Special Session 153: Stochastic computing and structure preserving methods

Symplectic Integrators from B-Series and Generating Functions for Stochastic Hamiltonian Systems

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

Abstract: We propose a new class of symplectic schemes for the strong approximation of the solutions of stochastic Hamiltonian systems. The proposed schemes are a trade-off between the stochastic Runge-Kutta methods and the symplectic methods based on generating functions. Symplectic stochastic Runge-Kutta methods use only first order derivatives of the Hamiltonians, but they satisfy many algebraic conditions, which makes strong orders above 1 hard to obtain for general stochastic Hamiltonian systems with multiplicative noise. Symplectic schemes of strong order 1.5 or higher can be obtained using generating functions, but they require higher order derivatives of the Hamiltonians. Here we propose a new family of symplectic schemes constructed by defining a generating function similar with the ones associated to symplectic stochastic Runge-Kutta methods. The proposed schemes are a generalization of the implicit midpoint rule, are symplectic by construction, and use derivatives of at most second order. We derive schemes of strong order 1.5 for general stochastic Hamiltonian systems and study their long time behaviour through numerical experiments.

Approximation of invariant measure via an explicit scheme for the stochastic Cahn-Hilliard equation

Wanrong Cao Southeast University Peoples Rep of China Co-Author(s):    Nan Deng, Yibo Wang

Abstract: This work investigates the stochastic Cahn--Hilliard equation (SCHE) driven by additive space--time white noise. We first refine the analytical ergodic theory by proving that the continuum equation admits a unique invariant measure on the more regular state space $H_\alpha$, extending the classical result of Da Prato & Debussche (1996) on the negative Sobolev space $\dot{H}^{-1}_\alpha$. To approximate long-time behaviour, we introduce an explicit fully discrete scheme that combines a finite-difference spatial discretization with a strongly tamed exponential Euler method in time. Uniform-in-time moment bounds in the $L^\infty$-norm are established for the numerical solution, and a uniform strong convergence estimate with an explicit rate is derived for the fully discrete approximation. Exploiting a mass-preserving minorization tailored to Neumann boundary conditions, we further show that the numerical scheme is geometrically ergodic and possesses a unique invariant measure, together with polynomial-order error bounds for approximating the exact invariant measure. Strong laws of large numbers are proved for both the continuous and discrete systems, ensuring almost-sure convergence of temporal averages to the corresponding ergodic limits. Numerical experiments corroborate the theoretical findings, including the long-time strong convergence and the accuracy of invariant measure approximation.

The law of iterated logarithm for numerical approximation of time-homogeneous Markov process

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

Abstract: The law of the iterated logarithm (LIL) for the time-homogeneous Markov process with a unique invariant measure characterizes the almost sure maximum possible fluctuation of time averages around the ergodic limit. Whether a numerical approximation can preserve this asymptotic pathwise behavior remains an open problem. In this work, we give a positive answer to this question and establish the LIL for the numerical approximation of such a process under verifiable assumptions. The Markov process is discretized by a decreasing time-step strategy, which yields the non-homogeneous numerical approximation but facilitates a martingale-based analysis. The key ingredient in proving the LIL for such numerical approximation lies in extracting a quasi-uniform time-grid subsequence from the original non-uniform time grids and establishing the LIL for a predominant martingale along it, while the remainder terms converge to zero. Finally, we illustrate that our results can be flexibly applied to numerical approximations of a broad class of stochastic systems, including SODEs and SPDEs.

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

Jianbo Cui The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Jianbo Cui; Liu Shu; 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 L_1 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.

Structure-preserving numerical methods for stochastic differential equations

Raffaele D`Ambrosio University of L`Aquila Italy Co-Author(s):

Abstract: This talk presents recent advances in the numerical preservation of qualitative and quantitative features in stochastic ordinary and partial differential equations, with focus on two main directions. The first concerns the geometric integration of stochastic Hamiltonian systems. Two different settings are considered: in the Ito case, the expected Hamiltonian typically displays a linear drift in time, while in the Stratonovich case the Hamiltonian is preserved pathwise. The talk discusses how suitable numerical methods reproduce these behaviors and addresses their long-time properties through backward error analysis. The second direction regards structure-preserving discretizations for dissipative stochastic problems. In this context, the focus is on the preservation of mean-square contractivity under time discretization, especially for stochastic theta-methods. The analysis shows that such contractivity can be retained under appropriate stepsize restrictions. A unifying perspective throughout the talk is the existence of a bridge between the numerical treatment of stochastic problems and the corresponding deterministic theory. This presentation is based on joint work with Helena Biscevic (Gran Sasso Science Institute), Chuchu Chen (Chinese Academy of Sciences), David Cohen (Chalmers University of Technology & University of Gothenburg), Stefano Di Giovacchino (University of L`Aquila), and Annika Lang (Chalmers University of Technology & University of Gothenburg).

