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In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
In real-world applications, it is important for machine learning algorithms to be robust against data outliers or corruptions. In this paper, we focus on improving the robustness of a large class of learning algorithms that are formulated…
The Partitioning Min-Max Weighted Matching (PMMWM) problem is an NP-hard problem that combines the problem of partitioning a group of vertices of a bipartite graph into disjoint subsets with limited size and the classical Min-Max Weighted…
The global spectral analysis (GSA) of numerical methods ensures that the dispersion relation preserving (DRP) property is calibrated in addition to ensuring numerical stability, as advocated in the von Neumann analysis. The DRP nature plays…
In this paper, we propose a stochastic method for solving equality constrained optimization problems that utilizes predictive variance reduction. Specifically, we develop a method based on the sequential quadratic programming paradigm that…
The ever-growing size of modern space-time data sets, such as those collected by remote sensing, requires new techniques for their efficient and automated processing, including gap-filling of missing values. CUDA-based parallelization on…
We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [SIAM J. Sci. Comput.,…
We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method…
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…
This paper proposes a novel Generalized Non-Standard Finite Difference (GNSFD) scheme for the numerical solution of a class of fractional partial differential equations (FrPDEs). The formulation of the method is grounded in optimization and…
We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave…
This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes. Non-uniform grids are essential to capture the singularities of "typical" solutions of time-fractional…
For many inference problems in statistics and econometrics, the unknown parameter is identified by a set of moment conditions. A generic method of solving moment conditions is the Generalized Method of Moments (GMM). However, classical GMM…
Generalized Benders decomposition (GBD) is a globally optimal algorithm for mixed integer nonlinear programming (MINLP) problems, which are NP-hard and can be widely found in the area of wireless resource allocation. The main idea of GBD is…
We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test…
In this paper, we propose an efficient implementation of deep policy gradient method (PGM) for optimal control problems in continuous time. The proposed method has the ability to manage the allocation of computational resources, number of…
Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention…
In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a…
The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell…
The time-dependent Schr\"odinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials…