Related papers: Efficient multi-level hp-finite elements in arbitr…
The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or adaptive refinement. We…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…
The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The…
The finite element method (FEM) is among the most commonly used numerical methods for solving engineering problems. Due to its computational cost, various ideas have been introduced to reduce computation times, such as domain decomposition,…
We describe a set of lower-level abstractions to improve performance on modern large scale heterogeneous systems. These provide portable access to system- and hardware-dependent features, automatically apply dynamic optimizations at run…
Modern GPGPUs provide massive arithmetic throughput, yet many scientific kernels remain limited by memory bandwidth. In particular, repeatedly loading precomputed auxiliary data wastes abundant compute resources while stressing the memory…
In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations posed on evolving hypersurfaces in $\mathbb{R}^d$, $d=2,3$. The method employs discontinuous piecewise…
Hardware accelerations of deep learning systems have been extensively investigated in industry and academia. The aim of this paper is to achieve ultra-high energy efficiency and performance for hardware implementations of deep neural…
Automated hyperparameter optimization (HPO) has gained great popularity and is an important ingredient of most automated machine learning frameworks. The process of designing HPO algorithms, however, is still an unsystematic and manual…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are…
Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently…
In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the…
We develop a method to compute $H^2$-conforming finite element approximations in both two and three space dimensions using readily available finite element spaces. This is accomplished by deriving a novel, equivalent mixed variational…
We introduce a high-performance virtual machine (VM) written in a numerically fast language like Fortran or C to evaluate very large expressions. We discuss the general concept of how to perform computations in terms of a VM and present…
In this work, we develop and analyze a higher-order finite element method for the multidimensional fragmentation equation. To the best of our knowledge, this is the first study to establish a rigorous, conforming finite element framework…
This book focuses on the use of algorithmic high-level synthesis (HLS) to build application-specific FPGA systems. Our goal is to give the reader an appreciation of the process of creating an optimized hardware design using HLS. Although…
In this paper we accomplish the development of the fast rank-adaptive solver for tensor-structured symmetric positive definite linear systems in higher dimensions. In [arXiv:1301.6068] this problem is approached by alternating minimization…
We consider a two-dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where…
We extend a distributed finite element method built upon model order reduction to arbitrary polynomial degree using a hybrid Nitsche scheme. The new method considerably simplifies the transformation of the finite element system to the…