Related papers: Infinite GMRES for parameterized linear systems
The GMRES algorithm of Saad and Schultz (1986) is an iterative method for approximately solving linear systems $A{\bf x}={\bf b}$, with initial guess ${\bf x}_0$ and residual ${\bf r}_0 = {\bf b} - A{\bf x}_0$. The algorithm employs the…
We consider an effective new method for solving trust-region and norm-regularization problems that arise as subproblems in many optimization applications. We show that the solutions to such subproblems effectively lie in a…
A new algorithm is presented for computing a canonical rank-R tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-R tensor consists of the sum of R rank-one components. Each…
Efficient algorithms for the sparse solution of under-determined linear systems $Ax = b$ are known for matrices $A$ satisfying suitable assumptions like the restricted isometry property (RIP). Without such assumptions little is known and…
We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse…
We present a complexity reduction algorithm for a family of parameter-dependent linear systems when the system parameters belong to a compact semi-algebraic set. This algorithm potentially describes the underlying dynamical system with…
We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of…
Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least…
This paper proposes a matrix-free residual evaluation technique for the hybridizable discontinuous Galerkin method requiring a number of operations scaling only linearly with the number of degrees of freedom. The method results from…
We compare two approaches to compute a portion of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale…
We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(\lambda)\mathbf{u}= 0$ on compact continua $\Omega \subset \mathbb{C}$. The method combines two key components: (1) a rational minimax…
In some cases, computational benefit can be gained by exploring the hyper parameter space using a deterministic set of grid points instead of a Markov chain. We view this as a numerical integration problem and make three unique…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
In this work, we introduce a highly efficient algorithm to address the nonnegative matrix underapproximation (NMU) problem, i.e., nonnegative matrix factorization (NMF) with an additional underapproximation constraint. NMU results are…
Performing Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances, as well as estimating model hyperparameters. Although direct matrix factorizations…
Modeling complex multiway relationships in large-scale networks is becoming more and more challenging in data science. The multilinear PageRank problem, arising naturally in the study of higher-order Markov chains, is a powerful framework…
In this work, the matrix-free solution of quasi-static phase-field fracture problems is further investigated. More specifically, we consider a quasi-monolithic formulation in which the irreversibility constraint is imposed with a…
We introduce a parallel algorithm to construct a preconditioner for solving a large, sparse linear system where the coefficient matrix is a Laplacian matrix (a.k.a., graph Laplacian). Such a linear system arises from applications such as…
We examine the problem of approximating a positive, semidefinite matrix $\Sigma$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$. This problem arises in sparse principal component analysis, where a decomposition of…
The Nonlinear GMRES (NGMRES) proposed by Washio and Oosterlee [Electron. Trans. Numer. Anal, 6(271-290), 1997] is an acceleration method for fixed point iterations. It has been demonstrated to be effective, but its convergence properties…