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Analysis of an exponential integrator for stochastic PDEs driven by Riesz noise

Charles-Edouard Bréhier, David Cohen, Lluís Quer-Sardanyons, Johan Ulander

Abstract

We present and study an explicit exponential integrator for parabolic SPDEs in any dimension driven by a Gaussian noise which is white in time and with spatial correlation given by a Riesz kernel. Under assumptions on the coefficients of the SPDE, we prove strong error bounds and exhibit how the rate of convergence depends on the exponent in the Riesz kernel. Finally, numerical experiments in spatial dimensions $1$ and $2$ are provided in order to confirm our convergence results.

Analysis of an exponential integrator for stochastic PDEs driven by Riesz noise

Abstract

We present and study an explicit exponential integrator for parabolic SPDEs in any dimension driven by a Gaussian noise which is white in time and with spatial correlation given by a Riesz kernel. Under assumptions on the coefficients of the SPDE, we prove strong error bounds and exhibit how the rate of convergence depends on the exponent in the Riesz kernel. Finally, numerical experiments in spatial dimensions and are provided in order to confirm our convergence results.
Paper Structure (20 sections, 9 theorems, 202 equations, 8 figures)

This paper contains 20 sections, 9 theorems, 202 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Notation
  4. Riesz noise
  5. Heat kernel
  6. Regularity assumptions
  7. Well-posedness and properties of the SPDE driven by Riesz noise
  8. Convergence analysis of the stochastic exponential integrator
  9. Numerical experiments
  10. Spatial discretization of the parabolic SPDE
  11. Time evolution and profile of the solution in dimension $\mathbf1$
  12. Strong convergence in dimension $\mathbf1$
  13. Computational times in dimension $\mathbf1$
  14. Strong convergence in dimension $\mathbf2$
  15. Computational times in dimension $\mathbf2$
  16. ...and 5 more sections

Key Result

Lemma 2

The heat kernel $G_d$, its spatial gradient $\nabla_{x} G_{d}$ and its temporal derivative $\partial_{t} G_{d}$ satisfy the following upper bounds: there exist $c_{d},C_{d}\in(0,\infty)$ such that for all $(t,x,y) \in (0,\infty) \times \overline{Q} \times \overline{Q}$, one has

Figures (8)

  • Figure 1: Time evolution (left) and profile at $T_{end}=0..5$ (right) for different values of the parameter $\alpha$.
  • Figure 2: Strong errors for the SPDE \ref{['prob']} with Riesz potential $f(r)=r^{-\alpha}$.
  • Figure 3: Strong errors for the SPDE \ref{['prob']} with Riesz potential $f(r)=r^{-\alpha}$ for $\alpha=0.2,0.4,0.6,0.8$.
  • Figure 4: Computational time as a function of the averaged final error for the three numerical methods.
  • Figure 5: Root mean square convergence of the stochastic exponential Euler scheme \ref{['sexp']} for $\alpha = 0.2, 0.4, 0.6, 0.8$ and for $n = 64$ space grid point in each direction.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 11 more