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We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…
PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also…
The Finite Difference Time Domain (FDTD) method is a widely used numerical technique for solving Maxwell's equations, particularly in computational electromagnetics and photonics. It enables accurate modeling of wave propagation in complex…
Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…
We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist…
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…
Solving Partial Differential Equations (PDEs) is fundamental to numerous scientific and engineering disciplines. A common challenge arises from solving the PDE families, which are characterized by sharing an identical mathematical structure…
In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…
Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus…
Molecular Dynamics (MD) codes predict the fundamental properties of matter by following the trajectories of a collection of interacting model particles. To exploit diverse modern manycore hardware, efficient codes must use all available…
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
The electrical and electronic engineering has used parallel programming to solve its large scale complex problems for performance reasons. However, as parallel programming requires a non-trivial distribution of tasks and data, developers…
We present an assignment for a full Parallel Computing course. Since 2017/2018, we have proposed a different problem each academic year to illustrate various methodologies for approaching the same computational problem using different…