252 research outputs found
Monte Carlo Determination of Multiple Extremal Eigenpairs
We present a Monte Carlo algorithm that allows the simultaneous determination
of a few extremal eigenpairs of a very large matrix without the need to compute
the inner product of two vectors or store all the components of any one vector.
The new algorithm, a Monte Carlo implementation of a deterministic one we
recently benchmarked, is an extension of the power method. In the
implementation presented, we used a basic Monte Carlo splitting and termination
method called the comb, incorporated the weight cancellation method of Arnow
{\it et al.}, and exploited a new sampling method, the sewing method, that does
a large state space sampling as a succession of small state space samplings. We
illustrate the effectiveness of the algorithm by its determination of the two
largest eigenvalues of the transfer matrices for variously-sized
two-dimensional, zero field Ising models. While very likely useful for other
transfer matrix problems, the algorithm is however quite general and should
find application to a larger variety of problems requiring a few dominant
eigenvalues of a matrix.Comment: 22 pages, no figure
On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity
We consider generalized operator eigenvalue problems in variational form with
random perturbations in the bilinear forms. This setting is motivated by
variational forms of partial differential equations with random input data. The
considered eigenpairs can be of higher but finite multiplicity. We investigate
stochastic quantities of interest of the eigenpairs and discuss why, for
multiplicity greater than 1, only the stochastic properties of the eigenspaces
are meaningful, but not the ones of individual eigenpairs. To that end, we
characterize the Fr\'echet derivatives of the eigenpairs with respect to the
perturbation and provide a new linear characterization for eigenpairs of higher
multiplicity. As a side result, we prove local analyticity of the eigenspaces.
Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful
Monte Carlo sampling strategy for multiple eigenvalues and develop an
uncertainty quantification perturbation approach. Numerical examples are
presented to illustrate the theoretical results
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A Randomized Proper Orthogonal Decomposition Method for Reducing Large Linear Systems
The proper orthogonal decomposition (POD) method is a powerful tool for reducing large data systems which can quickly overwhelm modern computing tools. In this thesis we provide a link between randomized projections and statistical methods by introducing the randomized POD method. We also apply the POD method to a heat transfer finite element model and image compression. In doing so we demonstrate the practical use and quantify the error introduced by the POD method.Aerospace Engineerin
Subsampling Algorithms for Semidefinite Programming
We derive a stochastic gradient algorithm for semidefinite optimization using
randomization techniques. The algorithm uses subsampling to reduce the
computational cost of each iteration and the subsampling ratio explicitly
controls granularity, i.e. the tradeoff between cost per iteration and total
number of iterations. Furthermore, the total computational cost is directly
proportional to the complexity (i.e. rank) of the solution. We study numerical
performance on some large-scale problems arising in statistical learning.Comment: Final version, to appear in Stochastic System
Multiple Extremal Eigenpairs by the Power Method
We report the production and benchmarking of several refinements of the power
method that enable the computation of multiple extremal eigenpairs of very
large matrices. In these refinements we used an observation by Booth that has
made possible the calculation of up to the 10 eigenpair for simple test
problems simulating the transport of neutrons in the steady state of a nuclear
reactor. Here, we summarize our techniques and efforts to-date on determining
mainly just the two largest or two smallest eigenpairs. To illustrate the
effectiveness of the techniques, we determined the two extremal eigenpairs of a
cyclic matrix, the transfer matrix of the two-dimensional Ising model, and the
Hamiltonian matrix of the one-dimensional Hubbard model.Comment: 29 papes, no figure
A perturbation analysis of spontaneous action potential initiation by stochastic ion channels
A stochastic interpretation of spontaneous action potential initiation is developed for the Morris- Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently-developed quasi-stationary perturbation method. Using the fact that the activating ionic channel’s random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT
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