Related papers: Backward error analysis for multisymplectic discre…
In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the…
This paper reports a development in the proper symplectic decomposition (PSD) for model reduction of forced Hamiltonian systems. As an analogy to the proper orthogonal decomposition (POD), PSD is designed to build a symplectic subspace to…
Explicit discretizations of stochastic differential equations often encounter instability when the coefficients are not globally Lipschitz. The truncated schemes and tamed schemes have been proposed to handle this difficulty, but truncated…
We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation.…
This work presents a data-driven approach to the identification of spatial and temporal truncation errors for linear and nonlinear discretization schemes of Partial Differential Equations (PDEs). Motivated by the central role of truncation…
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which…
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of…
In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a…
Hamiltonian simulation is a fundamental algorithm in quantum computing that has attracted considerable interest owing to its potential to efficiently solve the governing equations of large-scale classical systems. Exponential speedup…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…
Evolutionary partial differential equations play a crucial role in many areas of science and engineering. Spatial discretization of these equations leads to a system of ordinary differential equations which can then be solved by numerical…
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we propose and analyze a posteriori error estimators for an optimal control problem involving the stationary Navier--Stokes equations; control…
Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping…
Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence…
In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g., (spatial)…
There are several well-established approaches to constructing finite difference schemes that preserve global invariants of a given partial differential equation. However, few of these methods preserve more than one conservation law locally.…
We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents…
We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…