22,543 research outputs found
An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
An a posteriori verification method is proposed for the generalized
real-symmetric eigenvalue problem and is applied to densely clustered
eigenvalue problems in large-scale electronic state calculations. The proposed
method is realized by a two-stage process in which the approximate solution is
computed by existing numerical libraries and is then verified in a moderate
computational time. The procedure returns intervals containing one exact
eigenvalue in each interval. Test calculations were carried out for organic
device materials, and the verification method confirms that all exact
eigenvalues are well separated in the obtained intervals. This verification
method will be integrated into EigenKernel (https://github.com/eigenkernel/),
which is middleware for various parallel solvers for the generalized eigenvalue
problem. Such an a posteriori verification method will be important in future
computational science.Comment: 15 pages, 7 figure
Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems
We propose a verified computation method for partial eigenvalues of a
Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a
contour integral-type eigensolver, can reduce a given eigenproblem into a
generalized eigenproblem of block Hankel matrices whose entries consist of
complex moments. In this study, we evaluate all errors in computing the complex
moments. We derive a truncation error bound of the quadrature. Then, we take
numerical errors of the quadrature into account and rigorously enclose the
entries of the block Hankel matrices. Each quadrature point gives rise to a
linear system, and its structure enables us to develop an efficient technique
to verify the approximate solution. Numerical experiments show that the
proposed method outperforms a standard method and infer that the proposed
method is potentially efficient in parallel.Comment: 15 pages, 4 figures, 1 tabl
Localization theorems for nonlinear eigenvalue problems
Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function
that is analytic on some simply-connected domain \Omega \subset \bbC. A point
is an eigenvalue if the matrix is singular.
In this paper, we describe new localization results for nonlinear eigenvalue
problems that generalize Gershgorin's theorem, pseudospectral inclusion
theorems, and the Bauer-Fike theorem. We use our results to analyze three
nonlinear eigenvalue problems: an example from delay differential equations, a
problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure
Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices
Super-resolution is a fundamental task in imaging, where the goal is to
extract fine-grained structure from coarse-grained measurements. Here we are
interested in a popular mathematical abstraction of this problem that has been
widely studied in the statistics, signal processing and machine learning
communities. We exactly resolve the threshold at which noisy super-resolution
is possible. In particular, we establish a sharp phase transition for the
relationship between the cutoff frequency () and the separation ().
If , our estimator converges to the true values at an inverse
polynomial rate in terms of the magnitude of the noise. And when no estimator can distinguish between a particular pair of
-separated signals even if the magnitude of the noise is exponentially
small.
Our results involve making novel connections between {\em extremal functions}
and the spectral properties of Vandermonde matrices. We establish a sharp phase
transition for their condition number which in turn allows us to give the first
noise tolerance bounds for the matrix pencil method. Moreover we show that our
methods can be interpreted as giving preconditioners for Vandermonde matrices,
and we use this observation to design faster algorithms for super-resolution.
We believe that these ideas may have other applications in designing faster
algorithms for other basic tasks in signal processing.Comment: 19 page
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