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Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part I: Time discretization

Yoshihito Kazashi, Fabio Nobile, Fabio Zoccolan

TL;DR

This work develops and analyzes three time-discretization schemes for dynamical low-rank approximations of high-dimensional SDEs, focusing on a rank-$k$ DLRA representation with time-evolving deterministic and stochastic modes. The DLR Euler–Maruyama method requires a Δt restriction tied to the smallest Gramian singular value for convergence, while the projector-splitting variants (DLR-PS-EM and DLR-PS-SDE) achieve stability and convergence without this restriction under nondegenerate diffusion. The authors establish strong convergence of DLRA to the continuous DO solution, provide mean-square stability results, and prove Gramian nondegeneracy under elliptic diffusion, with numerical experiments validating the theoretical claims across basic SDEs and PDEs with additive and multiplicative noise. The findings highlight the practical advantage of projector-splitting DLRA schemes in high-dimensional stochastic contexts and set the stage for Part II, which will address stochastic discretization in probability. overall, the work advances robust, scalable DLRA approaches for uncertainty quantification and data assimilation in stochastic systems.

Abstract

In this work (Part I), we study three time-discretization procedures of the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO) method for DLRA proposed and analyzed in arXiv:2308.11581v4, which consists of a linear combination of products between deterministic orthonormal modes and stochastic modes, both time-dependent. The first strategy we consider for numerical time-integration is very standard, consisting in a forward discretization in time of both deterministic and stochastic components. Its convergence is proven subject to a time-step restriction dependent on the smallest singular value of the Gram matrix associated to the stochastic modes. Under the same condition on the time-step, this smallest singular value is shown to be always positive, provided that the SDE under study is driven by a non-degenerate noise. The second and the third algorithms, on the other hand, are staggered ones, in which we alternately update the deterministic and the stochastic modes in half steps. These approaches are shown to be more stable than the first one and allow us to obtain convergence results without the aforementioned restriction on the time-step. Computational experiments support theoretical results. In this work we do not consider the discretization in probability, which will be the topic of Part II.

Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part I: Time discretization

TL;DR

This work develops and analyzes three time-discretization schemes for dynamical low-rank approximations of high-dimensional SDEs, focusing on a rank- DLRA representation with time-evolving deterministic and stochastic modes. The DLR Euler–Maruyama method requires a Δt restriction tied to the smallest Gramian singular value for convergence, while the projector-splitting variants (DLR-PS-EM and DLR-PS-SDE) achieve stability and convergence without this restriction under nondegenerate diffusion. The authors establish strong convergence of DLRA to the continuous DO solution, provide mean-square stability results, and prove Gramian nondegeneracy under elliptic diffusion, with numerical experiments validating the theoretical claims across basic SDEs and PDEs with additive and multiplicative noise. The findings highlight the practical advantage of projector-splitting DLRA schemes in high-dimensional stochastic contexts and set the stage for Part II, which will address stochastic discretization in probability. overall, the work advances robust, scalable DLRA approaches for uncertainty quantification and data assimilation in stochastic systems.

Abstract

In this work (Part I), we study three time-discretization procedures of the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO) method for DLRA proposed and analyzed in arXiv:2308.11581v4, which consists of a linear combination of products between deterministic orthonormal modes and stochastic modes, both time-dependent. The first strategy we consider for numerical time-integration is very standard, consisting in a forward discretization in time of both deterministic and stochastic components. Its convergence is proven subject to a time-step restriction dependent on the smallest singular value of the Gram matrix associated to the stochastic modes. Under the same condition on the time-step, this smallest singular value is shown to be always positive, provided that the SDE under study is driven by a non-degenerate noise. The second and the third algorithms, on the other hand, are staggered ones, in which we alternately update the deterministic and the stochastic modes in half steps. These approaches are shown to be more stable than the first one and allow us to obtain convergence results without the aforementioned restriction on the time-step. Computational experiments support theoretical results. In this work we do not consider the discretization in probability, which will be the topic of Part II.
Paper Structure (18 sections, 18 theorems, 177 equations, 17 figures, 3 algorithms)

