Related papers: On Galerkin Approximations for the Zakai Equation …
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
The aim of this paper is to propose a new method for numerical approximations of the solution of the linear stochastic partial differential equation arising in non-linear filtering problems: the Zaka\"i equation. The approximation scheme is…
Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities…
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline…
We study an approximation method for the one-dimensional nonlinear filtering problem, with discrete time and continuous time observation. We first present the method applied to the Fokker-Planck equation. The convergence of the…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
The aim of the paper is to demonstrate the use of the Galerkin method for some kind of Volterra equations, determininistic and stochastic as well. The paper consists of two parts: the theoretical and numerical one. In the first part we…
A numerical method is proposed to compute a low-rank Galerkin approximation to the solution of a parametric or stochastic equation in a non-intrusive fashion. The considered nonlinear problems are associated with the minimization of a…
The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall…
The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of…
The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential…
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite…
The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution…
It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions…
In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter…
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions…