2,125 research outputs found
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
We present an efficient procedure for computing resonances and resonant modes
of Helmholtz problems posed in exterior domains. The problem is formulated as a
nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use
of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains.
We consider a variational formulation and show that the spectrum consists of
isolated eigenvalues of finite multiplicity that only can accumulate at
infinity. The proposed method is based on a high order finite element
discretization combined with a specialization of the Tensor Infinite Arnoldi
method. Using Toeplitz matrices, we show how to specialize this method to our
specific structure. In particular we introduce a pole cancellation technique in
order to increase the radius of convergence for computation of eigenvalues that
lie close to the poles of the matrix-valued function. The solution scheme can
be applied to multiple resonators with a varying refractive index that is not
necessarily piecewise constant. We present two test cases to show stability,
performance and numerical accuracy of the method. In particular the use of a
high order finite element discretization together with TIAR results in an
efficient and reliable method to compute resonances
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
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