Abstract : Nowadays, the stochastic partial differential equations (SPDEs) are widely accepted as suitable models to understand complex phenomena and have been successfully applied in a broad range of areas including acoustics, electromagnetic and fluid dynamics. It is highly desirable to build efficient and reliable numerical methods, and to analyze their qualitative and quantitative properties: convergence rates (in strong and weak senses), long time behavior, approximation of invariant distributions, preservation of invariants, etc. This Minisymposium aims to provide a platform to show the significance and recent developments in numerical methods for SPDEs, and to foster interactions between academic and industrial researchers.
Organizer(s) : Charles-Edouard Bréhier, Jianbo Cui
[05603] Convergence Analysis of splitting up method for nonlinear filtering problem
Format : Talk at Waseda University
Author(s) :
Yanzhao Cao (Auburn University)
Abstract : Abstract: We consider a nonlinear filtering model where correlated Wiener processes and point processes drive observations. We first derive a Zakai equation that provides the filter solution's unnormalized probability density function. We then use a splitting-up technique to decompose the Zakai equation into three stochastic differential equations. Based on this, we construct a splitting-up approximate solution. We will present a half-order convergence result. Additionally, we will present a finite difference method to create a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.
[04327] CLT for approximating ergodic limit of SPDEs via a full discretization
Format : Talk at Waseda University
Author(s) :
Chuchu Chen (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Tonghe Dang (Academy of Mathematics and Systems Science，Chinese Academy of Sciences)
Jialin Hong (Academy of Mathematics and Systems Science，Chinese Academy of Sciences)
Tau Zhou (Academy of Mathematics and Systems Science，Chinese Academy of Sciences)
Abstract : The approximation of the ergodic limit is of fundamental importance in many applications. In this talk, we focus on characterizing quantitatively the fluctuations between the ergodic limit and the time-averaging estimator of the full discretization for the parabolic stochastic partial differential equation. We establish a central limit theorem, which shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence rate of the considered full discretization.
[04355] Energy regularized approximations for stochastic logarithmic Schrodinger equation
Format : Talk at Waseda University
Author(s) :
Jianbo Cui (Hong Kong Polytechnic University )
Jialin Hong (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Liying Sun (Capital Normal University)
Abstract : In this talk, we first prove the global existence and uniqueness of the solution of the stochastic logarithmic Schroedinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schroedinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of RSlogS equations. Then we present energy regularized numerical schemes and their strong convergence rates.
[05599] Density approximation for stochastic heat equation
Format : Online Talk on Zoom
Author(s) :
Derui Sheng (The Hong Kong Polytechnic University)
Chuchu Chen (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Jianbo Cui (Department of Applied Mathematics, The Hong Kong Polytechnic University)
Jialin Hong (Academy of Mathematics and Systems Science，Chinese Academy of Sciences)
Abstract : This talk presents the numerical approximation of the density of the stochastic heat equation via the accelerated
exponential Euler scheme. The existence and smoothness of the density of the numerical solution are proved through Malliavin calculus. By presenting a test-function-independent weak convergence analysis, we show that the
convergence orders of the density in uniform convergence topology are 1/2 and nearly 1 in the nonlinear drift case and in the affine drift case, respectively.
[03909] Space-time Discontinuous Galerkin Methods for the $\varepsilon$-dependent Stochastic Allen-Cahn Equation with mild noise
Format : Online Talk on Zoom
Author(s) :
Dimitra Antonopoulou (University of Chester)
Abstract : We consider the $\varepsilon$-dependent stochastic Allen-Cahn equation with mild space-time noise posed on a bounded domain in $\mathbb{R}^d$, $d\geq 1$. The noise tends to rough on the sharp interface limit. This equation is numerically approximated by a space-time discontinuous in time nonlinear Galerkin scheme for which we prove existence and uniqueness. A priori and a posteriori error
analysis is applied and error estimates are established.
[05156] Finite differences method for stochastic heat equation with singular drifts.
Format : Online Talk on Zoom
Author(s) :
Ludovic Michel Goudenège (CNRS)
El Mehdi Haress (Paris-Saclay University)
Alexandre Richard (Paris-Saclay University)
Abstract : I will present the numerical approximation of the unique solution to a stochastic heat equation in dimension 1 with distributional drifts under Besov regularity and additive space-time white noise.
The approximation is based on a tamed Euler finite-difference scheme with mollified drift.
The rate of convergence of the numerical approximation towards the unique strong solution is related to the regularity of the drift.
When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the rate of convergence 1/2 in space and 1/4 in time.
Some numerical simulations of the stochastic heat equation with Dirac drift or penalization drift will be presented.
[04362] Numerical schemes and related qualitative properties for degenerate PDEs driven by L𝑒́𝑣𝑦 noise.
Format : Online Talk on Zoom
Author(s) :
Ananta Kumar Majee (Indian Institute of Technology Delhi)
Soumya Ranjan Behera (Indian Institute of Technology Delhi)
Abstract : In this talk, we consider an operator splitting scheme and semi-discrete finite difference
scheme for fractional degenerate conserva1on laws driven by L𝑒́𝑣𝑦 noise and degenerate parabolic-
hyperbolic PDE with L𝑒́𝑣𝑦 noise respectively. By using necessary a-priori bounds for approximate
solutions, generated by splitting scheme, and average time continuity of regularized viscous solutions
together with a variant of classical Kru𝑧̌kov’s doubling of variables approach, we prove convergence of
approximate solutions to the unique BV entropy solution of the underlying problem. Moreover, the
convergence analysis is illustrated by several numerical examples. Furthermore, for compactly supported
initial data, we prove that the expected value of the 𝐿1 -difference between the unique entropy solution
and the approximate solutions, generated by finite difference scheme, converges at a rate of order 1/7.
[04784] Linear implicit time-stepping schemes for SPDEs with super-linearly growing coefficients
Format : Online Talk on Zoom
Author(s) :
Xiaojie Wang (Central South University)
Mengchao Wang (Central South University)
Abstract : The present talk is on strong approximations of stochastic partial differential equations (SPDEs) with polynomially growing nonlinearity and multiplicative trace-class noise. We propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linear implicit Euler-type scheme for the temporal discretization. We recover the strong convergence rates of the fully discrete scheme.
