MINISTERO
DELL'ISTRUZIONE, DELL'UNIVERSITÀ E DELLA RICERCA
DIPARTIMENTO PER L'UNIVERSITÀ, L'ALTA FORMAZIONE ARTISTICA,
MUSICALE E
COREUTICA E PER LA RICERCA SCIENTIFICA E TECNOLOGICA
PROGRAMMI DI RICERCA SCIENTIFICA DI RILEVANTE INTERESSE NAZIONALE
RICHIESTA DI COFINANZIAMENTO
(DM n. 30 del
12 febbraio 2004)
PROGRAMMA DI RICERCA  MODELLO A
Anno 2004  prot. 2004015437
Parte: I
1.1 Programma di Ricerca di tipo: interuniversitario
Area Scientifico Disciplinare: Scienze
Matematiche e informatiche
1.2 Titolo del
Programma di Ricerca
1.3 Abstract
del Programma di Ricerca
Large part of problems in scientific computing
is reduced to solving numerical linear algebra problems. The large
scale and
the complexity of the mathematical models turn into large amount of
data and
high complexity of the computations. General solution methods are not
practically applicable for their large cost, and it is demanding to
design and
analyze specific algorithms strongly based on the peculiarity of the
model.
These peculiarities, once restated in terms of the linear algebra
language, can
be read as structure and sparsity patterns of the matrices involved in
the
model.
Many patterns are recurrent in diverse scientific contexts and
applications. For
instance, band matrices are encountered in contexts like the numerical
treatment of boundary value problems, in difference equations, in
spline
interpolation; shift invariant properties encountered in signal and
image
processing, statistic analysis, Markov chains, lead to Toeplitz
matrices.
Multidimensional problems lead to multilevel structures and block
matrices.
Often the properties of the original model turn into structures which
are not
so evident, like displacement structure, local structures and
semiseparable or
quasiseparable matrices, which look like "general" matrices.
The analysis of structures and the design of specific algorithms for
solving
problems in numerical linear algebra has received a great attention and
has
lead to important advances in the last years. On one hand, important
theoretical advances have been reached in the study of theoretical
properties
of frequently encountered structured matrices, on the other hand the
design of
specific algorithms has allowed the effective solution of important
problems in
different applicative fields. For instance, algorithms for solving
matrix
equations and queueing models, based on Toeplitz computations, have
lead to
strong reductions of the cpu time in solving network problems in
engineering.
THE PROJECT:
This project is the continuation of a previous MIUR project 20022003,
its goal
is to enlarge the range of structured tools and applications under
investigation by expanding and integrating the results obtained in
20022003
(see http://bezout.dm.unipi.it). The aim of the project consists of
A Analysis of algebraic, analytic and computational properties of
classes of
structured matrices encountered in theoretical and in applied problems,
including linear and nonlinear structures. In particular, displacement
structures (Toeplitz, Hankel, Cauchy, Bezout, etc), semiseparable
matrices,
multilevel (multiindex) matrices, locally structured matrices,
parametric
systems.
B Design of fast and numerically reliable (stable and robust)
algorithms for
solving computational problems involving structured matrices, like
total least
squares, linear systems, eigenvalue computations. Adaptation of known
techniques like multigrid, PCG, GMRES, QR iteration, to specific
structures. Use
of structured tools for locally structured or even unstructured
(minimization)
problems.
C Application of the results to the solution of computational problems
from
scientific computing, including: image processing, nonlinear inverse
problems,
queueing models, matrix equations, integral and differential equations,
inverse
scattering problems, approximating polynomial zeros, computer aided
geometric
design, human metabolism models, information retrieval, neural networks.
D Implementation of the algorithms in a prototype code, numerical
comparisons
and validation.
STATE OF THE ART:
There are several international research groups which are very active
in the
field. The Italian group is actively working in the area since many
years. There
is a strong theoretical and algorithmic basis made up by many tools and
powerful techniques developed in the years by many researchers. There
are
different research lines where the interest is growing up at the
international
level. Some new lines stemmed from the previous MIUR project.
1.4 Durata del
Programma di Ricerca: 24
Mesi
1.5 Settori
scientificodisciplinari interessati dal Programma di Ricerca
·
MAT/08 
Analisi Numerica
·
MAT/07 
Fisica Matematica
·
INF/01 –
Informatica
·
MAT/02 –
Algebra
·
MAT/05 
Analisi Matematica
1.6 Parole chiave
STRUCTURED MATRICES ; TOEPLITZ MATRICES ;
PRECONDITIONING FOR STRUCTURED MATRICES ; WIENERHOPF FACTORIZATION ;
SEMISEPARABLE MATRICES ; IMAGE RESTORATION ; POLYNOMIAL COMPUTATIONS ;
NUMERICAL METHODS FOR MARKOV CHAINS ; MATRIX EQUATIONS
1.7
Coordinatore Scientifico del Programma di Ricerca
DARIO ANDREA BINI,
Professore ordinario,
Dipartimento di
Matematica "Leonida Tonelli", Università di PISA,
Facoltà di Sienze
Matematiche Fisiche e Naturali bini@dm.unipi.it
1.8 Curriculum
scientifico
Dario Bini is full
professor of numerical analysis since 1986. He is author of more than
70
papers, published in international journals and in proceedings of
international
conferences, and of several books in numerical analysis and
computational
mathematics. He is member of the editorial board of the international
journal
"Calcolo", associate editor of SIAM J. Matrix Anal. Appl., and has a
wide and long expertise in the research fields of structured matrix
computations, design and analysis of numerical algorithms and
polynomial
computations. He has organized international conferences on these
research
topics. He has been advisor of several PhD thesis. His main research
interests
include, Toeplitz matrix computations, displacement operators,
polynomial
computations, numerical solution of Markov chains, solution of matrix
equations.
1.9 Pubblicazioni scientifiche
più
significative del Coordinatore del Programma di Ricerca
1.
BINI D.A.;
BOETTCHER A (2003). Polynomial
factorization through Toeplitz matrix computations LINEAR ALGEBRA
AND ITS
APPLICATIONS. (vol. 366 pp.
2537)
2.
BINI D.A.; L.
GEMIGNANI; B. MEINI (2003). Solving
certain matrix equations by means of Toeplitz computations: algorithms
and
applications CONTEMPORARY MATHEMATICS. (vol. 323 pp. 151167)
3.
BINI D.A.; G.
LATOUCHE; B. MEINI (2002). Solving
matrix polynomial equations arising in queueing problems LINEAR
ALGEBRA AND
ITS APPLICATIONS. (vol. 340
pp. 225244)
4.
BINI D.A.; L.
GEMIGNANI; B. MEINI (2001). Computations
with infinite Toeplitz matrices and polynomials LINEAR ALGEBRA AND
ITS
APPLICATIONS. (vol. 343344
pp. 2161)
5.
BINI
D.A.; MEINI B. (1999). Effective methods for solving banded
Toeplitz
systems SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. (vol. 20
pp.
700719)
1.10 Elenco delle
Unità di Ricerca
Nº 
Responsabile scientifico 
Qualifica 
Settore 
Università 
Dipart./Istituto 
Mesi 
1. 
Prof. ordinario 
MAT/08 
PISA 
MATEMATICA "Leonida Tonelli" 
16 

2. 
Prof. associato 
MAT/08 
GENOVA 
MATEMATICA 
14 

3. 
