84 research outputs found
On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications
Pencils of Hankel matrices whose elements have a joint Gaussian distribution
with nonzero mean and not identical covariance are considered. An approximation
to the distribution of the squared modulus of their determinant is computed
which allows to get a closed form approximation of the condensed density of the
generalized eigenvalues of the pencils. Implications of this result for solving
several moments problems are discussed and some numerical examples are
provided.Comment: 30 pages, 16 figures, better approximations provide
A Shift Selection Strategy for Parallel Shift-invert Spectrum Slicing in Symmetric Self-consistent Eigenvalue Computation
© 2020 ACM. The central importance of large-scale eigenvalue problems in scientific computation necessitates the development of massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the world's largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing massive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions that will be explored in future work
Computational aspects and applications of a new transform for solving the complex exponentials approximation problem
Many real life problems can be reduced to the solution of a complex
exponentials approximation problem which is usually ill posed. Recently a new
transform for solving this problem, formulated as a specific moments problem in
the plane, has been proposed in a theoretical framework. In this work some
computational issues are addressed to make this new tool useful in practice. An
algorithm is developed and used to solve a Nuclear Magnetic Resonance
spectrometry problem, two time series interpolation and extrapolation problems
and a shape from moments problem.Comment: 28 pages, 20 figure
A new transform for solving the noisy complex exponentials approximation problem
The problem of estimating a complex measure made up by a linear combination
of Dirac distributions centered on points of the complex plane from a finite
number of its complex moments affected by additive i.i.d. Gaussian noise is
considered. A random measure is defined whose expectation approximates the
unknown measure under suitable conditions. An estimator of the approximating
measure is then proposed as well as a new discrete transform of the noisy
moments that allows to compute an estimate of the unknown measure. A small
simulation study is also performed to experimentally check the goodness of the
approximations.Comment: 42 pages, 5 figure
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