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Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems…

In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the…

Sparse matrices are favorable objects in machine learning and optimization. When such matrices are used, in place of dense ones, the overall complexity requirements in optimization can be significantly reduced in practice, both in terms of…

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz…

In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…

Numerous practical medical problems often involve data that possess a combination of both sparse and non-sparse structures. Traditional penalized regularizations techniques, primarily designed for promoting sparsity, are inadequate to…

Large sparse symmetric linear systems appear in several branches of science and engineering thanks to the widespread use of the finite element method (FEM). The fastest sparse linear solvers available implement hybrid iterative methods.…

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel…

This note addresses identification of the $A$-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge can be used to simplify the…

For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for…

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of…

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $\mathcal{H}$-matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that…

We introduce a universal sparse preconditioner that accelerates geometry optimisation and saddle point search tasks that are common in the atomic scale simulation of materials. Our preconditioner is based on the neighbourhood structure and…

We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric…

A sequence of approximations for the determinant and its logarithm of a complex matrixis derived, along with relative error bounds. The determinant approximations are derived from expansions of det(X)=exp(trace(log(X))), and they apply to…

In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a…

We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious…

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the…

In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries…

In this study, we propose the lopsided HSS (LHSS) iteration method for solving a class of complex symmetric indefinite systems of linear equations. This method employs an alternating iterative scheme, where each iteration entails solving…