An accelerated Newton algorithm based on Krylov subspaces is applied to solving nonlinear equations of structural equilibrium. The algorithm uses a low-rank least-squares analysis to advance the search for equilibrium at the degrees of freedom DOFs where the largest changes in structural state occur; then it corrects for smaller changes at the remaining DOFs using a modified Newton computation....

This paper presents two new algorithms to compute sparse solutions of large-scale linear discrete ill-posed problems. The proposed approach consists in constructing a sequence quadratic problems approximating an $\ell_2$-$\ell_1$ regularization scheme (with additional smoothing ensure differentiability at the origin) and partially solving each problem using flexible Krylov--Tikhonov methods. Th...

Given a pair of matrices and starting vectors, we present a procedure to generate the biorthonormal basis of the second-order right and left Krylov subspaces. The application is to solve the large-scale quadratic eigenvalue problems via oblique projection technique. This method can take full advantage of the sparseness of large-scale system as well as the superior convergence behavior of Krylov...

We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients. Karhunen-Loève expansions are used to represent the stochastic variables and the stochastic Galerkin method with double orthogonal polynomials is used to derive a sequence of uncoupled determi...

For a given nonderogatory matrix A, formulas are given for functions of A in terms of Krylov matrices of A. Relations between the coefficients of a polynomial of A and the generating vector of a Krylov matrix of A are provided. With the formulas, linear transformations between Krylov matrices and functions of A are introduced, and associated algebraic properties are derived. Hessenberg reductio...

High-dimensional data requires scalable algorithms. We propose and analyze three scalable and related algorithms for semi-supervised discriminant analysis (SDA). These methods are based on Krylov subspace methods which exploit the data sparsity and the shift-invariance of Krylov subspaces. In addition, the problem definition was improved by adding centralization to the semi-supervised setting. ...

We propose a block Krylov subspace version of the GCRO-DR method proposed in [Parks et al. SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented block Krylov subspace. We offer a clean derivation of the method and discuss methods of selecting recycling subspaces at restart as well as implementation decisions in the context of high...

A fully automatic macromodeling methodology to generate scalable reduced-order models for microelectromechanical systems is presented in this paper. Krylov subspace methods are used to generate reduced-order models from detailed higher-order models of the device under consideration. The entire methodology is implemented in a symbolic computation environment to preserve dependencies on physical ...

Since version 2.4 of the RKToolbox, the rat krylov function can simulate the parallel construction of a rational Krylov basis. This is done by imposing various nonzero patterns in a so-called continuation matrix T . Simply speaking, the j-th column of this matrix contains the coefficients of the linear combination of j rational Krylov basis vectors which have been used to compute the next (j + ...

In recent years, model order reduction techniques based on Krylov subspaces have become the methods of choice for generating small-scale macromodels of the large-scale multi-port RCL networks that arise in VLSI interconnect analysis. A difficult and not yet completely resolved issue is how to ensure that the resulting macromodels preserve all the relevant structures of the original large-scale ...