Prof. ordinario 
MAT/08 
CAGLIARI 
MATEMATICA 
12 
1.11 Mesi uomo
complessivi dedicati al programma


Numero 
Mesi uomo 
Mesi uomo 
Totale mesi uomo 
Personale
universitario dell'Università sede dell'Unità di Ricerca 
14 
83 
73 
156 

Personale
universitario di altre Università 
10 
69 
67 
136 

Titolari di
assegni di ricerca 
1 
9 
7 
16 

Titolari di borse 
Dottorato 
1 
8 
8 
16 
Postdottorato 
0 




Scuola di
Specializzazione 
0 




Personale a
contratto 
Assegnisti 
1 
9 
7 
16 
Borsisti 
0 




Dottorandi 
0 




Altre
tipologie 
1 
4 
6 
10 

Personale
extrauniversitario 
2 
8 
8 
16 

TOTALE 

30 
190 
176 
366 
Parte:
II
2.1 Obiettivo del Programma di Ricerca
The project is based on the following guidelines and
underlying ideas:
·
Structured
matrices play a fundamental role in a great part of mathematical models
in
theory and applications.
·
The analysis
of structured matrices, besides its theoretical interest, is a
fundamental step
for the design of specific solution algorithms, where general
algorithms fail
for their large cost.
·
The
structured matrix technology,
developed along the years, can be used to effectively solve large scale
problems of great importance in the applications, which could be hardly
solved
with general algorithms.
·
The structured
matrix technology may provide tools for solving apparently unstructured
problems.
·
The
design and the analysis of
algorithms taylored on the specific structures is not enough in itself.
Great
attention must be payed to robustness and numerical stability of
algorithms.
Therefore, at the basis of this project there is the analysis of
theoretical
and computational properties of matrix structures in order to improve
the
knowhow in structured matrix technology. The aim is designing highly
efficient
and reliable algorithms for solving computational problems from
different areas
in theoretical and in scientific computing.
For the sake of clarity, the program is ideally subdivided in the
following
(possibly overlapping) parts:
A To perform the analysis of theoretical and computational properties
of
structured matrices.
B To devise structured matrix tools for designing efficient solution
algorithms for the most important computational problems.
C To apply the new algorithms to problems coming from applications.
D To implement, in terms of prototype software, the algorithms
designed in
this research and to perform experimental validation and comparison.
The topics which will be considered are problems in Structured
Numerical Linear
Algebra (SNLA) which are the natural continuation of the issues of
major
interest investigated in the previous MIUR project 20022003 titled
"Structured matrix analysis: numerical methods and applications", or
problems, which although not in the program 20022003, have shown to be
particularly interesting in themselves and for their applications. In
fact, in
the new project more interest is addressed to applications. Applicative
topics
are a source of very interesting theoretical problems and of structured
matrix
classes, and, at the same time they constitute an important benchmark
where to
test the effectiveness of the algorithms designed with SNLA tools.
The class of matrices which we consider in this research are Toeplitz
and
displacement structured matrices (including Hankel, Cauchy, Pick,
Vandermonde,
Sylvester, Bezout, Frobenius, generalized companion matrices), locally
Toeplitz, banded, semiseparable and quasiseparable matrices,
multiindex
(multilevel) matrices, sequences of shifted matrices and their
parametric
generalizations, trigonometric algebras.
The computational problems include solving linear systems, computing
eigenvalues/eigenvectors, solving matrix equations, solving total least
squares
problems, minimization problems, computing matrix factorizations and
WienerHopf factorization, polynomial computations.
Computational issues encountered in this study which will be
investigated are
the analysis of preconditioners, algebraic multigrid techniques, matrix
factorizations, analysis of Krylov sequences, QR iteration,
regularization
problems, projection methods, numerical conditioning and numerical
stability
issues, integration of symbolic and numeric tools, interplay between
polynomial
and structured matrix computations.
The main application areas involve problems in image processing as LBT
imaging
and image restoration, queueing problems modeled by Markov chains of
the G/M/1,
M/G/1 and QBD type, acoustics and inverse scattering problems, computer
aided
geometric design, differential models of human metabolism, search
engines in
the web, neural networks and optimal learning, integral equations.
More precisely, the classes of structured matrices and the related
problems
which are studied in parts A,B,C,D are outlined below. For a more
detailed
description we refer to section 2.4 and to the "Modello B" of each
research unit.
A1 Toeplitz and locally Toeplitz matrices
A2 Semiseparable and quasiseparable matrices
A3 Parametric matrices
A4 New matrix algebras related to trigonometric transforms
A5 Generalized companion and resultant matrices
A6 Block Toeplitz matrices
A7 Multilevel (multiindex) matrices
A8 Infinite Toeplitz matrices
A9 Nonconventional structures
The computational problems in part B will be:
B1 Linear systems and total least squares problems
B2 Eigenvalue problems
B3 Functional approximation
B4 Inverse problems and regularization
B5 PDEs
B6 Multigrid
B7 Preconditioning
B8 Polynomial computations
B9 Matrix equations
B10 WienerHopf factorization
B11 Minimization problems
B12 Integral equations with structured kernels
B13 GMRES and Krylov methods
B14 Projection methods in weighted Wiener algebras
The Applications in part C will be:
C1 Web search engines
C2 Human metabolism models
C3 Image restoration
C4 Inverse scattering, nonlinear inverse problems, acoustics and
remote
sensing
C5 Identification problems
C6 Markov chains and queuing models
C7 Computer Aided Geometric Design
C8 Neural networks
C9 Astronomy problems
Concerning part D
D1 Test matrices
D2 MatLab Toolbox
D3 Prototype software
Finally, one aim of the project is the organization of the fifth
edition of the
International Workshop
Matrix Analytic Method in Stochastic Models MAM5, Pisa, June 2005
(http://www.dm.unipi.it/~mam5).
The conference, which covers topics in applied probability, engineering
and
numerical analysis, is an international forum where the state of the
art in the
field of numerical methods in applied probability will be traced and
discussed.
2.2 Base di
partenza scientifica nazionale o internazionale
Many important problems in pure and applied mathematics
involve classes
of structured matrices which are frequently encountered in different
contexts. Matrix
structures are the algebraic translation of the specific properties of
the
problem. General properties as the shift invariance, the compact
support, the
separability of variables, which are shared by many problems in
different areas
turn into matrix structures that are almost pervasive like Toeplitz
matrices,
band matrices, semiseparable and quasiseparable matrices. They are
encountered in many different problems in numerical analysis,
statistics,
probability, computer algebra, signal and image processing, acoustics,
astronomy, just to quote a few. Other related structures like Hankel,
Vandermonde, Cauchy, Pick, Bezout Sylvester matrices are not less
relevant.
The analysis of matrix structure is extremely important in order to
devise
tools for designing fast and reliable solution algorithms for many
important
problems which, for their large size, hardly could be solved by means
of
general algorithms.
The interest on structured matrices, already alive in the 60's [GS] and
in the
70's [Tr], [GoS] with the direct methods for Toeplitz systems and
implicitly
with the fast Poisson solvers [BGN], has increased much its importance
in the
years due to the work of several very active international research
groups. A
wide literature has been consolidated on different related mathematical
problems and many international conferences specially devoted to
structured
matrix analysis and its applications have been organized.
The theoretical and algorithmic basis on which this project relies is
very wide
and rich of results and of tools.
Historically, the main advances in the field of structured matrix
analysis are
reported in several books, some of which are milestones in this field.
Here for
space reason we may cite only a few of them: the books of Iohvidov [I]
and of
Heinig and Rost [HR] on Toeplitz and Hankel structures, the book of
Bultheel
and Van Barel [BvB] on the relations between structures and orthogonal
polynomials, the book of Bini and Pan [BP] on the relations between
structures
and polynomial computations, the books of Grenander and Szego [GS] and
of
Boettcher and Silbermann [BS] on asymptotic spectral properties of
Toeplitz
matrices, the books of Gohberg, Goldberg and Kaashoek [GGK] and of
Gohberg
Lancaster and Rodman [GLR] and the book series Operator Theory and
Applications
by Birkhauser, on more theoretical issues, The book [DV] of Dewilde and
van der
Veen related to semiseparable matrices, the books edited by Kailath
and Sayed
[KS], by Bini, Tyrtyshnikov and Yalamov [BTY], and by V. Olshevsky [O]
which
contain recent advances in the field, and the review paper by Chan and
Ng [CN]
on Toeplitz preconditioning.
