124 research outputs found
A variational approach to modeling slow processes in stochastic dynamical systems
The slow processes of metastable stochastic dynamical systems are difficult
to access by direct numerical simulation due the sampling problem. Here, we
suggest an approach for modeling the slow parts of Markov processes by
approximating the dominant eigenfunctions and eigenvalues of the propagator. To
this end, a variational principle is derived that is based on the maximization
of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be
estimated from statistical observables that can be obtained from short
distributed simulations starting from different parts of state space. The
approach forms a basis for the development of adaptive and efficient
computational algorithms for simulating and analyzing metastable Markov
processes while avoiding the sampling problem. Since any stochastic process
with finite memory can be transformed into a Markov process, the approach is
applicable to a wide range of processes relevant for modeling complex
real-world phenomena
On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the Rayleigh-Ritz method projects the large quadratic
eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar
to the Rayleigh-Ritz method for the linear eigenvalue problem, the
Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP
with respect to the projection subspace. We analyze the convergence of the
method when the angle between the subspace and the desired eigenvector
converges to zero. We prove that there is a Ritz value that converges to the
desired eigenvalue unconditionally but the Ritz vector converges conditionally
and may fail to converge. To remedy the drawback of possible non-convergence of
the Ritz vector, we propose a refined Ritz vector that is mathematically
different from the Ritz vector and is proved to converge unconditionally. We
construct examples to illustrate our theory.Comment: 20 page
Spectral discretization errors in filtered subspace iteration
We consider filtered subspace iteration for approximating a cluster of
eigenvalues (and its associated eigenspace) of a (possibly unbounded)
selfadjoint operator in a Hilbert space. The algorithm is motivated by a
quadrature approximation of an operator-valued contour integral of the
resolvent. Resolvents on infinite dimensional spaces are discretized in
computable finite-dimensional spaces before the algorithm is applied. This
study focuses on how such discretizations result in errors in the eigenspace
approximations computed by the algorithm. The computed eigenspace is then used
to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff
distance between the computed and exact eigenvalue clusters are obtained in
terms of the discretization parameters within an abstract framework. A
realization of the proposed approach for a model second-order elliptic operator
using a standard finite element discretization of the resolvent is described.
Some numerical experiments are conducted to gauge the sharpness of the
theoretical estimates
Two-sided Grassmann-Rayleigh quotient iteration
The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a
pair of corresponding left-right eigenvectors of a matrix . We propose a
Grassmannian version of this iteration, i.e., its iterates are pairs of
-dimensional subspaces instead of one-dimensional subspaces in the classical
case. The new iteration generically converges locally cubically to the pairs of
left-right -dimensional invariant subspaces of . Moreover, Grassmannian
versions of the Rayleigh quotient iteration are given for the generalized
Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian
eigenproblem.Comment: The text is identical to a manuscript that was submitted for
publication on 19 April 200
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