Numerical Approximation of the Invariant Measure for Parabolic SPDEs with Singular Drift

Ludovic Goudenege CNRS France Co-Author(s):    Charles-Edouard Brehier, El Mehdi Haress, Jonathan Naffrichoux, Alexandre Richard

Abstract: We study a time-space discretization scheme for the non-linear stochastic heat equation driven by space-time white noise, with a singular reaction term b modeled as a distribution. Our numerical procedure combines a finite difference method in space with an Euler scheme, together with a taming of the singular reaction term. Under suitable scaling of all numerical parameters, we obtain strong convergence results with explicit rate that allow not only the visualization of trajectories but also the exploration of long-time statistical properties. In particular, we will focus on the approximation of the invariant measure associated with the equation. By combining dissipative Lipschitz drift with a singular drift of small Besov norm, we can study the long-time behavior and provide numerical schemes that capture the invariant distribution. This is joint work with Charles-Edouard Brehier, El Mehdi Haress, Jonathan Naffrichoux, and Alexandre Richard

Strong Convergence of Positivity Preserving Explicit Numerical Approximation for n-Dimensional Superlinear SDEs

Qian Guo Shanghai Normal University Peoples Rep of China Co-Author(s):    Yongmei Cai, Xuerong Mao

Abstract: Stochastic Differential Equations (SDEs) are intensively employed across various fields, including biology, finance, and engineering. Many prominent models in these disciplines, such as the $n$-dimensional Lotka-Volterra population systems and the Heston stochastic volatility model, possess unique positive solutions. However, simulating multidimensional SDEs with superlinear coefficients presents significant challenges. Standard numerical methods, like the classical Euler-Maruyama (EM) method, are known to diverge in finite time when coefficients grow super-linearly. Furthermore, most existing explicit methods fail to preserve the structural positivity of the exact solution.This talk proposes and analyzes an explicit, cost-effective numerical scheme: the Positivity Preserving Truncated Euler-Maruyama (PPTEM) method. The core of this method lies in combining a truncation technique to manage superlinear growth with a positivity mapping to ensure the numerical solution remains within the positive cone $\mathbb{R}^n_+$.

Strong Convergence Order of an Operator-Splitting Finite Difference Fully Discrete Scheme for Stochastic Gross-Pitaevskii Equations

Linghua Kong Jiangxi Normal University Peoples Rep of China Co-Author(s):    Rong Gao

Abstract: For the one-dimensional stochastic Gross-Pitaevskii equation, we propose a fully discrete finite difference scheme based on operator splitting, and establish that the scheme attains a strong convergence order of $O(\tau^2+h^2)$, where $\tau$ and $h$ denote the temporal and spatial step sizes, respectively. First, under suitable assumptions, we analyze the regularity of both the exact solution and the splitting solution. Subsequently, we employ the Crank-Nicolson scheme for temporal discretization and the finite difference method for spatial discretization, and establish the convergence theory of the fully discrete scheme under the condition $h^2$

C0 Finite Element Methods for Biharmonic and Triharmonic Problems

Hengguang Li Wayne State University USA Co-Author(s):    Peimeng Yin and Zhimin Zhang

Abstract: Consider the biharmonic and triharmonic problems in a general polygonal domain with the simply supported boundary condition. The finite element approximation to such high-order problems often involves sophisticated construction of the bilinear form and of the finite element spaces. We propose a new C0 finite element algorithm to solve these problems. Our methods are intuitive, easy to implement, and applicable to both convex and nonconvex domains. We give finite element error analysis and report numerical results to validate the algorithms.

