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Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load…
We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these…
Multigrid methods were invented for the solution of discretized partial differential equations in ordered systems. The slowness of traditional algorithms is overcome by updates on various length scales. In this article we discuss…
During the last decade, Neural Networks (NNs) have proved to be extremely effective tools in many fields of engineering, including autonomous vehicles, medical diagnosis and search engines, and even in art creation. Indeed, NNs often…
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and…
This paper provides a rounding-error analysis for two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the…
We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means…
In the present paper we concentrate on an important issue in constructing a good multigrid solver: the choice of an efficient smoother. We will introduce all-at-once multigrid solvers for optimal control problems which show robust…
Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the…
A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on…
The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to…
A multigrid framework is described for multiphysics applications. The framework allows one to construct, adapt, and tailor a monolithic multigrid methodology to different linear systems coming from discretized partial differential…
Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The…
We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Gr\"unwald difference in space leads to dense…
We propose an adaptive refinement algorithm to solve total variation regularized measure optimization problems. The method iteratively constructs dyadic partitions of the unit cube based on i) the resolution of discretized dual problems and…
A multigrid method is proposed for solving nonlinear eigenvalue problems by the finite element method. With this new scheme, solving nonlinear eigenvalue problem is decomposed to a series of solutions of linear boundary value problems on…
We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of…
We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements (CutFEM). Three of these unfitted finite element methods known from the literature are studied. All three…
Nonlinear regression has been extensively employed in many computer vision problems (e.g., crowd counting, age estimation, affective computing). Under the umbrella of deep learning, two common solutions exist i) transforming nonlinear…
The aim of this paper is to develop an algebraic multigrid method to solve eigenvalue problems based on the combination of the multilevel correction scheme and the algebraic multigrid method for linear equations. Our approach uses the…