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Piecewise linear interpolation of noise in finite element approximations of parabolic SPDEs

Gabriel Lord, Andreas Petersson

TL;DR

This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel using piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain.

Abstract

Efficient simulation of stochastic partial differential equations (SPDE) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires non-trivial techniques like Hilbert--Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include: noise interpolation does not introduce additional errors for Matérn kernels in $d\ge2$; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.

Piecewise linear interpolation of noise in finite element approximations of parabolic SPDEs

TL;DR

This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel using piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain.

Abstract

Efficient simulation of stochastic partial differential equations (SPDE) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires non-trivial techniques like Hilbert--Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include: noise interpolation does not introduce additional errors for Matérn kernels in ; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.
Paper Structure (21 sections, 7 theorems, 84 equations, 5 figures)

This paper contains 21 sections, 7 theorems, 84 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and mathematical background
  3. Operator theory and entropy numbers
  4. Reproducing kernel Hilbert spaces
  5. Fractional Sobolev spaces
  6. Fractional powers of elliptic operators
  7. Parabolic SPDEs driven by cylindrical Wiener processes in RKHSs
  8. Mild solutions of SPDEs
  9. Cylindrical Wiener processes in RKHSs
  10. Non-linear multiplicative noise operator
  11. Piecewise linear interpolation of noise in SPDE approximations
  12. Discretization setting
  13. The main result
  14. Example kernels and numerical simulation
  15. Example kernels
  16. ...and 6 more sections

Key Result

Proposition 3.2

Let $F$ be given by eq:F1-F2-def, where for some constant $C < \infty$, Then $F$ extends to an operator on $H$ and fulfills Assumption assumption:regularityassumption:regularity:F for all $r \in [0,1]$

Figures (5)

  • Figure 1: Noise discretization idea as in Example \ref{['ex:numerical-matern']} for multiplicative noise, with $\mathcal{D}$ a dodecagon.
  • Figure 2: Strong error approximations for $X_{h,\Delta t}$.
  • Figure 3: Approximate strong errors for circulant embedding approximation and eigentruncation approximation. For both methods we have used a reference solution with $\Delta t = 2^{-11}, h = 2^{-12}$ and a total of $N = 10$ Monte Carlo samples, with the same realization of $W$ used for every $h$.
  • Figure 4: Offline costs for computing the covariance integrals \ref{['eq:covariance-integrals']}, the eigendecomposition and the extension for the circulant embedding matrix.
  • Figure 5: Average cost for obtaining one path of $X_{h,\Delta t}$ for fixed $\Delta t = 2^{-9}$ for the eigendecomposition method and the circulant embedding method.

Theorems & Definitions (23)

  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 13 more