The theory of displacement operators by Kailath et al [KS] and the
large
literature originated around it constitute an important and solid
basis. We
recall the distributional results pioneered by Szego (see [BS]) and
culminated
in the work of Tyrtyshnikov [TZ] concerning the "global behaviour" of
the spectra of Toeplitz sequences. Extensions to matrix valued
generating
functions and to sequences arising in the preconditioning theory as
well as
some applications to approximation and quadrature, can be found in
literature. A
second important extension concerns the case of locally Toeplitz
matrices,
i.e., structures which are of Toeplitz type "only locally" or that
possess different "local structures" [Ti] and including, as special
instance, discretizations of differential operators with boundary
conditions.
Asymptotic spectral properties play an important role in the design of
preconditioners
for conjugate gradient. The circulant class associated with the
discrete
Fourier transform which Strang [St] proposed first in 1986, has been
complemented with other trigonometric matrix algebras. A systematic
treatment
of this topic has been performed by many researchers like T.Chan,
R.Chan,
S.Serra, F. DiBenedetto, G.Fiorentino, E.Tyrtyshnikov, T.Huckle, Potts,
Steidl,
Nagy, Plemmons, in the onelevel and multilevel case. Other
trigonometric
algebras and related classes have been studied by Kailath, Olshevsky,
and by
the Italian group.
The GMRES, Lanczos and conjugate gradient methods are important tools
in this
project. To this regard there exists a lot of consolidated literature
in many
paper and classic books.
For multiindex Toeplitz matrices there is not a lot of literature
because of
the comparative nonexistence of an algebraic theory of multiindex
matrix
operators, in spite of the objective interest in the subject in many
applied
fields. Contributions in this direction are given by the group of van
der Mee,
Rodriguez, Seatzu, Ehrardt. Exclusively in the scalar case, two
spectral
factorization methods in weighted Banach algebras with respect to a
fixed total
order have been developed. In the multiindex block Toeplitz case there
only
exist partial results on spectral factorization.
An interesting extension of the band extension method that seems
promising for
solving multiindex systems has recently been obtained in [GW].
Matrix structures play an important role in many polynomial
computations where
several problems related to polynomial and rational approximation can
be described
in terms of structured matrices, including Cauchy, Pick, Vandermonde,
locally
Toeplitz and semiseparable. We cite the book [BP] and the wide existing
literature with the contributions of Bini, Boettcher, Gemignani, Meini,
Fiorentino. Concerning the computation of WienerHopf factorization,
Laurent
polynomials and biinfinite Toeplitz matrix inversion, we refer to
[BGM1, GMRS,
MRS]. The interplay between numerical and symbolic computations is
important in
this context. To this regard the European project FRISCO (FRamework for
the
Integration of Symbolic and numeric Computing) constitutes a solid
basis for
our research, a nice integration of numeric and symbolic tools is made
in
[CGTW].
Semiseparable and quasiseparable matrices have a particular role in
our
project. There is a wide and solid literature on this subject, besides
the
classical results on semiseparable structures of Gohberg, Eidelman,
Dewilde,
Chandrasekaran, Ming Gu, recently some interesting results have been
obtained
by the group of Van Barel and by the Italian group. In particular in
[VBM1]
interesting similarity transformations into semiseparable form are
shown, in
[BGT] the properties of semiseparable matrices are used for the design
of
effective tridiagonal eigenvalue solvers.
Another structure arising in several frameworks (image processing,
timedependent PDEs, iterative eigensolvers, etc.), concerns shifted
linear
systems as (A+eI)x=b. Sometimes I is replaced by a more general
diagonal shift
[BeBe].
Finally, application of structured matrix analysis to unstructured
minimization
problems and total least squares has recently shown to be very
effective see
[DFLZ,LMV1] where some applications to automatic learning and speech
compression are shown.
