相关论文: Fast integration of DPG matrices based on tensoriz…

The Random Dot Product Graph (RDPG) is a generative model for relational data, where nodes are represented via latent vectors in low-dimensional Euclidean space. RDPGs crucially postulate that edge formation probabilities are given by the…

A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion…

In this paper, we introduce the Phantom Domain Finite Element Method (PDFEM), a novel computational approach tailored for the efficient analysis of heterogeneous and composite materials. Inspired by fictitious domain methods, this method…

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades…

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…

This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the $d$-dimensional integral of a given function. The new algorithm, called the MDI-SG ({\em multilevel…

We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8), a recently introduced mesh-independent algorithm, to obtain a high-order finite element…

In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving…

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is…

We report an efficient algorithm for calculating momentum-space integrals in solid state systems on modern graphics processing units (GPUs). Our algorithm is based on the tetrahedron method, which we demonstrate to be ideally suited for…

This research explored a novel explicit total Lagrangian Fragile Points Method (FPM) for finite deformation of hyperelastic materials. In contrast to mesh-based methods, where mesh distortion may pose numerical challenges, meshless methods…

I present an accurate and efficient technique for numerical evaluation of singular 6-dimensional integrals over tetrahedon-product domains, with applications to calculation of Galerkin matrix elements for discretized…

Unstructured-mesh ocean models are increasingly used for coastal applications due to their ability to represent complex geometries and apply local grid refinement where needed. However, their broader use has been hindered by their high…

The goal of this paper is to develop a numerical algorithm that solves a two-dimensional elliptic partial differential equation in a polygonal domain using tensor methods and ideas from isogeometric analysis. The proposed algorithm is based…

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…

The Hilbert-space Gaussian Process (HGP) approach offers a hyperparameter-independent basis function approximation for speeding up Gaussian Process (GP) inference by projecting the GP onto M basis functions. These properties result in a…

Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…

Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical…

Hidden Markov models (HMMs) are general purpose models for time-series data widely used across the sciences because of their flexibility and elegance. However fitting HMMs can often be computationally demanding and time consuming,…

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for…