相关论文: Learning, Solving and Optimizing PDEs with TensorG…

In this article we investigate a finite element formulation of strongly monotone quasi-linear elliptic PDEs in the context of fixed-point iterations. As opposed to Newton's method, which requires information from the previous iteration in…

Hybrid finite element methods such as hybridizable discontinuous Galerkin, hybrid high-order and weak Galerkin have emerged as powerful techniques for solving partial differential equations on general polytopal meshes. Despite their diverse…

A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials…

We present the design and optimization of a linear solver on General Purpose GPUs for the efficient and high-throughput evaluation of the marginalized graph kernel between pairs of labeled graphs. The solver implements a preconditioned…

This paper proposes a matrix-free residual evaluation technique for the hybridizable discontinuous Galerkin method requiring a number of operations scaling only linearly with the number of degrees of freedom. The method results from…

The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using…

Piezoelectric Micromachined Ultrasonic Transducers (PMUTs) are essential for next-generation ultrasonic sensing and imaging due to their bidirectional electromechanical behavior, compact design, and compatibility with low-voltage…

We analyze a novel multi-level version of a recently introduced compressed sensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab: Compressive Sensing Petrov-Galerkin approximation of high-dimensional parametric operator…

This paper develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known…

This work presents a novel methodology for speeding up the assembly of stiffness matrices for laminate composite 3D structures in the context of isogeometric and finite element discretizations. By splitting the involved terms into their…

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear…

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is…

Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…

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 describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator…

High-order Discontinuous Galerkin (DG) methods promise to be an excellent discretisation paradigm for partial differential equation solvers by combining high arithmetic intensity with localised data access. They also facilitate dynamic…

We introduce a simultaneous and meshfree topology optimization (TO) framework based on physics-informed Gaussian processes (GPs). Our framework endows all design and state variables via GP priors which have a shared, multi-output mean…

We present the latest developments of our High-Order Spectral Element Solver (HORSES3D), an open source high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including compressible flows (with or…

We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous…