173 research outputs found
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
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
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
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