Yin, Shengwen, Yuan Gao, Xiaohan Zhu, and Zhonggang Wang. "Anisotropy-Based Adaptive Polynomial Chaos Method for Hybrid Uncertainty Quantification and Reliability-Based Design Optimization of Structural-Acoustic System." Mathematics 11, no. 4 (February 7, 2023): 836. http://dx.doi.org/10.3390/math11040836.

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Анотація:

The evaluation of objective functions and component reliability in the optimisation of structural-acoustic systems with random and interval variables is computationally expensive, especially when strong nonlinearity exhibits between the response and input variables. To reduce the computational cost and improve the computational efficiency, a novel anisotropy-based adaptive polynomial chaos (ABAPC) expansion method was developed in this study. In ABAPC, the anisotropy-based polynomial chaos expansion, namely the retained order of polynomial chaos expansion (PCE) differs from each variable, is used to construct the initial surrogate model instead of first-order polynomial chaos expansion in conventional methods. Then, an anisotropy-based adaptive basis growth strategy was developed to reduce the estimation of the coefficients of the polynomial chaos expansion method and increase its computational efficiency. Finally, to solve problems with probabilistic and interval parameters, an adaptive basis truncation strategy was introduced and implemented. Using the ABAPC method, the computational cost of reliability-based design optimisation for structural-acoustic systems can be efficiently reduced. The effectiveness of the proposed method were demonstrated by solving two numerical examples and optimisation problems of a structural-acoustic system.

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22

SEPAHVAND, K., S. MARBURG, and H. J. HARDTKE. "NUMERICAL SOLUTION OF ONE-DIMENSIONAL WAVE EQUATION WITH STOCHASTIC PARAMETERS USING GENERALIZED POLYNOMIAL CHAOS EXPANSION." Journal of Computational Acoustics 15, no. 04 (December 2007): 579–93. http://dx.doi.org/10.1142/s0218396x07003524.

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Анотація:

This paper presents a numerical algorithm which is using generalized polynomial chaos combined with the finite difference method for the solution of the one-dimensional wave equation with stochastic physical parameters. The stochastic parameters are represented by the Hermite polynomial chaos. A spectral–finite difference model for the numerical solution is introduced using generalized polynomial chaos expansion. The general conditions for convergence and stability of numerical algorithms are derived. Finally, the method is applied to a vibrating string. Results are compared with those of a Monte Carlo simulation.

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23

Chang, K., and A. R. Ko. "TURBULENT FLOW SIMULATION AROUND FLAT PLATE BASED ON POINT-COLLOCATION NON-INTRUSIVE POLYNOMIAL CHAOS METHOD." Journal of Computational Fluids Engineering 23, no. 2 (June 30, 2018): 94–100. http://dx.doi.org/10.6112/kscfe.2018.23.2.094.

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24

Kwapien, Stanislaw. "Decoupling Inequalities for Polynomial Chaos." Annals of Probability 15, no. 3 (July 1987): 1062–71. http://dx.doi.org/10.1214/aop/1176992081.

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25

Gerritsma, Marc, Jan-Bart van der Steen, Peter Vos, and George Karniadakis. "Time-dependent generalized polynomial chaos." Journal of Computational Physics 229, no. 22 (November 2010): 8333–63. http://dx.doi.org/10.1016/j.jcp.2010.07.020.

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26

Zhao, Wei, and Ji Ke Liu. "Stochastic Finite Element Method Using Polynomial Chaos Expansion." Advanced Materials Research 199-200 (February 2011): 500–504. http://dx.doi.org/10.4028/www.scientific.net/amr.199-200.500.

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Анотація:

We present a new response surface based stochastic finite element method to obtain solutions for general random uncertainty problems using the polynomial chaos expansion. The approach is general but here a typical elastostatics example only with the random field of Young's modulus is presented to illustrate the stress analysis, and computational comparison with the traditional polynomial expansion approach is also performed. It shows that the results of the polynomial chaos expansion are improved compared with that of the second polynomial expansion method.

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27

Yin, Shengwen, Xiaohan Zhu, and Xiang Liu. "A Novel Sparse Polynomial Expansion Method for Interval and Random Response Analysis of Uncertain Vibro-Acoustic System." Shock and Vibration 2021 (September 23, 2021): 1–15. http://dx.doi.org/10.1155/2021/1125373.

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For the vibro-acoustic system with interval and random uncertainties, polynomial chaos expansions have received broad and persistent attention. Nevertheless, the cost of the computation process increases sharply with the increasing number of uncertain parameters. This study presents a novel interval and random polynomial expansion method, called Sparse Grids’ Sequential Sampling-based Interval and Random Arbitrary Polynomial Chaos (SGS-IRAPC) method, to obtain the response of a vibro-acoustic system with interval and random uncertainties. The proposed SGS-IRAPC retains the accuracy and the simplicity of the traditional arbitrary polynomial chaos method, while avoiding its inefficiency. In the SGS-IRAPC, the response is approximated by the moment-based arbitrary polynomial chaos expansion and the expansion coefficient is determined by the least squares approximation method. A new sparse sampling scheme combined the sparse grids’ scheme with the sequential sampling scheme which is employed to generate the sampling points used to calculate the expansion coefficient to decrease the computational cost. The efficiency of the proposed surrogate method is demonstrated using a typical mathematical problem and an engineering application.

