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Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics

Federico Cornalba, Julian Fischer

TL;DR

This work develops and analyzes multilevel Monte Carlo (MLMC) methods to approximate weak statistics of the Dean–Kawasaki equation, a highly singular SPDE arising in fluctuating hydrodynamics. It introduces two noise-coupling schemes—Fourier coupling and Right-Most Nearest Neighbours (NN) coupling—to couple discretizations across levels, and proves cross-level variance and systematic error bounds that decay with level, enabling MLMC complexity gains. The results show substantial computational savings over standard Monte Carlo, quantified as dimension- and coupling-dependent cost bounds and explicit variance-reduction factors that depend on the average particle density $N h^d_{min}$. Numerical simulations in two dimensions validate the theory, demonstrating variance decay, accurate mean estimates, and practical speed-ups up to around 23x for challenging density regimes. Overall, the paper provides a rigorous weak-convergence-based MLMC framework for a singular SPDE, with concrete guidance on coupling strategies and density requirements for achieving significant computational gains in fluctuating hydrodynamics simulations.

Abstract

Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean--Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean--Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density, and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean--Kawasaki equation being highly singular.

Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics

TL;DR

This work develops and analyzes multilevel Monte Carlo (MLMC) methods to approximate weak statistics of the Dean–Kawasaki equation, a highly singular SPDE arising in fluctuating hydrodynamics. It introduces two noise-coupling schemes—Fourier coupling and Right-Most Nearest Neighbours (NN) coupling—to couple discretizations across levels, and proves cross-level variance and systematic error bounds that decay with level, enabling MLMC complexity gains. The results show substantial computational savings over standard Monte Carlo, quantified as dimension- and coupling-dependent cost bounds and explicit variance-reduction factors that depend on the average particle density . Numerical simulations in two dimensions validate the theory, demonstrating variance decay, accurate mean estimates, and practical speed-ups up to around 23x for challenging density regimes. Overall, the paper provides a rigorous weak-convergence-based MLMC framework for a singular SPDE, with concrete guidance on coupling strategies and density requirements for achieving significant computational gains in fluctuating hydrodynamics simulations.

Abstract

Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean--Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean--Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density, and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean--Kawasaki equation being highly singular.
Paper Structure (25 sections, 9 theorems, 74 equations, 4 figures, 1 table)

This paper contains 25 sections, 9 theorems, 74 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Structure of the paper
  4. Basic notation
  5. MLMC framework
  6. First coupling: Fourier coupling
  7. Second coupling: Right-Most Nearest Neighbours (NN) coupling
  8. Assumptions
  9. Cross-level variance bound, and variance reduction result
  10. Proofs of Proposition \ref{['PropBoundVarLevels']}, Theorem \ref{['main_res']}, and Proposition \ref{['PropVarFactor']}
  11. Proof of Proposition \ref{['PropBoundVarLevels']} with Fourier coupling
  12. Proof of Theorem \ref{['main_res']}
  13. Proof of Proposition \ref{['PropVarFactor']}
  14. Numerical simulations
  15. Types of experiments
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\varepsilon > 0$ be a given accuracy. Consider a sequence of levels for which the finest grid size $h_{\min}$ satisfies $h^2_{\min} \lesssim \varepsilon$. Under the technical assumptions N_h_scaling--reg_mfl--ass_stability--reg_test_func--init_datum, which in particular require the average part we can set up a Multilevel Monte Carlo estimator $\mu_{MLMC}$ for the simulation of DK which achiev

Figures (4)

  • Figure 3: 3d snapshots at time $T = 1024\,\, ms$ of a trajectory of the discrete Dean--Kawasaki equation \ref{['FullyDiscreteDK']} started from $\overline{\rho}_{0,reg}$, and with $N=2\cdot 10^6$ particles, on different levels ($h=2\pi\cdot 2^{-7}, \tau =0.001$ for Left plot, $h=2\pi\cdot 2^{-5}, \tau =0.001\cdot 4^2$ for Right plot).
  • Figure 4: Computational results ($N=2\cdot 10^9$, $\overline{\rho}_{0} = \overline{\rho}_{0,reg}$).
  • Figure 5: Computational results ($N=2\cdot 10^5$, $\overline{\rho}_{0} = \overline{\rho}_{0,reg}$).
  • Figure 6: Computational results ($N=2\cdot 10^9$, $\overline{\rho}_{0} = \overline{\rho}_{0,irreg}$).

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: Space-time discretised Dean--Kawasaki model approximating an underlying system of $N$ non-interacting Brownian particles in ${\mathbb{T}^d}$
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.6
  • Remark 4.7
  • Remark 4.8
  • ...and 20 more