3,257 research outputs found
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Random matrix theory and symmetric spaces
In this review we discuss the relationship between random matrix theories and
symmetric spaces. We show that the integration manifolds of random matrix
theories, the eigenvalue distribution, and the Dyson and boundary indices
characterizing the ensembles are in strict correspondence with symmetric spaces
and the intrinsic characteristics of their restricted root lattices. Several
important results can be obtained from this identification. In particular the
Cartan classification of triplets of symmetric spaces with positive, zero and
negative curvature gives rise to a new classification of random matrix
ensembles. The review is organized into two main parts. In Part I the theory of
symmetric spaces is reviewed with particular emphasis on the ideas relevant for
appreciating the correspondence with random matrix theories. In Part II we
discuss various applications of symmetric spaces to random matrix theories and
in particular the new classification of disordered systems derived from the
classification of symmetric spaces. We also review how the mapping from
integrable Calogero--Sutherland models to symmetric spaces can be used in the
theory of random matrices, with particular consequences for quantum transport
problems. We conclude indicating some interesting new directions of research
based on these identifications.Comment: 161 pages, LaTeX, no figures. Revised version with major additions in
the second part of the review. Version accepted for publication on Physics
Report
Simulating Cosmic Microwave Background maps in multi-connected spaces
This article describes the computation of cosmic microwave background
anisotropies in a universe with multi-connected spatial sections and focuses on
the implementation of the topology in standard CMB computer codes. The key
ingredient is the computation of the eigenmodes of the Laplacian with boundary
conditions compatible with multi-connected space topology. The correlators of
the coefficients of the decomposition of the temperature fluctuation in
spherical harmonics are computed and examples are given for spatially flat
spaces and one family of spherical spaces, namely the lens spaces. Under the
hypothesis of Gaussian initial conditions, these correlators encode all the
topological information of the CMB and suffice to simulate CMB maps.Comment: 33 pages, 55 figures, submitted to PRD. Higher resolution figures
available on deman
Numerical Analysis
Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud
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In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud
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With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright
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