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28

Di Persio, Luca, Gregorio Pellegrini, and Michele Bonollo. "Polynomial Chaos Expansion Approach to Interest Rate Models." Journal of Probability and Statistics 2015 (2015): 1–24. http://dx.doi.org/10.1155/2015/369053.

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The Polynomial Chaos Expansion (PCE) technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantityξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.

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29

Liu, Qian, Xufang Zhang, and Xianzhen Huang. "A sparse surrogate model for structural reliability analysis based on the generalized polynomial chaos expansion." Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 233, no. 3 (October 8, 2018): 487–502. http://dx.doi.org/10.1177/1748006x18804047.

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The reliability analysis of a structural system is typically evaluated based on a multivariate model that describes the underlying mechanistic relationship between the system’s input and output random variables. This is the need to develop an effective surrogate model to mimic the input–output relationship as the Monte Carlo simulation–based on the mechanistic model might be computationally intensive. In this regard, the article presents a sparse regression method for structural reliability analysis based on the generalized polynomial chaos expansion. However, results from the global sensitivity analysis have justified that it is unnecessary to contain all polynomial terms in the surrogate model, instead of comprising a rather small number of principle components only. One direct benefit of the sparse approximation allows utilizing a small number of training samples to calibrate the surrogate model, bearing in mind that the required sample size is positively proportional to the number of unknowns in regression analysis. Therefore, by utilizing the standard polynomial chaos basis functions to constitute an explanatory dictionary, an adaptive sparse regression approach characterized by introducing the most significant explanatory variable in each iteration is presented. A statistical approach for detecting and excluding spuriously explanatory polynomials is also introduced to maintain the high sparsity of the meta-modeling result. Combined with a variety of low-discrepancy schemes in generating training samples, structural reliability and global sensitivity analysis of originally true but computationally demanding models are alternatively realized based on the sparse surrogate method in conjunction with the brutal Monte Carlo simulation method. Numerical examples are carried out to demonstrate the applicability of the sparse regression approach to structural reliability problems. Results have shown that the proposed method is an effective, non-intrusive approach to deal with uncertainty analysis of various structural systems.

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30

Fox, Jamie, and Giray Ökten. "Brownian Path Generation and Polynomial Chaos." SIAM Journal on Financial Mathematics 12, no. 2 (January 2021): 724–43. http://dx.doi.org/10.1137/20m1343154.

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31

Williams, M. M. R. "Polynomial Chaos Functions and Neutron Diffusion." Nuclear Science and Engineering 155, no. 1 (January 2007): 109–18. http://dx.doi.org/10.13182/nse05-73tn.

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32

PÉREZ, GABRIEL. "ROBUST CHAOS IN POLYNOMIAL UNIMODAL MAPS." International Journal of Bifurcation and Chaos 14, no. 07 (July 2004): 2431–37. http://dx.doi.org/10.1142/s0218127404010722.

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Анотація:

Simple polynomial unimodal maps which show robust chaos, that is, a unique chaotic attractor and no periodic windows in their bifurcation diagrams, are constructed. Their invariant distributions and Lyapunov exponents are examined.

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33

Pulch, Roland, and Cathrin van Emmerich. "Polynomial chaos for simulating random volatilities." Mathematics and Computers in Simulation 80, no. 2 (October 2009): 245–55. http://dx.doi.org/10.1016/j.matcom.2009.05.008.

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34

Crestaux, Thierry, Olivier Le Maıˆtre, and Jean-Marc Martinez. "Polynomial chaos expansion for sensitivity analysis." Reliability Engineering & System Safety 94, no. 7 (July 2009): 1161–72. http://dx.doi.org/10.1016/j.ress.2008.10.008.

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35

Beck, André Teófilo, and Wellison José de Santana Gomes. "Stochastic fracture mechanics using polynomial chaos." Probabilistic Engineering Mechanics 34 (October 2013): 26–39. http://dx.doi.org/10.1016/j.probengmech.2013.04.002.

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36

Lucor, D., C. H. Su, and G. E. Karniadakis. "Generalized polynomial chaos and random oscillators." International Journal for Numerical Methods in Engineering 60, no. 3 (May 5, 2004): 571–96. http://dx.doi.org/10.1002/nme.976.

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37

Borell, Christer. "Real polynomial chaos and absolute continuity." Probability Theory and Related Fields 77, no. 3 (March 1988): 397–400. http://dx.doi.org/10.1007/bf00319296.

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38

Chen, Kung Yu, Shouh Jung Liu, and H. M. Srivastava. "Some new results for the Lagrange polynomials in several variables." ANZIAM Journal 49, no. 2 (October 2007): 243–58. http://dx.doi.org/10.1017/s1446181100012815.

