The multi-index Monte Carlo method for semilinear stochastic partial differential equations

TL;DR

This work develops a multi-index Monte Carlo framework (MIMC) tailored to semilinear parabolic SPDEs with multiplicative noise, leveraging an exponential integrator for the mild solution discretization. It proves a multiplicative convergence property for coupled MI-differences of EI solutions and derives asymptotic cost bounds as the target tolerance $\varepsilon \to 0$, showing substantial efficiency gains over MLMC in low-regularity settings. Theoretical results characterize the decay rates of MI-differences across a two-parameter index grid and establish cost scalings dependent on regularity parameters $\kappa$ and $\nu$, while numerical experiments on linear and nonlinear drift problems corroborate the theory and demonstrate clear speedups relative to MLMCEI. Overall, the paper provides a robust and tractable tool for weak approximation of statistics of semilinear SPDEs on bounded domains, with potential extensions to filtering and advanced splitting schemes.

Abstract

Stochastic partial differential equations (SPDEs) are often difficult to solve numerically due to their low regularity and high dimensionality. These challenges limit the practical use of computer-aided studies and pose significant barriers to statistical analysis of SPDEs. In this work, we introduce a highly efficient multi-index Monte Carlo method (MIMC) designed to approximate statistics of mild solutions to semilinear parabolic SPDEs. Key to our approach is the proof of a multiplicative convergence property for coupled solutions generated by an exponential integrator numerical solver, which we incorporate with MIMC. We further describe theoretically how the asymptotic computational cost of MIMC can be bounded in terms of the input accuracy tolerance, as the tolerance goes to zero. Notably, our methodology illustrates that for an SPDE with low regularity, MIMC offers substantial performance improvements over other viable methods. Numerical experiments comparing the performance of MIMC with the multilevel Monte Carlo method on relevant test problems validate our theoretical findings. These results also demonstrate that MIMC significantly outperforms state-of-the-art multilevel Monte Carlo, thereby underscoring its potential as a robust and tractable tool for solving semilinear parabolic SPDEs.

Figures (7)

• Figure 1: Left: Estimate of the double difference convergence rate for the SPDE \ref{['eq:ex1_linSpde']}. Right: A function $p(\ell_1,\ell_2) \approx \min(2^{- \beta_1 \ell_1/2 - \beta_2 \ell_2/2 +c_1}, 2^{- \beta_2 \ell_2+c_2})$ that appears to be a good global approximation of $e_F$.
• Figure 2: Comparison of performance of MIMCEI and MLMC for the SPDE \ref{['eq:ex1_linSpde']} with $\nu =4/3$. Left: Approximation error. Right: Computational cost.
• Figure 3: Comparison of performance of MIMCEI and MLMC for the SPDE \ref{['eq:ex1_linSpde']} with $\nu =2$. Left: Approximation error. Right: Computational cost.
• Figure 4: Comparison of performance of standard MIMCEI (labeled MIMC1), reduced-samples MIMCEI (labeled MIMC2), and MLMC applied to the SPDE \ref{['eq:ex1_linSpde']} with $\nu =10/9$. Left: Approximation error. Right: Computational cost.
• Figure 5: Left: Estimate of the double difference convergence rate for the SPDE \ref{['eq:ex2_nonlinearSpde']}. Right: A function $p(\ell_1,\ell_2) \approx \min(2^{- \beta_1 \ell_1/2 - \beta_2 \ell_2/2 +c_1}, 2^{- \beta_2 \ell_2 +c_2})$ that is a good global approximation of $e_F$.

Theorems & Definitions (36)

• Example 2.1
• Remark 2.2
• Example 2.4
• Lemma 2.5
• proof
• Corollary 2.6: Mixed difference convergence
• Theorem 2.7
• proof
• Theorem 2.8
• proof