相关论文: Low-Dimensional Galerkin Approximations of Nonline…

In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of local discontinuous Galerkin (LDG) method and…

Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical…

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…

We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme)…

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…

We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the…

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…

In this work, we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties. First, we…

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed…

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i.e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…

We review different (reduced) models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists in the minimization of a fourth order energy, potentially subject to a nonconvex constraint.…

We propose two new alternative numerical schemes to solve the coupled Einstein-Euler equations in the Generalized Harmonic formulation. The first one is a finite difference (FD) Central Weighted Essentially Non-Oscillatory (CWENO) scheme on…

We present a new line-based discontinuous Galerkin (DG) discretization scheme for first- and second-order systems of partial differential equations. The scheme is based on fully unstructured meshes of quadrilateral or hexahedral elements,…

Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a…

This manuscript is devoted to the study of a class of nonlinear non-instantaneous impulsive first order abstract retarded type functional differential equations in an arbitrary separable Hilbert space H. A new set of sufficient conditions…

Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…