A periodic Krylov-Schur algorithm for large matrix products

Stewart's recently introduced Krylov-Schur algorithm is a modification of the implicitly restarted Arnoldi algorithm which employs reordered Schur decompositions to perform restarts and deflations in a numerically reliable manner. This paper describes a variant of the Krylov-Schur algorithm suitable for addressing eigenvalue problems associated with products of large and sparse matrices. It performs restarts and deflations via reordered periodic Schur decompositions and, by taking the product structure into account, it is capable of achieving qualitatively better approximations to eigenvalues of small magnitude

A domain decomposition strategy to efficiently solve structures containing repeated patterns

This paper presents a strategy for the computation of structures with repeated patterns based on domain decomposition and block Krylov solvers. It can be seen as a special variant of the FETI method. We propose using the presence of repeated domains in the problem to compute the solution by minimizing the interface error on several directions simultaneously. The method not only drastically decreases the size of the problems to solve but also accelerates the convergence of interface problem for nearly no additional computational cost and minimizes expensive memory accesses. The numerical performances are illustrated on some thermal and elastic academic problems

Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method

The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iteration, where the infinite Arnoldi method for a particular type of structured functions is appropriately restarted. The restarting exploits the structure and is inspired by the well-known implicitly restarted Arnoldi method for standard eigenvalue problems. The final algorithm is applied to examples from a benchmark collection, showing that both processing time and memory consumption can be considerably reduced with the restarting technique

Efficient computation of two-dimensional steady free-surface flows

We consider a family of steady free-surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli's equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally employed for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian-free Newton-Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface-piercing object and bottom topographies.Comment: 20 pages, 13 figures, under revie

Some numerical challenges in control theory

We discuss a number of novel issues in the interdisciplinary area of numerical linear algebra and control theory. Although we do not claim to be exhaustive we give a number of problems which we believe will play an important role in the near future. These are: sparse matrices, structured matrices, novel matrix decompositions and numerical shortcuts. Each of those is presented in relation to a particular (class of) control problems. These are respectively: large scale control systems, polynomial system models, control of periodic systems, and normalized coprime factorizations in robust control