Related papers: Improved approximation algorithm for solution of n…

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…

The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…

Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the…

The Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) methods, introduced in [42], were originally designed for solving linear partial differential equations. In this paper, we extend the LRNN-DG methods to solve…

The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…

We introduce a new numerical method to approximate the solutions of a class of stationary Hamilton-Jacobi (HJ) partial differential equations arising from minimum time optimal control problems. We rely on nested grid approximations, and…

We implement the adaptive step size scheme from the optimization methods AdaGrad and Adam in a novel variant of the Proximal Gradient Method (PGM). Our algorithm, dubbed AdaProx, avoids the need for explicit computation of the Lipschitz…

In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets…

Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This…

This dissertation explores block decomposable methods for large-scale optimization problems. It focuses on alternating direction method of multipliers (ADMM) schemes and block coordinate descent (BCD) methods. Specifically, it introduces a…

We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial…

We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782--S808) in several…

We consider a time fractional differential equation of order $\alpha$, $0<\alpha<1$, $$ \frac{\partial c(x,t)}{\partial t}={}^C_0\mathcal{D}_t^{\alpha}[(Ac)(x,t)]+q(x,t) ,\quad x > 0, t > 0, \quad c(x,0)=f(x). $$ where…

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…

Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions,…

The finite-difference time-domain (FDTD) method is a well established method for solving the time evolution of Maxwell's equations. Unfortunately the scheme introduces numerical dispersion and therefore phase and group velocities which…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter (e.g., the temperature) that depends on integrals of the unknown…

The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational…