Computational methodologies of bilinear forms and applications

Doctoral Dissertation uoadl:2918920 120 Read counter

Department of Mathematics
Library of the School of Science
Deposit date:
Roupa Paraskevi
Dissertation committee:
Μαριλένα Μητρούλη, Καθηγήτρια, Τμήμα Μαθηματικών, ΕΚΠΑ
Βασίλειος Δουγαλής, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Μιχαήλ Δρακόπουλος, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Δημήτριος Θηλυκός, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Σωτήριος Νοτάρης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Παναγιώτης Ψαρράκος, Καθηγητής, ΣΕΜΦΕ, ΕΜΠ
Ιωάννης Κολέτσος, Αναπληρωτής Καθηγητής, ΣΕΜΦΕ, ΕΜΠ
Original Title:
Μεθοδολογίες υπολογισμού διγραμμικών μορφών και εφαρμογές
Translated title:
Computational methodologies of bilinear forms and applications
The computation of bilinear forms is a mathematical problem with many applications. Specifically, they arise in network analysis in order to determine the importance of the nodes and the ease of travelling between the nodes of a graph.
They also arise naturally for the computation of parameters in some numerical methods for solving least squares and in Tikhonov regularization for solving ill-posed problems, etc.
Moreover, the action of a matrix function f(A) on a real vector of length p, i.e. the product f(A) b, often appears in applications which are originated from partial differential equations, in lattice quantum chromodynamics computations in chemistry and physics, in sampling from a Gaussian process distribution, and so forth.
If the matrix A is sufficiently large, the direct computation is not possible. Therefore, it is necessary to derive efficient estimates for these forms.
In the present PhD thesis, we derive estimates for the bilinear forms of the type xT f(A) y, where A is a given matrix with specific structure, x,y given vectors and f is a smooth function defined on the spectrum of the matrix A. The developed method based on an extrapolation procedure. Specifically, we generate estimates for the bilinear form xT A^{-1} y and vector estimates for the quantity f(A) b.
Furthermore, we adjust some numerical methods based on extrapolation and Gaussian quadrature rules for estimating the diagonal of matrix functions, without approximating the whole matrix f(A).
We are also concerned with the estimation and minimization of the generalized cross-validation (GCV) function by using a combination of an extrapolation procedure and a statistical approach.
Finally, we apply the generated estimates for the aforementioned quantities in problems which deal with network analysis, the solution of discrete ill-posed problems and the computation of the regression parameter of a statistical model.
Main subject category:
bilinear form, diagonal, f(A)b, GCV function, extrapolation
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