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

A rational QZ method

We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom of selecting poles. In this article we study Hessenberg, Hessenberg pencils, link them to rational Krylov subspaces, propose a direct reduction method to such a pencil, and introduce the implicit rational QZ step. The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method. In the proofs, we operate directly on the pencil instead of rephrasing it all in terms of a single matrix. Numerical experiments are included to illustrate competitiveness in terms of speed and accuracy with the classical approach. Two other types of experiments exemplify new possibilities. First we illustrate that good pole selection can be used to deflate the original problem during the reduction phase, and second we use the rational QZ method to implicitly filter a rational Krylov subspace in an iterative method

Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt

The ParaOpt algorithm was recently introduced as a time-parallel solver for optimal-control problems with a terminal-cost objective, and convergence results have been presented for the linear diffusive case with implicit-Euler time integrators. We reformulate ParaOpt for tracking problems and provide generalized convergence analyses for both objectives. We focus on linear diffusive equations and prove convergence bounds that are generic in the time integrators used. For large problem dimensions, ParaOpt's performance depends crucially on having a good preconditioner to solve the arising linear systems. For the case where ParaOpt's cheap, coarse-grained propagator is linear, we introduce diagonalization-based preconditioners, inspired by recent advances in the ParaDiag family of methods. These preconditioners not only lead to a weakly-scalable ParaOpt version, but are themselves invertible in parallel, making maximal use of available concurrency. They have proven convergence properties in the linear diffusive case that are generic in the time discretization used, similarly to our ParaOpt results. Numerical results confirm that the iteration count of the iterative solvers used for ParaOpt's linear systems becomes constant in the limit of an increasing processor count. The paper is accompanied by a sequential MATLAB implementation

Frequency extraction for BEM-matrices arising from the 3D scalar Helmholtz equation

The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and H-matrix based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an H-matrix. We show that the frequency dependency of this H-matrix can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine