Future of tensor computations: theory, algorithms and applications in many and few dimensions
Lecturer: E. E. Tyrtyshnikov
Matrix structures and applications
Lecturers: D. Bertaccini, C. Di Fiore, P. Zellini
Condensed forms for matrices under unitary similarities and congruences: theoretical problems and computational implications
Lecturer: K. Ikramov
A unified tight-frame approach for missing data recovery in images
Lecturer: R. H. Chan
Abstract
In many practical problems in image
processing, the observed data sets are often incomplete in the sense
that features of interest in the image are missing partially or
corrupted by noise. The recovery of missing data from incomplete data
is an essential part of any image processing procedures whether the
final image is utilized for visual interpretation or for automatic
analysis.
In this course, we start with an introduction of tight-frame. Then we present our tight-frame algorithm for missing data recovery. We illustrate how to apply the idea to different image processing applications such as: inpainting, impulse noise removal, super-resolution image reconstruction, video enhancement, and parallel MRI. We end by proving the convergence of the tight-frame algorithm.
Eigenvalues of large Toeplitz matrices: asymptotic approach
Lecturer: S. Grudsky
Abstract
The main goal of this course is to give an
introduction to the modern state of the art in the asymptotic analysis
of eigenvalues (and other spectral characteristics) of large Toeplitz
matrices. We discuss the finite section method and stability, Szegö’s
limit theorems, and asymtotics of determinants. We also embark on the
description of the limiting sets of the spectrum in some cases and on
the asymptotics of the extreme eigenvalues. Eventually we turn to
theorems about the asymptotics of individual eigenvalues and
eigenvectors in several important special cases. In conclusion we
formulate some open problems that might be of impact for further
research.
A detailed description of
the contents of the course and of the desiderable knowledge to
attend the lessons can be found here.
Place: Rome and Moscow
Place: Rome and Moscow
Place: Rome and Moscow
Place: Rome
Place: Moscow
The Rome-Moscow school of
Matrix Methods and Applied Linear Algebra
19th September - 3th
October 2010, University of Rome “Tor Vergata”
10th - 24th October 2010, Lomonosov Moscow State University
Courses
Abstract
General classes of spaces and algebras of nxn matrices, matrix projections and positive definite matrix projections, algebras of matrices not diagonalizable, diagonalizable, diagonalizable by unitary transforms; fast real unitary transforms; applications in defining efficient direct solvers of structured linear systems, and in improving the rate of convergence and in reducing the complexity per step of iterative schemes for solving n-dimensional structured and non structured minimization problems (image restoration, neural networks) or ordinary/partial differential problems (biomedicine).
The students do not need to
understand wavelets fully in order to understand the algorithm
presented. They will only have to remember that wavelet is an
orthonormal basis. It will take them a long time to understand wavelet
but then most of them materials in a wavelet book will not be needed. Actually something like
that in wikipedia will be more than sufficient.
Copies of the detailed power point slides used by the lecturer (pdf) (ppt)
Program of the
lecture course
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1.The Lanczos and Arnoldi algorithms, the conjugate gradient method, and GMRES. Matrix reduction to condensed forms via unitary similarities.
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2.Normal matrices: what are they good for? How the condensed form of a normal matrix under unitary similarities looks like? The generalized Lanczos algorithm. Some important particular cases. Methods for solving linear systems with normal coefficient matrices based on the constructed condensed form. Low-rank perturbations of normal coefficient matrices and their condensed forms under unitary similarities. Solving linear systems with coefficient matrices of this type.
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3.Tridiagonalization of a complex symmetric matrix via unitary congruences and the CSYM algorithm. Unitary congruences as a particular case of consimilarities. The basics of consimilarity theory: coneigenvalues, coneigenvectors, coninvariant subspaces, condensed and canonical forms. Semilinear matrix equations: solvability and uniqueness.
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4.Conjugate-normal matrices as counterparts to normal matrices with respect to unitary congruences. Some of their remarkable properties. Canonical forms. Sensitivity of coneigenvalues.
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5.Condensed form of a conjugate-normal matrix under unitary congruences. Methods for solving linear systems with conjugate-normal coefficient matrices based on this condensed form. Low-rank perturbations of conjugate-normal coefficient matrices and their condensed forms under unitary congruences. Solving linear systems with coefficient matrices of this type.
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6.Some open problems: criteria for unitary congruence, decomplexification of complex matrices via unitary similarities and congruences, simultaneous decomplexification of a pair of matrices.
The only
preliminary knowledge required is a standard course of linear algebra
including the spectral theory and Jordan form, although the familiarity
with the very basics of numerical linear algebra (such as the Lanczos
and Arnoldi algorithms, the conjugate gradient method, and GMRES) will
be helpful.
To a considerable extent, this lecture course is based on lecturer's research. PDF-files of the corresponding publications (in English) will be available to the audience.
The course will be
self-contained, and will require only standard math knowledge, as the
basics of linear algebra, calculus and a little bit of numerical
analysis.
Some writings (covering some topics but not everything) will be available.
Below is a list of the
main courses, relative abstracts and some references for preliminary
knowledges.
Note that further local expert researchers, in Rome and in Moscow, will hold seminars on topics related with the school theme. Link