Concerning other applications:
Numerical methods for Markov chains is a field where recently the
structured
matrix technology has produced great advances. We refer to the books
[Ne], [LR]
for the description of the problems and to the papers by Bini, Meini
and
Latouche, for a sample of algorithms designed on Toeplitz tools. In
particular
the cyclic reduction algorithm designed in [BGN] has recently revealed
a
powerful tool also in the field of M/G/1 and G/M/1 Markov chains [BM3].
Another
important problem strictly related to queueing models, encountered in
many
other applications where structured matrices play an important role,
concerns
the solution of nonlinear matrix equations for which there exists a
wide
literature [HK,BGM2].
A very recent imaging device that requires the solution of challenging
reconstruction problems is the Large Binocular Telescope (LBT) [AHSS].
All the
numerical methods used in image reconstruction must take into account
that some
regularization approach is needed. The reconstruction obtained by a
basic
method can be further improved with the help of two advanced tools:
boundary
conditions (see [J], [NCT] for classical proposals and [S3] for the new
antireflective idea) and wavelets [CCSS].
Various applied problems stemming from astronomy such as, in
particular, the
study of light transfer in planetary atmospheres lead to the treatment
of
various types of matrices with nonconventional structure. Some of these
have
been studied in [HMD]. A large amount of research on inverse
scattering, in
particular, acoustics, optics and electromagnetism strictly depends on
the
solution of integral equations with structured kernel and their
discrete
analogs. The results on the functional analysis and operator theory
most
relevant to the project are to be found in the book series Operator
Theory and
Applications by Birkhauser.
Many new contributions have been recently given in the MIUR project
20022003
(see the web page http://bezout.dm.unipi.it) of which this proposal is
a
continuation, and of which we report below the list of the main papers
with the
most important achievements. These are in the fields of spectral and
computational analysis of wide classes of structured matrices including
Toeplitz, locally Toeplitz and semiseparable matrices, of the
regularization of
inverse problems arising in several applications, of applied areas
including
partial differential equations, image processing and functional
approximation,
in the field of polynomial computations and CAGD, of Markov chains and
queuing
problems, in matrix equations, in minimization problems, least squares
and
applications to neural networks and speech compression, on multiindex
matrices
and on spectral factorization, on ill conditioned systems.
Many of these results have been presented at numerous conferences
organized in
this area, among which the Fourteenth IWOTA Conference, on Operator
Theory and
Applications, Cagliari, June 2003, whose proceedings will be published
in the
Birkhauser OT series, edited by Kaashoek, van der Mee and S. Seatzu.
The members of this research project have a long tradition in
structured matrix
analysis with a strong basic expertise and a different specialization
in
diverse fields. The expertise of the members of the Units includes
theoretical
and computational issues, design and analysis of algorithms and
software, and
applications in different fields like image processing, differential
and
integral equations, queueing models and Markov chains, polynomial
computations
and computer algebra.
[ADS] A.Aricò, M.Donatelli, S.Serra, Multigrid optimal
convergence for certain
(multilevel) structured linear systems. SIMAX, in press.
[BBCR] M.Bertero, P.Boccacci, A.Custo, M.Robberto, A Fourierbased
method for
the restoration of chopped and nodded images, Astron. Astroph. 406,2003
[BCV] D.Bini,G.Codevico,M.Van Barel, Solving Toeplitz Least Squares
Problems by
Means of Newton's Iteration, Num.Algo.,33,2003
[BDFZ] A.Bortoletti C.Di Fiore S.Fanelli P.Zellini, A new class of
quasinewtonian methods for optimal learning in MLPnetworks, IEEE
Trans.Neur.Netw. 14,2003
[BDG] D.Bini F.Daddi L.Gemignani. On the shifted QR iteration applied
to
Frobenius matrices, Dip.Mat. Univ.Pisa 2003
[BeBe] M.Benzi,D.Bertaccini, Approximate inverse preconditioning for
shifted
linear systems. BIT 43,2003
[BG1] D.Bini L.Gemignani. BernsteinBezoutian matrices. Theor.Comp.Sci
2004.
