Krylov subspace recycling is a powerful tool when solving long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these appear naturally, as typically hundreds or thousands optimization steps are needed with small changes in the geometry. this setting, however, applying can be difficult task. As geometry evolves, general, so does fin...

A novel formulation of approximate truncated balanced realization (TBR) is introduced to unify three approaches: two iterative methods for solving the underlying Lyapunov equations – the alternating directions implicit (ADI) iteration and the rational Krylov subspace method (RKSM) – and a two-step procedure that performs a Krylov-based projection and subsequently direct TBR. The framework allow...

Many problems in scientific computing involving a large sparse matrix A are solved by Krylov subspace methods. This includes methods for the solution of large linear systems of equations with A, for the computation of a few eigenvalues and associated eigenvectors of A, and for the approximation of nonlinear matrix functions of A. When the matrix A is non-Hermitian, the Arnoldi process commonly ...

We show how the quasi-Newton least squares method (QN-LS) relates to Krylov subspace methods in general and to GMRes in particular.

In the numerical simulation of inviscid and viscous compressible fluid flow, implicit Newton-Krylov methods are frequently used. A crucial ingredient of Krylov subspace methods is the evaluation of the product of the Jacobian matrix of the spatial operator, e.g., fluxes, and a Krylov vector. In this article we consider a matrix-free implementation of the Jacobian-vector product within the flow ...

We compare two approaches to compute a fraction 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 applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism o...

The discretization of second-order elliptic stochastic partial differential equations in primal or mixed formulation, where the diffusion coefficient is modelled via a truncated Karhunen-Loève expansion, results in a sequence of linear systems. We employ Krylov subspace recycling methods which utilize information generated during the solution of a previous system in order to improve the converg...