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Numerical solution of kinetic SPDEs via stochastic Magnus expansion

Kevin Kamm, Stefano Pagliarani, Andrea Pascucci

TL;DR

This paper tackles the efficient numerical solution of kinetic SPDEs with two spatial variables by discretizing in space to obtain a large sparse SDE $dX_t = B_t X_t \,dt + A_t X_t \,dW_t$, then applying the Itô-stochastic Magnus expansion $X_t \approx e^{Y_t}$ to propagate in time. By iterating the Magnus expansion on subintervals and exploiting sparsity with GPU-accelerated expmv techniques, the authors demonstrate substantial speedups over Euler–Maruyama while maintaining high accuracy. They validate the method first against a constant-coefficient Langevin SPDE with an explicit solution and then extend to variable coefficients, noting that two or three Magnus terms suffice for good accuracy. The results indicate strong practical impact for solving high-dimensional SPDEs in physics and finance, with speedups ranging from 20 to 200× (up to ~280× in some benchmarks) depending on resolution and tolerance. Overall, the work provides a scalable, parallel, and accurate framework for SPDEs using stochastic Magnus expansion on modern hardware.

Abstract

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity. As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler-Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler-Maruyama scheme, depending on the accuracy target and the spatial resolution.

Numerical solution of kinetic SPDEs via stochastic Magnus expansion

TL;DR

This paper tackles the efficient numerical solution of kinetic SPDEs with two spatial variables by discretizing in space to obtain a large sparse SDE , then applying the Itô-stochastic Magnus expansion to propagate in time. By iterating the Magnus expansion on subintervals and exploiting sparsity with GPU-accelerated expmv techniques, the authors demonstrate substantial speedups over Euler–Maruyama while maintaining high accuracy. They validate the method first against a constant-coefficient Langevin SPDE with an explicit solution and then extend to variable coefficients, noting that two or three Magnus terms suffice for good accuracy. The results indicate strong practical impact for solving high-dimensional SPDEs in physics and finance, with speedups ranging from 20 to 200× (up to ~280× in some benchmarks) depending on resolution and tolerance. Overall, the work provides a scalable, parallel, and accurate framework for SPDEs using stochastic Magnus expansion on modern hardware.

Abstract

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity. As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler-Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler-Maruyama scheme, depending on the accuracy target and the spatial resolution.
Paper Structure (7 sections, 2 theorems, 58 equations, 9 figures, 2 tables)

This paper contains 7 sections, 2 theorems, 58 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. SPDEs with two space-variables
  3. Numerical experiments
  4. The Magnus expansion for the stochastic Langevin equation with constant coefficients
  5. The Magnus expansion for the stochastic Langevin equation with variable coefficients
  6. Conclusions
  7. Magnus expansion formulas for constant coefficients

Key Result

Theorem 1.1

Let $A_t$ and $B_t$ be bounded progressively measurable matrices in $\mathbb{R}^{d\times d}$ and let $(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_t)_{t\geq 0})$ be a filtered probability space equipped with a standard Brownian motion $W$. For $T>0$ let also $X=(X_t)_{t\in[0,T]}$ be the unique strong

Figures (9)

  • Figure 1: Sparsity patterns of $A$, $B$ and commutators of the SPDE from Section \ref{['sec:stochasticLangevin']} with coefficients \ref{['eq:coeffsLangevinConst']} and $d=50$. nz stands for the number of non-zero entries.
  • Figure 2: Schematic of the iterative Magnus scheme
  • Figure 3: Graphical representation of $U_t^{d,\kappa}$ compared to $U_t^d$ for the error analysis to disregard boundary effects.
  • Figure 4: Constant coefficients as in \ref{['eq:coeffsLangevinConst']}: Computational times and errors of the Magnus expansion for varying number step-size $\Delta_t$ with fixed spatial dimension $d=200$.
  • Figure 5: Constant coefficients as in \ref{['eq:coeffsLangevinConst']}: Absolute Errors of m3 compared to exact using $d=300$ grid points at $t=0.25$ (left) and $t=1$ (right).
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 1.1: KPP2021
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma A.1
  • proof