Related papers: Multigrid with Linear Storage Complexity

The accurate assembly of the system matrix is an important step in any code that solves partial differential equations on a mesh. We either explicitly set up a matrix, or we work in a matrix-free environment where we have to be able to…

Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present…

Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…

Many of the most fundamental laws of nature can be formulated as partial differential equations (PDEs). Understanding these equations is, therefore, of exceptional importance for many branches of modern science and engineering. However,…

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable…

Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for partial differential equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the…

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix based approach within the Finite Element Method requires…

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that…

Fully connected multilayer perceptrons are used for obtaining numerical solutions of partial differential equations in various dimensions. Independent variables are fed into the input layer, and the output is considered as solution's value.…

Diffraction drastically limits the bit density in optical data storage. To increase the storage density, alternative strategies involving supplementary recording dimensions and robust read-out schemes must be explored. Here, we propose to…

Matrix-free finite element implementations for large applications provide an attractive alternative to standard sparse matrix data formats due to the significantly reduced memory consumption. Here, we show that they are also competitive…

In distributed optimization, a popular technique to reduce communication is quantization. In this paper, we provide a general analysis framework for inexact gradient descent that is applicable to quantization schemes. We also propose a…

The resource requirements of deep neural networks (DNNs) pose significant challenges to their deployment on edge devices. Common approaches to address this issue are pruning and mixed-precision quantization, which lead to latency and memory…

This work presents a matrix-free finite element solver for finite-strain elasticity adopting an $hp$-multigrid preconditioner. Compared to classical algorithms relying on a global sparse matrix, matrix-free solution strategies significantly…

Time-parallel algorithms seek greater concurrency by decomposing the temporal domain of a Partial Differential Equation (PDE), providing possibilities for accelerating the computation of its solution. While parallelisation in time has…

Multigrid methods have proven to be an invaluable tool to efficiently solve large sparse linear systems arising in the discretization of partial differential equations (PDEs). Algebraic multigrid methods and in particular adaptive algebraic…

An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and…

With the increasing number of components and further miniaturization the mean time between faults in supercomputers will decrease. System level fault tolerance techniques are expensive and cost energy, since they are often based on…

A parallel and nested version of a frequency filtering preconditioner is proposed for linear systems corresponding to diffusion equation on a structured grid. The proposed preconditioner is found to be robust with respect to jumps in the…

We consider the problem of representing, in a compressed format, a bit-vector $S$ of $m$ bits with $n$ 1s, supporting the following operations, where $b \in \{0, 1 \}$: $rank_b(S,i)$ returns the number of occurrences of bit $b$ in the…