Symplectic methods for stochastic Hamiltonian systems: asymptotic error distributions and Hamiltonian-specific analysis

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

Abstract: We investigate the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems and further provide Hamiltonian-specific analysis that clarifies the superiority of symplectic methods. Our contribution is threefold. First, we derive the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems with multiplicative noise and additive noise, respectively, and show that the obtained limiting stochastic processes satisfy equations retaining the Hamiltonian formulations. Second, we propose a new approach for calculating the asymptotic error distribution, revealing the connection between the stochastic modified equation and the asymptotic error distribution. Third, we characterize the limiting distribution of the normalized Hamiltonian deviation, thereby illustrating through test equations the superiority of symplectic methods for long-time simulations of the Hamiltonians, even in the limit as the step size tends to zero.

Old and new numerical methods for SPDEs: non-asymptotic uniform in time error bounds

Michela Ottobre Heriot Watt University and Maxwell Institute, Edinburgh Scotland Co-Author(s):    Can Huang, Gideon Simpson

Abstract: We discuss numerical methods to sample from the invariant measure of a class of SPDEs with non-Lipshitz drift, focusing in particular on tamed schemes. The methods we present approximate the SPDE dynamics uniformly in time; that is, they approximate correctly both the transient and the long time behaviour. We will show this fact by providing bounds which are non-asymptotic. Moreover, we will discuss a general approach to prove non-asymptotic, uniform in time bounds for approximations of Markovian dynamics. This approach has been shown to work for a range of Markovian dynamics and associated approximations( whether the approximation is produced numerically, via particle systems or via other methods).

Geometric collocation methods for stochastic multisymplectic PDEs

Linyu Peng Keio University Japan Co-Author(s):    Ruiao Hu

Abstract: Extending the stochastic generalization of the variational framework originally developed for deterministic multisymplectic partial differential equations, we introduce a stochastic variational formulation that guarantees the preservation of stochastic 1-form and 2-form conservation laws, together with stochastic counterparts of Noether`s theorem. Motivated by this formulation, we develop a family of stochastic collocation schemes that inherently preserve geometric structure. In particular, these methods maintain the stochastic multisymplectic 2-form at the discrete level. For linear problems, the proposed approach additionally ensures the preservation of discrete momentum.

Controllability of stochastic Maxwell system: Theory and numerical simulation

Liying Sun Capital Normal University Peoples Rep of China Co-Author(s):

Abstract: In this tale, we study the exact controllability of stochasticMaxwell equations. First, the observability inequality for backward stochastic Maxwell equations is established via the\r\nmultiplier method. Combined with duality theory, the exact controllability result of the forward\r\nequations is further proved. It is also shown that the control imposed on the diffusion term cannot\r\nbe weakened. Meanwhile, a numerical method is given for the solution of such an exact control problem to verify the theoretical result

Learning deterministic and stochastic forced Hamiltonian systems

Tomasz Tyranowski University of Twente Netherlands Co-Author(s):    Benedikt Brantner

Abstract: We present a neural network architecture capable of learning the parameter-dependent flow of a Hamiltonian system subject to external forcing, while preserving the underlying Lagrange-d`Alembert structure. We demonstrate that this architecture can learn the flows of time-dependent systems---both deterministic and stochastic---and more accurately emulate the system`s energy evolution compared to general-purpose, non-structure-preserving neural networks. This results in more stable and higher-quality solutions. We also discuss prospective applications to structure-preserving model reduction of stochastic Hamiltonian systems.

Preservation of random attractor and SRB measure under numerical discretization

Yibo Wang 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 investigate whether the random attractor and the Sinai--Ruelle--Bowen (SRB) measure can be preserved under numerical discretization for the stochastic Hopf bifurcation modeled by a nonlinear stochastic differential equation. To this end, we establish that the stochastic Hopf bifurcation under discretization induces a discrete random dynamical system. Further, we prove that this discrete system possesses a random attractor, and then derive the existence of an SRB measure by demonstrating a strictly positive numerical Lyapunov exponent. Numerical experiments visualize the retained random attractor and SRB measure for the discrete random dynamical system, revealing structures consistent with the theoretical chaotic phase.

Convergence rate of two classes of error processes for stochastic differential equations

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

Abstract: By the Cauchy problem as a bridge, this paper establishes the convergence rate of the asymptotic error process for the Euler scheme of stochastic differential equations (SDEs). When the noise is additive, the convergence rate of the stronger error process is obtained. These results fill in the gap of the current literature on the convergence of the asymptotic error process.