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Ett informationsblad från matematiska institutionen vid Linköpings universitet. Material till Lite Mat lämnas till Maud Lindström litemat@mai.liu.se senast torsdagar kl 12.00.
Vi har dessutom ett arkiv av gamla nummer.
Ett informationsblad från matematiska institutionen vid Linköpings universitet
v48 1998
Stabilitet för sobolevrum med randvärden noll.
Sammanfattning: Jag skall tala om ett gemensamt arbete med Tero Kilpeläinen, som inspirerats av resultat av Peter Lindqvist om ''egenvärden'' till p-laplaceoperatorn och dessas beroende av p. Inga speciella förkunskaper behövs.
Mapping properties of composition operators in Besov and Sobolev spaces.
Punktvisa interpolationsolikheter.
Sammanfattning: Jag kommer att diskutera nya interpolationsolikheter för derivator och potentialer.
Tillämpningar på Sobolevmultiplikatorer kommer att ges.
Lokal: MAI:s seminarierum Beurling.
Onsdagen den 25 november kl 13.15-14.30 talar Magnus Arnér, SAAB Trollhättan, om Statistikens historia.
Lokal: S26 (hus C)
Annica Isaksson
Tisdag den 1 december kl 13.15 talar Prof. Oleg Burdakov, University of Campinas, Brazil, om A greedy algorithm for the optimal basis problem and its applications.
Abstract:: The following problem is considered. Given m+1 points in which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form . This problem is present in, e.g., stable variants of the secant and interpolation methods, where it is required to approximate the Jacobian matrix of a nonlinear mapping f by using values of f computed at m+1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider the Hadamard condition number which is minimized to find an optimal combination of m pairs . We show that the problem is not NP-hard, but can be reduced to the minimum spanning tree problem, which is solved by the greedy algorithm in time. The complexity of this reduction is equivalent to one matrix-matrix multiplication, and according to the Coppersmith- Winograd estimate, is below for m=n. Applications of the algorithm to interpolation methods and to matrix balancing are discussed.
Lokal: MAI:s seminarierum Beurling
Rank-reduction algorithms in signal processing
A numerical analysts perspective
Modern iteration methods for linear eigenvalue problems.
Abstract: Many physical problems associated with stability lead to eigenvalue problems. Suppose we want to compute one or more eigenvalues and their corresponding eigenvectors of the n by n matrix A. In this context n may be rather large, say 100.000 or more. Several iterative methods are known: Jacobi's method, the power method, the method of Lanczos, Arnoldi's method, and Davidson's method. The latter method has been reported as being quite successful, most notably in connection with certain symmetric problems in computational chemistry. The success of the method seems to depend quite heavily on (strong) diagonal dominance of A. However, the method is unable to recognize the eigenvalues if A is a diagonal matrix. This made the method very suspicious in the numerical linear algebra community.
Some recent convergence results, as well as numerical experiments, have been reported in Saad's book on eigenvalue problems, but in spite of these relations the occasional success of the method was not well understood.
However, as we will show, Davidson's method has an interesting connection with an old and almost forgotten method of Jacobi (1846). This leads to another view on the method of Davidson, that helps us to explain the behaviour of the method. It also leads to new efficient algorithms for general matrices.
Båda seminarierna i Seminarierummet Beurling, ing 25, 1 tr.
Välkomna!
Lars Eldén
Fredagen den 27 november kl. 10.15 försvarar Eva Lundström sin avhandling med titeln Singular Value Computations for Toeplitz Matrices and Subspace Tracking.
Abstract: This thesis addresses the problem of computing the largest singular values and corresponding singular vectors of a Toeplitz matrix. These are often requested in signal processing and system identification to extract the signal from the noise.
Toeplitz matrices resemble sparse matrices in the sense that the matrix-vector multiplication can be computed efficiently. For Toeplitz matrices this can be done by using the FFT. These similarities make it reasonable to study methods developed for sparse matrices. We compare different algorithms, mainly based on the Lanczos procedure, for computing a few singular values. We give a theoretical review of the methods and perform numerical tests to compare their performance.
A Toeplitz matrix can easily be transformed into a Hankel matrix, which is symmetric. We derive a new algorithm for computing a few singular triplets of a complex symmetric matrix. The algorithm resembles the Lanczos procedure for Hermitian matrices, but computes the partial singular value decomposition directly. Theoretical results are deduced concerning singular value bounds and relations to the block Lanczos method and numerical experiments show the method to be powerful.
We also address the subspace tracking problem. This problem involves computing several partial singular value decompositions successively, and arises, for instance, when a system varies with time. A new Lanczos like algorithm is derived, which computes the partial eigendecomposition of matrices which differ by a rank-one matrix. The method explicitly utilizes the structure in the update and uses a modified version of the Implicitly Restarted Lanczos (IRL) to speed up the convergence. The method is compared to IRL numerically and shows good behavior, especially when the matrices have multiple or clustered eigenvalues.
The subspace tracking problem for more general updates can be solved by using the Newton method on the Grassman manifold. We give a theoretical review of this method and discuss implementation aspects such as how to solve the Sylvester equation involved.
Opponent är professor Henk van der Vorst, Utrecht Universitet, Holland.
Lokal: Planck, Fysikhuset.
Optimeringslära gästas av Prof. Krzysztof Kiwiel, System Research Institute, Polish Academy, Warsaw, Polen, under tiden 29/11-12/12.
Martin Joborn, doktorand i Optimeringslära, har tilldelats ett stipendium på 3000 kr av Ekonomiska klubben i Linköping. Stipendiet utgår till "förtjänstfull doktorand vid Linköpings universitet".
Beloppsgränsen för intern representation är höjd till 250 kr exkl. moms per person. Avdragsrätten för moms är 90 kr/person. Beloppsgränsen för extern representation är höjd till 400 kr exkl. moms per person. Avdragsrätten för moms är 90 kr/person. Dessutom kan vid extern representation måttfull förtäring av vin eller öl utan samband med annan förtäring räknas som enklare måltid.
Jag ber att få påminna om att bonuspoäng som genereras av tjänsteresor ej kan utnyttjas privat utan enbart i tjänsten.
Mer information om MAI finns på under MAIs hemsida
Material till Lite Mat lämnas till Maud Lindström senast
torsdagar kl 12.00.
Tel 013-281405, Fax 013-100746, Email: litemat@mai.liu.se
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Senast ändrad: Thu 2010-03-18; 18:16 MET