In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av = λv. Our analysis is designed to apply to the case when A is large and sparse and where iterative methods are used to solve the shifted linear systems (A − σI)y = x which arise. We rst present a general convergence theory that is independent of the nature of the inexact solver used. Next we consider...

We present a novel monolithic divergence-conforming HDG scheme for linear fluid-structure interaction problem with thick structure. A pressure-robust optimal energy-norm estimate is obtained the semidiscrete scheme. When combined Crank--Nicolson time discretization, our fully discrete energy stable and produces an exactly divergence-free fluid velocity approximation. The resulting system, which...

We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. prove central limit theorem norms the residual vectors that are produced by conjugate gradient and MINRES algorithms when applied to wide class sample covariance matrices satisfying some standard moment conditions. The proof involves establishing four so-called spectral measure, implying, in particula...

Projected Krylov methods are full-space formulations of Krylov methods that take place in a nullspace. Provided projections into the nullspace can be computed accurately, those methods only require products between an operator and vectors lying in the nullspace. We provide systematic principles for obtaining the projected form of any well-defined Krylov method. Projected Krylov methods are math...

We study numerical methods for solving nonlinear elliptic eigenvalue problems which contain folds and bifurcation points. First we present some convergence theory for the MINRES, a variant of the Lanczos method. A multigrid-Lanczos method is then proposed for tracking solution branches of associated discrete problems and detecting singular points along solution branches. The proposed algorithm ...

Circulant preconditioners that cluster eigenvalues are well established for linear systems involving symmetric positive definite Toeplitz matrices. For these preconditioners rapid convergence of the preconditioned conjugate gradient method is guaranteed. Since circulant preconditioners can be applied quickly using the fast Fourier transform, preconditioned CG with circulant preconditioning is e...

An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ‖Ax−b‖2, with A being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation ATAx = ATb, so that the quantities ‖Ark‖ are monotonically decreasing (where rk = b−Axk is the re...