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In this article, we present a parallel recursive algorithm based on multi-level domain decomposition that can be used as a precondtioner to a Krylov subspace method to solve sparse linear systems of equations arising from the discretization…
In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov…
In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference…
Preconditioning techniques are crucial for enhancing the efficiency of solving large-scale linear equation systems that arise from partial differential equation (PDE) discretization. These techniques, such as Incomplete Cholesky…
Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a…
In this paper, we propose a tensor type of discretization and optimization process for solving high dimensional partial differential equations. First, we design the tensor type of trial function for the high dimensional partial differential…
It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving…
We develop a unified Petrov-Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form $ _{0}{\mathcal{D}}_{t}^{2\tau}u^{} + \sum_{i=1}^{d}$ $[c_{l_i}$…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial…
A trademark of nonlinear, time-dependent, convection-dominated problems is the spontaneous formation of non-smooth macro-scale features, like shock discontinuities and non-differentiable kinks, which pose a challenge for high-resolution…
In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of…
We introduce a new spectral method for image segmentation that incorporates long range relationships for global appearance modeling. The approach combines two different graphs, one is a sparse graph that captures spatial relationships…
Tensor B-spline methods are a high-performance alternative to solve partial differential equations (PDEs). This paper gives an overview on the principles of Tensor B-spline methodology, shows their use and analyzes their performance in…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional…