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We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our…
We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step…
Recent literature has advocated the use of randomized methods for accelerating the solution of various matrix problems arising throughout data science and computational science. One popular strategy for leveraging randomization is to use it…
Linear inverse problems are ubiquitous in various science and engineering disciplines. Of particular importance in the past few decades, is the incorporation of sparsity based priors, in particular $\ell_1$ priors, into linear inverse…
We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are…
We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many…
We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze…
We introduce a new general purpose multiresolution preconditioner for symmetric linear systems. Most existing multiresolution preconditioners use some standard wavelet basis that relies on knowledge of the geometry of the underlying domain.…
This study develops a framework for a class of constant modulus (CM) optimization problems, which covers binary constraints, discrete phase constraints, semi-orthogonal matrix constraints, non-negative semi-orthogonal matrix constraints,…
We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret…
We propose an augmented Lagrangian-based preconditioner to accelerate the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure such as those arising from mixed finite element…
Many works on inverse problems in the imaging sciences consider regularization via one or more penalty functions or constraint sets. When the models/images are not easily described using one or a few penalty functions/constraints, additive…
We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…
We construct and analyze a preconditioner of the linear elastiity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on discretization of a…
Estimating equations arise in a wide range of statistical applications, including longitudinal and clustered data analysis, survival analysis, econometrics, and semiparametric inference. In high-dimensional settings, adding…
Large linear systems are ubiquitous in modern computational science and engineering. The main recipe for solving them is the use of Krylov subspace iterative methods with well-designed preconditioners. Recently, GNNs have been shown to be a…
Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential…
We present preconditioning techniques to solve linear systems of equations with a block two-by-two and three-by-three structure arising from finite element discretizations of the fictitious domain method with Lagrange multipliers. In…
This paper considers the regularization continuation method and the trust-region updating strategy for the nonlinearly equality-constrained optimization problem. Namely, it uses the inverse of the regularization quasi-Newton matrix as the…
Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES)…