Related papers: Semi-Lagrangian one-step methods for two classes o…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an…
Two-step predictor/corrector methods are provided to solve three classes of problems that present themselves as systems of ordinary differential equations (ODEs). In the first class, velocities are given from which displacements are to be…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
This paper studies the time-dependent test-function error in the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method caused by numerical integration errors of the characteristic ODE solver, and its…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
A semi-Lagrangian discontinuous finite element scheme based on the characteristic Galerkin method (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of the transport equation and its…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science…
In this paper we propose a Lagrangian method for solving Lane-Emden equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on a Modified generalized Laguerre functions Lagrangian…
A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and…
We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solve a new adjoint PDE at each optimization iteration, which…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
Randomness is ubiquitous in modern engineering. The uncertainty is often modeled as random coefficients in the differential equations that describe the underlying physics. In this work, we describe a two-step framework for numerically…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward…