Special Session 116: Stochastic computing and structure preserving methods

Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients

Cristina Anton MacEwan University Canada Co-Author(s):

Abstract: Under the uniform Hormander`s hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity condition. We obtain estimates for the Malliavin covariance matrix and its inverse, and to avoid non-integrability problems we use results about Malliavin differentiability based on the concepts of Ray Absolute Continuity and Stochastic Gateaux differentiability.

Diffusion model for generative learning

Yanzhao Cao Auburn University USA Co-Author(s):    Feng Bao, Guanan Zhang

Abstract: Abstract: We present a supervised learning framework for training generative models for density estimation. Generative models, including generative adversarial networks, normalizing flows, and variational auto-encoders, are usually considered unsupervised learning models because labeled data are generally unavailable for training. Despite the success of the generative models, there are several issues with unsupervised training, e.g., the requirement of reversible architectures, vanishing gradients, and training instability. We utilize the score-based diffusion model to generate labeled data to enable supervised learning in generative models. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE) corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer high accuracy and substantial time savings in neural network training. Both algorithm development and convergence analysis will be presented.

Superiority of stochastic symplectic methods via the law of iterated logarithm

Xinyu Chen Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Chuchu Chen, Tonghe Dang, Jialin Hong

Abstract: The superiority of stochastic symplectic methods over non-symplectic counterparts has been verified by plenty of numerical experiments, especially in capturing the asymptotic behaviour of the underlying solution process. This talk aims to theoretically investigate the superiority from the perspective of the law of iterated logarithm, taking the linear stochastic Hamiltonian system in Hilbert space as a test model. Based on the time-change theorem for martingales and the Borell--TIS inequality, we first prove that the upper limit of the exact solution with a specific scaling function almost surely equals some non-zero constant, thus confirming the validity of the law of iterated logarithm. Then, we prove that stochastic symplectic fully discrete methods asymptotically preserve the law of iterated logarithm, but non-symplectic ones do not. This reveals the good ability of stochastic symplectic methods in characterizing the almost sure asymptotic growth of the utmost fluctuation of the underlying solution process. Applications of our results to the linear stochastic oscillator and the linear stochastic Schr\{o}dinger equation are also presented.

A supervised learning scheme for computing Hamilton--Jacobi equation via density coupling

Jianbo Cui Hong Kong Polytechnic University Hong Kong Co-Author(s):    Shu Liu and Haomin Zhou

Abstract: We propose a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on L1 residual of the proposed scheme based on the coupling density. Furthermore, the proposed scheme can be used to describe the behaviors of Hamilton--Jacobi PDEs beyond the singularity formations on the support of coupling density. Several numerical examples with different Hamiltonians are provided to support our findings.

Long-time strong convergence of one-step methods for McKean-Vlasov SDEs with non-globally Lipschitz continuous coefficients

Siqing Gan Central South University Peoples Rep of China Co-Author(s):

Abstract: This talk focuses on strong error analysis for long-time approximations of McKean-Vlasov SDEs with super linear growth coefficients. Under certain non-globally Lipschitz conditions, the propagation of chaos over infinite time is derived. The long-time mean-square convergence theorem is then established for general one-step methods. As applications of the obtained convergence theorem, the mean-square convergence rate of two numerical schemes such as the split-step backward Euler method and the projected Euler method is investigated. Numerical examples are finally provided to validate our theoretical findings.

Numerical approximation of the invariant measure for a class of stochastic damped wave equations

Ziyi Lei Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Charles-Edouard Brehier, Siqing Gan

Abstract: We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge, this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

A second-order Langevin sampler preserving positive volume for isothermal-isobaric ensemble

Lei Li Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: We propose a second-order Langevin sampler for the isothermal-isobaric ensemble (the NPT ensemble) for molecular simulations, preserving a positive volume for the simulation box throughout the simulation. The equations of motion are obtained by taking the cell mass to zero in the equations in our previous work. We show the well-posedness of the new system of equations. Choosing a suitable friction, the equation for the volume can be converted into an SDE with additive noise, based on which we design the second-order scheme.

Long-time weak convergence analysis of a semi-discrete scheme for stochastic Maxwell equations

Ge Liang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Chuchu Chen, Jialin Hong

Abstract: In this talk, we focus on investigating the weak convergence of the implicit Euler scheme for stochastic Maxwell equations on the infinite time horizon. Based on the properties of the Maxwell operator, we first analyze the regularities of transformed Kolmogorov equation associated to the stochastic Maxwell equations. Then by constructing an adapted continuous auxiliary process of the implicit Euler scheme, we prove that the long-time weak convergence order of the scheme is one, which is twice the strong convergence order. At last, we give some applications of the weak convergence result. This is a joint work with Prof. Chuchu Chen and Prof. Jialin Hong.

The stochastic scalar auxiliary variable approach for stochastic nonlinear Klein--Gordon equation

Liying Sun Capital Normal University Peoples Rep of China Co-Author(s):    Jianbo Cui, Jialin Hong, Liying Sun

Abstract: In this talk, we propose and analyze energy-preserving numerical schemes for the stochastic nonlinear wave equation.These numerical schemes, called stochastic scalar auxiliary variable (SAV) schemes, are constructed by transforming the considered equation into a higher dimensional stochastic system with a stochastic SAV. We prove that they can be solved explicitly, and preserve the modiified energy evolution law and the regularity structure of the original system. These structure-preserving properties are the keys to overcoming the mutual effect of noise and nonlinearity. By providing new regularity estimates of the introduced SAV, we obtain the strong convergence rate of stochastic SAV schemes under Lipschitz conditions. Furthermore, based on the modified energy evolution laws, we derive the exponential moment bounds and sharp strong convergence rate of the proposed schemes for equation with a non-globally Lipschitz nonlinearity in the additive noise case. To the best of our knowledge, this is the first result on the construction and strong convergence of semi-implicit energy-preserving schemes for stochastic nonlinear wave equations.

Asymptotic error distribution for the Euler scheme of stochastic delay differential equation with locally Lipschitz coefficients

Fuke Wu Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Huagui Liu, Ya Wang

Abstract: This paper primarily concentrates on Euler schemes for stochastic delay differential equations (SDDEs) with locally Lipschitz coefficients. The convergence in probability and the weak limit process of the normalized error process are derived. Furthermore, this paper consider a specific degenerate stochastic system and obtain the associated weak limit process. In contrast to ``non-degenerate`` systems considered earlier, the normalized error parameter of such degenerate systems is n instead of \sqrt{n}. This caused some challenges, as there are additional terms in the weak limit process. These results are new even for stochastic differential equations without delay.

A new class of splitting methods that preserve ergodicity and exponential integrability for stochastic Langevin equation

Fengshan Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Chuchu Chen, Tonghe Dang, Jialin Hong

Abstract: In this talk, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the original equation, which is inspired by the inheritance of the Lyapunov structure for obtaining the ergodicity. We prove that the exponential moment of the numerical solution is bounded, thus validating the exponential integrability of the proposed methods. Further, we show that under moderate verifiable conditions, the methods have the first-order convergence in both strong and weak senses, and we present several concrete splitting schemes based on the methods. The splitting strategy of methods can be readily extended to construct conformal symplectic methods and high-order methods that preserve both the ergodicity and the exponential integrability, as demonstrated in numerical experiments. Our numerical experiments also show that the proposed methods have good performance in the long-time simulation. This work was completed in collaboration with Chuchu Chen, Tonghe Dang and Jialin Hong.

Strong Stability Preserving Multistep Schemes for FBSDEs

Weidong Zhao Shandong University Peoples Rep of China Co-Author(s):    Shuixin Fang, Tao Zhou

Abstract: This talk concerns with strong stability preserving (SSP) multistep schemes for forward backward stochastic differential equations (FBSDEs). We first perform a comprehensive analysis on a general type of multistep schemes for FBSDEs, then we present new sufficient conditions on the coefficients such that the associated schemes are stable and enjoy certain order of consistency. Upon these results, we propose a practical way to design high-order SSPM schemes for FBSDEs. Some numerical experiments are carried out to show the merits of our SSP multist schemes.