This paper contains 18 sections, 18 theorems, 177 equations, 17 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. General Setting of DLRA for SDEs
  3. General Notation
  4. Time integration schemes
  5. DLR Euler--Maruyama discretization
  6. DLR Projector Splitting for Euler-Maruyama (EM)
  7. DLR Projector Splitting for SDEs
  8. Convergence of the DLR Euler-Maruyama
  9. Analysis of the DLR Projector Splitting for EM
  10. Mean Square Stability of the DLR Projector Splitting for EM
  11. Analysis of the DLR Projector Splitting for SDEs
  12. Mean Square Stability of the DLR Projector Splitting for SDEs
  13. Numerical Experiments
  14. Basic SDE examples
  15. Mean-square stability
  16. ...and 3 more sections

Key Result

Lemma 3.2

Let $X_n= U_n^{\top}Y_n$, $n=0,\dots,N$ be the solution produced by Algorithm alg: Eva Proj Splitt SDE algorithm with an arbitrary step-size $\Delta t_n$ such that $\sum_{n=1}^{N} \Delta t_n= T$. Then, the following relations hold: and

Figures (17)

  • Figure 1: For $\Delta t=0.1,0.05,0.02$, $k=2$, $M=10000$, $\sigma_{B}=10^{-8}$ for \ref{['ex: toy example']}. (Top) Smallest singular values of $\mathbb{E}[Y_nY_n^{\top}]$ for the DLR Euler-Maruyama, (Center) for the DLR Projector Splitting for EM, (Bottom) for the DLR Projector Splitting for SDEs. Lower bounds of $\sigma_k$ refer to \ref{['eq: cov 2 EM']}, \ref{['eq: cov 2 KNV']}, and \ref{['eq: cov 2 Stoch DLR Proj']} for all the three algorithms, respectively.
  • Figure 2: (Left) Singular values for the process $X^{\mathrm{true}}(t)$, solution of equation \ref{['ex: toy example']}. (Right) Relative errors for the DLR Euler-Maruyama, DLR Projector Splitting for EM, and the DLR Projector Splitting for SDEs, with respect to the true solution $X^{\mathrm{true}}$ and the continuous DLRA of rank $k=2$. $M=10000$ paths are used in all methods to approximate expectations.
  • Figure 3: For $\Delta t=0.1,0.05,0.02$, $k=2$, $M=10000$, $\sigma_{B}=10^{-19}$. (Top) Smallest singular values of $\mathbb{E}[Y_nY_n^{\top}]$ for the DLR Euler-Maruyama, (Center) for the DLR Projector Splitting for EM, (Bottom) for the DLR Projector Splitting for SDEs for Problem \ref{['ex: toy example 2']}.
  • Figure 4: Relative errors for the DLR Euler-Maruyama, DLR Projector Splitting for EM, and the DLR Projector Splitting for SDEs, with respect to the true solution $X^{\mathrm{true}}$ and the continuous DLRA of rank $k=2$ for Problem \ref{['ex: toy example 2']}. $M=10000$ paths are used in all methods to approximate expectations.
  • Figure 5: For $\Delta t=0.1,0.05,0.02$, $k=2$, $M=10000$, $\sigma_{B}=10^{-19}$. For Problem \ref{['ex: toy example 3']} (Top) smallest singular values of $\mathbb{E}[Y_nY_n^{\top}]$ for the DLR Euler-Maruyama, (Center) for the DLR Projector Splitting for EM, (Bottom) for the DLR Projector Splitting for SDEs
  • ...and 12 more figures

Theorems & Definitions (44)

  • Remark 3.1: Well posedness of equation \ref{['eq: DLR EM 1v']}
  • Lemma 3.2: Auxiliary result for DLR PS EM computations
  • proof
  • Remark 3.3: Independence of the choice of solution for Projector Splitting schemes
  • Proposition 4.1
  • proof
  • Lemma 4.2: $L^2$-norm bound of DLR-EM solution
  • proof
  • Remark 4.3
  • Lemma 4.4: Boundedness of the stochastic modes
  • ...and 34 more