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AbstractIn some recent investigations involving certain differential operators for a general family of Lagrange polynomials, Chan el al. encountered and proved a certain summation identity for the Lagrange polynomials in several variables. In the present paper, we derive some generalizations of this summation identity for the Chan-Chyan-Srivastava polynomials in several variables. We also discuss a number of interesting corollaries and consequences of our main results.

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39

Sinha, Alok. "Computation of the Statistics of Forced Response of a Mistuned Bladed Disk Assembly via Polynomial Chaos." Journal of Vibration and Acoustics 128, no. 4 (June 15, 2005): 449–57. http://dx.doi.org/10.1115/1.2215620.

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The method of polynomial chaos has been used to analytically compute the statistics of the forced response of a mistuned bladed disk assembly. The model of the bladed disk assembly considers only one mode of vibration of each blade. Mistuning phenomenon has been simulated by treating the modal stiffness of each blade as a random variable. The validity of the polynomial chaos method has been corroborated by comparison with the results from numerical simulations.

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40

Drakos, S., and G. N. Pande. "Stochastic Finite Element Analysis using Polynomial Chaos." Studia Geotechnica et Mechanica 38, no. 1 (March 1, 2016): 33–43. http://dx.doi.org/10.1515/sgem-2016-0004.

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Abstract This paper presents a procedure of conducting Stochastic Finite Element Analysis using Polynomial Chaos. It eliminates the need for a large number of Monte Carlo simulations thus reducing computational time and making stochastic analysis of practical problems feasible. This is achieved by polynomial chaos expansion of the displacement field. An example of a plane-strain strip load on a semi-infinite elastic foundation is presented and results of settlement are compared to those obtained from Random Finite Element Analysis. A close matching of the two is observed.

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41

Kaur, Navjot, and Kavita Goyal. "Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations." International Journal of Modern Physics C 31, no. 09 (August 8, 2020): 2050128. http://dx.doi.org/10.1142/s0129183120501284.

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The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

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42

Slika, Wael, and George Saad. "A practical polynomial chaos Kalman filter implementation using nonlinear error projection on a reduced polynomial chaos expansion." International Journal for Numerical Methods in Engineering 112, no. 12 (June 27, 2017): 1869–85. http://dx.doi.org/10.1002/nme.5586.

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43

Gayrard, Emeline, Cédric Chauvière, Hacène Djellout, and Pierre Bonnet. "MODELING EXPERIMENTAL DATA WITH POLYNOMIALS CHAOS." Probability in the Engineering and Informational Sciences 34, no. 1 (August 14, 2018): 14–26. http://dx.doi.org/10.1017/s026996481800030x.

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Given a raw data sample, the purpose of this paper is to design a numerical procedure to model this sample under the form of polynomial chaos expansion. The coefficients of the polynomial are computed as the solution to a constrained optimization problem. The procedure is first validated on samples coming from a known distribution and it is then applied to raw experimental data of unknown distribution. Numerical experiments show that only five coefficients of the Chaos expansions are required to get an accurate representation of a sample.

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44

Gibson, Nathan L. "A Polynomial Chaos Method for Dispersive Electromagnetics." Communications in Computational Physics 18, no. 5 (November 2015): 1234–63. http://dx.doi.org/10.4208/cicp.230714.100315a.

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AbstractElectromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider “polydispersive” materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for such problems involving Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion analyzes are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.

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45

Rajendran, Karthikeyan, Andreas C. Tsoumanis, Constantinos I. Siettos, Carlo R. Laing, and Ioannis G. Kevrekidis. "MODELING HETEROGENEITY IN NETWORKS USING POLYNOMIAL CHAOS." International Journal for Multiscale Computational Engineering 14, no. 3 (2016): 291–302. http://dx.doi.org/10.1615/intjmultcompeng.2016015897.

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46

Shi, Wenjie, and Daniel M. Tartakovsky. "Polynomial Chaos Expansions for Stiff Random ODEs." SIAM Journal on Scientific Computing 44, no. 3 (May 2, 2022): A1021—A1046. http://dx.doi.org/10.1137/21m1432545.

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47

Che, Yiming, Ziqi Guo, and Changqing Cheng. "Generalized polynomial chaos-informed efficient stochastic Kriging." Journal of Computational Physics 445 (November 2021): 110598. http://dx.doi.org/10.1016/j.jcp.2021.110598.

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48

Fox, Jamie, and Giray Ökten. "Polynomial Chaos as a Control Variate Method." SIAM Journal on Scientific Computing 43, no. 3 (January 2021): A2268—A2294. http://dx.doi.org/10.1137/20m1336515.

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49

Dodson, Michael, and Geoffrey T. Parks. "Robust Aerodynamic Design Optimization Using Polynomial Chaos." Journal of Aircraft 46, no. 2 (March 2009): 635–46. http://dx.doi.org/10.2514/1.39419.

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50

Levajkovic, Tijana, Hermann Mena, and Lena-Maria Pfurtscheller. "Solving Stochastic LQR Problems by Polynomial Chaos." IEEE Control Systems Letters 2, no. 4 (October 2018): 641–46. http://dx.doi.org/10.1109/lcsys.2018.2844730.

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