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A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In…

We discretize the Lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods. A priori and a posteriori error estimates are derived and numerically verified.

Quantum optimization algorithms hold the promise of solving classically hard, discrete optimization problems in practice. The requirement of encoding such problems in a Hamiltonian realized with a finite -- and currently small -- number of…

We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R.…

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling…

This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a…

We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner.…

A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…

In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular,…

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…

This paper addresses the efficient solution of linear systems arising from curl-conforming finite element discretizations of $H(\mathrm{curl})$ elliptic problems with heterogeneous coefficients. We first employ the discrete form of a…

This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by N\'ed\'elec finite element methods of fixed order on meshes…

Multitrace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. We…

For the solution of discrete ill-posed problems, in this paper a novel preconditioned iterative method based on the Arnoldi algorithm for matrix functions is presented. The method is also extended to work in connection with Tikhonov…

In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem…

Presented in this paper is a new sparse linear solver methodology motivated by multigrid principles and based around general local transformations that diagonalize a matrix while maintaining its sparsity. These transformations are…

This paper studies chance-constrained stochastic optimization problems with finite support. It presents an iterative method that solves reduced-size chance-constrained models obtained by partitioning the scenario set. Each reduced problem…

A study is presented on the convergence of the computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even types are assigned in two subdomains, and Schwarz iteration…

The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell…

We investigate iterative methods with randomized preconditioners for solving overdetermined least-squares problems, where the preconditioners are based on a random embedding of the data matrix. We consider two distinct approaches: the…