[BG2] D.Bini L.Gemignani. Solving quadratic matrix equations and
factoring
polynomials: new fixed point iterations based on Schur complements of
Toeplitz
matrices. Numer.Lin.Alg.Appl.
2004.
[BGP] D.Bini L.Gemignani V.Pan. Inverse Power and DurandKerner
Iterations for
Univariate Polynomial RootFinding, Comp.Math.Appl.
[BFGM] D.Bini, G.Fiorentino L.Gemignani B.Meini. Effective fast
algorithms for
polynomial spectral factorization. Num.Algo.
34,2003
[BGM2] D.Bini L.Gemignani B.Meini, Solving certain matrix equations by
means of
Toeplitz computations: algorithms and applications. Cont.Math.323, 2003
[BGW] D.Bini L.Gemignani J.Winkler, Structured matrix methods for CAGD,
Dip.Mat. Univ. Pisa 2003
[BLM] D.A.Bini G.Latouche B.Meini, Solving nonlinear matrix equations
arising
in treelike stochastic processes,Lin.Alg.Appl. 366, 2003
[BM] D.Bini, B.Meini. Nonskipfree M/G/1 type Markov chains and
Laurent matrix
power series, Fourth Int.Conf. Num. Sol. of Markov Chains, vol. 1, 2003
[BN] D.Bertaccini M.Ng, BandToeplitz Preconditioned GMRES Iterations
for
timedependent PDEs. BIT 43,2003
[D] C.DiFiore, Structured matrices in unconstrained minimization
methods, Cont.
Math. 323,2003
[DiB] F.DiBenedetto, The mth difference operator applied to L2
functions on a
finite interval Lin.Alg.Appl. 366,2003
[DLZ] C.DiFiore F.Lepore P.Zellini, Hartleytype algebras in
displacement and
optimization strategies, Lin.Alg.Appl.366,2003
[ET] C.Estatico A.Tamasan, A Newton linearization approach for solving
the
atmospheric retrieval problem. Tech. Rep., UCLA, 2003
[FG1] D.Fasino L.Gemignani. Fast and stable solution of
bandedplussemiseparable linear systems. CALCOLO 39,2002
[FG3] D.Fasino L.Gemignani Direct and inverse eigenvalue problems for
diagonalplussemiseparable matrices, Num.Algo.34,2003
[FMB] D.Fasino N.Mastronardi M.Van Barel, Fast and stable algorithms
for
reducing diagonal plus semiseparable matrices to tridiagonal and
bidiagonal
form. Cont.Math. 323,2003
[FMV] D.Fasino N.Mastronardi M.VanBarel Fast and Stable Algorithms for
Reducing
Diagonal plus Semiseparable Matrices to Tridiagonal and Bidiagonal
Form,
Cont.Math. 323,2003
[FS] D.Fasino, S.Serra, From Toeplitz matrix sequences to zero
distribution of
orthogonal polynomials, Cont.Math. 323,2003
[GL] L.Gemignani G.Lotti. Efficient and stable solution of Mmatrix
linear
systems of (block) Hessenberg form. SIMAX 24,2003
[HMD] J.W.Hovenier C.van der Mee H.Domke. Transfer of polarized light
in
planetary atmospheres. Basic concepts and practical methods. Kluwer,
Dordrecht,
2004,to appear
[Ia] B.Iannazzo, A note on computing the matrix square root. CALCOLO,
2004.
[LMV1] P.Lemmerling,N.Mastronardi,S.VanHuffel, Efficient implementation
of a
structured total least squares based speech compression method,
Lin.Alg.Appl.
366,2003
[M4] B.Meini, The matrix square root from a new functional perspective:
theoretical results and computational issues , SIMAX to appear
[MNS] C.van der Mee M.Z.Nashed S.Seatzu, Sampling expansions and
interpolation
in unitarily translation invariant reproducing kernel Hilbert spaces,
Adv.Comp.Math.,19,2003.
