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

Abstract

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.

Kernel-based probability measures for generalized interpolations: A deterministic or stochastic problem?

“…Thus, the kernel-based probability measures on Banach spaces in Definition 2.2 can be viewed as a generalization of original kernel-based probability measures on reproducing kernel Hilbert spaces in [3,7,8] or Sobolev spaces in [24,25,26]. Proposition 2.5.…”

Kernel-based Methods for Stochastic Partial Differential Equations

“…Remark 8.1. In our original papers [14,31,32,33], we call the kernel-based methods the kernel-based collocation methods. But, some people may confuse the kernel-based collocation and the stochastic collocation in [5].…”

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

“…27,28 In addition, time-dependent and time-independent SPDEs have been solved by using the meshless method in Cialenco et al 29 and Fasshauer and Ye. 30 The general form of nonlinear time-fractional stochastic sine-Gordon equation is given by…”

“…Problems in financial mathematics have been solved using the RBFs‐based approaches in Ballestra and Pacelli 27,28 . In addition, time‐dependent and time‐independent SPDEs have been solved by using the meshless method in Cialenco et al 29 and Fasshauer and Ye 30 …”