Approximation of stochastic partial differential equations by a kernel-based collocation method

Cialenco, Igor; Fasshauer, Gregory E.; Ye, Qi

doi:10.1080/00207160.2012.688111

Cited by 50 publications

(48 citation statements)

References 19 publications

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“…The reader may note that the form of the expression for the variance σ(x) 2 is analogous to that of the power function [5,10], and we can therefore use the same techniques as in the proofs from [4,5,10,11] to obtain a formula for the order of σ(x), i.e.,…”

Section: Convergence Analysismentioning

confidence: 99%

“…The random coefficients are obtained solving by system of linear equations that is slightly different from [4]. However the main ideas and techniques are the same as in [4].…”

Section: Approximation Of Spdesmentioning

confidence: 99%

“…In the papers [4,11] we only consider Gaussian noises. However, one may generalize the kernel-based collocation idea to other problems with colored noise.…”

Section: Competitive Advantagesmentioning

confidence: 99%

“…In this paper, we compare the use of a kernel-based collocation method [4,11] (meshfree approximation method) and a Galerkin finite element method [1,2] to approximate the solution of elliptic SPDEs. For kernel-based collocation, we directly simulate the Gaussian noise at a set of collocation points.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, current numerical methods usually show limited success for highdimensional problems and in complex domains. In our recent papers [4,11], we use a kernel-based collocation method to approximate the solution of highdimensional SPDE problems. Since parabolic SPDEs can be transformed into elliptic SPDEs using, e.g., an implicit time stepping scheme, solution of the latter represents a particularly important aspect of SPDE problems.…”

Section: Introductionmentioning

confidence: 99%

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Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Fasshauer

2012

Meshfree Methods for Partial Differential Equations VI

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Summary. We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noise. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random coefficients are obtained by solving a system of linear equations with multiple random right-hand sides. The finite element solution is given as a tensor product of triangular finite elements and Lagrange polynomials defined on a finite-dimensional probability space. Its coefficients are obtained by solving several deterministic finite element problems. For the kernel-based collocation method, we directly simulate the (infinite-dimensional) Gaussian noise at the collocation points. For the finite element method, however, we need to truncate the Gaussian noise into finite-dimensional noises. According to our numerical experiments, the finite element method has the same convergence rate as the kernel-based collocation method provided the Gaussian noise is truncated using a suitable number terms.

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Section: Convergence Analysismentioning

confidence: 99%

“…The random coefficients are obtained solving by system of linear equations that is slightly different from [4]. However the main ideas and techniques are the same as in [4].…”

Section: Approximation Of Spdesmentioning

confidence: 99%

“…In the papers [4,11] we only consider Gaussian noises. However, one may generalize the kernel-based collocation idea to other problems with colored noise.…”

Section: Competitive Advantagesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Fasshauer

2012

Meshfree Methods for Partial Differential Equations VI

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Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Eidinejad,

Saadati,

Vahidi

et al. 2023

Int J Numerical Modelling

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We study the two‐dimensional stochastic time‐fractional Sine–Gordon equation (2D ST‐FS‐G) and provide a solution for it. We use the new clique polynomial method to obtain a numerical solution to the 2D TFS‐G equation. In this technique, the clique polynomial is considered as a basic function for operational matrices. By converting the 2D ST‐FS‐G equation to algebraic equations, a solution is obtained for the desired equation, which clearly shows that this approach is suitable and accurate for dealing with the equation. We further present the error boundary for the obtained approximation of the desired three‐variable function based on the clique polynomial. Finally, we compare some numerical results obtained with the exact ones by a few practical examples.

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Solution of time‐fractional stochastic nonlinear sine‐Gordon equation via finite difference and meshfree techniques

Mirzaee

Rezaei

Samadyar

2021

Math Methods in App Sciences

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In this article, we introduce a numerical procedure to solve time‐fractional stochastic sine‐Gordon equation. The suggested technique is based on finite difference method and radial basis functions interpolation. By using this algorithm, first time‐fractional stochastic nonlinear sine‐Gordon equation is converted to elliptic stochastic differential equations. Then, the meshfree method based on radial basis functions (RBFs) is used to approximate the obtained equation. In fact, the finite difference method is used to approximate the unknown function in the time direction and generalized Gaussian RBF is applied to estimate the obtained equation in the space direction. The most important advantage of this method is that the noise terms are simulated directly at the collocation points at each time step. By employing this method, the equation decreased to a nonlinear system of algebraic equations which can be solved simply. The obtained results of solving three examples confirm the validity and capability of the proposed solution.

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Approximation of stochastic partial differential equations by a kernel-based collocation method

Cited by 50 publications

References 19 publications

Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Solution of time‐fractional stochastic nonlinear sine‐Gordon equation via finite difference and meshfree techniques

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