Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

Abstract

The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R d is considered. The differential operator is given by the fractional power L β , β ∈ (0, 1), of an integer order elliptic differential operator L and is therefore non-local. Its inverse L −β is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator L −β is approximated by a weighted sum of non-fractional resolvents (I + t 2 j L) −1 at certain quadrature nodes t j > 0. The resolvents are then discretized in space by a standard finite element method.This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = κ 2 − ∆, κ > 0, with homogeneous Dirichlet boundary conditions on the unit cube (0, 1) d in d = 1, 2, 3 spatial dimensions for varying β ∈ (0, 1) attest the theoretical results.where · is the Euclidean norm on R d and Γ, K ν denote the gamma function and the modified Bessel function of the second kind, respectively. Via the positive parameters σ, ν, and κ the most important characteristics of the random field u

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

“…Recent studies on the finite element discretization and error analysis of semilinear and fractional elliptic SPDEs with white noise forcing can be found in [60] and [6,7,8], respectively. For the spatial discretization, classical piecewise linear [7,8,16,40] or mixed finite elements [43,44] have been employed. However, if the white noise is discretized by finite elements, then the load vector follows a Gaussian distribution with the finite element mass matrix as covariance matrix.…”

Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes

“…Theorem 5.1 (Theorem 2.10 and Corollary 2.4 in [5]). Let u be the solution of (11) (k = 1 case) and let u h be its FEM approximation obtained by using continuous Lagrange elements over a mesh of maximum element size h. Then there exist constants c 1 and c 2 such that…”

“…The estimated covariances match each other and the exact expression closely, demonstrating that our coupling technique is accurate also in practice. As a final verification step, we check that the coupled fields are consistent with the telescoping sum in (5) and (6) (15) and (16) for three different values of ν in the h-refinement case. The exact covariance C(r) is given by (1).…”

“…Different choices of bases yield different methods. Common choices are the basis of the eigenfunctions of C(x, y) (Karhunen-Loève expansion), the Fourier basis (circulant embedding [10]), or a finite element basis ( [5,11,27,40]). The former method is the most flexible as it can be used to sample Gaussian vectors with arbitrary covariance structure.…”

“…However this is generally not feasible and R d is in practice truncated to a bounded domain D. In this case, artificial boundary conditions must be prescribed on ∂D. Homogeneous Dirichlet or Neumann boundary conditions are often chosen [5,27], although it usually does not matter for practical purposes as the error in the covariance of the field decays rapidly away from the boundary [33]. After the meshing of D, (3) can be solved in linear time with the finite element method (FEM) and an optimally preconditioned Krylov solver.…”