Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs

TL;DR

This work addresses sampling the Gibbs-type invariant distribution μ⋆ of ergodic, gradient-structured parabolic semilinear SPDEs by applying a preconditioning operator that enhances temporal regularity without altering μ⋆. It develops first-order and second-order schemes for μ⋆ sampling, including a postprocessed Leimkuhler–Matthews-type method that achieves second-order accuracy, with exactness in the Gaussian case F=0. The analysis hinges on Kolmogorov and Poisson equations for the preconditioned dynamics and leverages spectral Galerkin discretizations to bridge infinite-dimensional theory with computable finite-dimensional schemes. Numerical experiments on the 1D heat equation with space-time white noise corroborate the theoretical orders and show substantial efficiency gains from preconditioning. Overall, the paper extends high-order sampling techniques to infinite-dimensional Gibbs measures arising from SPDEs and provides rigorous guarantees and practical methods for accurate invariant-measure estimation.

Abstract

For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach.

Figures (3)

• Figure 1: Error for sampling the invariant distribution of the preconditioned SPDE \ref{['eq:precondSPDE']} in the spatial domain $[0,1]$ with $A=\Delta$ and $F(x)=0$ using the explicit and implicit Euler methods EE and IE (\ref{['eq:thetascheme']} with $\theta=0$ and $\theta=1$), their postprocessed counterparts the Leimkuhler-Matthews method LM \ref{['eq:EulerExplicitePostprocessed']} and pIE \ref{['eq:EulerImplicitePostprocessed']}, the RK2 method \ref{['eq:RK2']} and the Crank-Nicholson scheme CN \ref{['eq:thetascheme']} with $\theta=\frac{1}{2}$, with the test function $\varphi(x)=\exp(-\left\Vert x\right\Vert_{L^2}^2)$ and $M=10^8$ trajectories.
• Figure 2: Error for sampling the invariant distribution of the preconditioned SPDE \ref{['eq:precondSPDE']} in the spatial domain $[0,1]$ with $A=\Delta$ and $F(x)=-x+\cos(x)$ using the explicit and implicit Euler methods EE and IE (\ref{['eq:thetascheme']} with $\theta=0$ and $\theta=1$), their postprocessed counterparts the Leimkuhler-Matthews method LM \ref{['eq:EulerExplicitePostprocessed']} and pIE \ref{['eq:EulerImplicitePostprocessed']}, the RK2 method \ref{['eq:RK2']} and the Crank-Nicholson scheme CN \ref{['eq:thetascheme']} with $\theta=\frac{1}{2}$, with the test function $\varphi(x)=\exp(-\left\Vert x\right\Vert_{L^2}^2)$ and $M=10^8$ trajectories.
• Figure 3: Error of the integrator \ref{['equation:linear_implicit_Euler_general_preconditioning']} for sampling the invariant distribution of the preconditioned SPDE \ref{['eq:precondSPDE-general']} in the spatial domain $[0,1]$ with $A=\Delta$, $F(x)=-x+\cos(x)$, $\mathcal{P}=(-A)^{-\alpha}$ for different $\alpha$, the test function $\varphi(x)=\exp(-\left\Vert x\right\Vert_{L^2}^2)$ and $M=10^7$ trajectories.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• proof : Proof of Lemma \ref{['propo:invariant-general']}
• Remark 3.4
• Proposition 3.5
• proof
• Corollary 3.6