[MRS1] C.van der Mee G.Rodriguez S.Seatzu. Semiinfinite multiindex
perturbed
block Toeplitz systems. Lin.Alg.App. 366,2003
[MS] C.van der Mee S.Seatzu. A method for generating infinite positive
definite
selfadjoint test matrices and Riesz bases. Preprint 2004
[NSV] D.Noutsos, S.Serra, P.Vassalos, Spectral equivalence and matrix
algebra
preconditioners for multilevel Toeplitz systems: a negative result,
Cont.Math.,
323,2003
[S3] S.Serra, A note on antireflective boundary conditions and fast
deblurring
models. SIAM J.Sci.Comp. 25,2003
[S4] S.Serra, Generalized Locally Toeplitz sequences: spectral analysis
and
applications to discretized Partial Differential equations,
Lin.Alg.Appl.
366,2003
[ST1] S.Serra, C.Tablino Possio, Analysis of preconditioning strategies
for
collocation linear systems. Lin.Alg.App.369,2003
[ST2] S.Serra, C.Tablino Possio, Superlinear preconditioners for Finite
Differences linear systems. SIMAX 25,2003
[STy1] S.Serra E.Tyrtyshnikov, How to prove that a preconditioner can
not be
superlinear, Math.Comp.72 2003
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2.3 Numero di
fasi del Programma di Ricerca: 1
2.4
Descrizione del Programma di Ricerca
Fase 1
Durata: 24 mesi Costo
previsto: 187.000 Euro
Descrizione:
Several collaborations already
exist among the different Research Units, throughout denoted as
U1,U2,U3. In
order to better articulate the research activity in a more organic way
and to
create synergies, we will form a coordination committee formed by the 3
responsibles of the Research Units. This committee will meet
periodically to
trace the state of the art of the research, to exchange information
about the
achievements and to program further meetings among the units.
At the beginning of the project we plan to make an initial meeting of
the
researchers of all the units in order to coordinate the activity. A
workshop is
planned after about one year to trace the state of the art of the
project. A
final workshop is planned at the end where the results obtained in the
project
are presented. The latter workshop is opened to external researchers,
belonging
to national and international research groups.
A strategic line of the project is to promote the relations between the
more
applicative aspects (see parts C and D of section 2.1) and the
theoretical
issues (part A), by supporting and improving the interactions between
the
researchers who have more experience in the applications with the ones
who have
a long expertise in structured numerical linear algebra. This will
create
synergies and benefits for the whole project. In fact, we expect that,
on one
hand this strategy will favor the use and the application of
theoretical and
computational tools to the solution of specific problems with the
consequent
development of effective software and with the substantial reduction of
the
computational costs. On the other hand, this strategy will allow the
researchers to more deeply investigate specific applicative problems
and,
consequently, to favor the development of new theoretical and
computational
tools of general use for structured matrix analysis. This strategy will
be
implemented by periodically organizing short workshops on very specific
subjects of interest which, if needed, will be opened to researchers of
international groups who can be invited by the single Research Units.
Therefore
we plan frequent contacts of researchers of the different units and
frequent
contacts with researchers of other international research groups to be
obtained
through short visits. An international conference on the specific
subject of
numerical methods for Markov chains will be organized within the
project in
June 2005.
Each Research Unit will provide autonomously to creating prototype
software
concerning all the algorithms designed and analyzed in the specific
program. However,
in order to make more significant comparisons and in order that each
unit can
easily use and improve the software made by other units, in the
coordinating
meetings we will state the common features which all the different
pieces of
software must satisfy. We will create a network of workstations which
constitute the hardware structure where the prototype software made by
the
different units is kept, tested, elaborated, compared and integrated.
This
hardware structure will store also the set of test problems which will
be
constructed in the research activity. The access to the information is
opened
and is realized by means of web pages.
Concerning the costs of the project we planned some specific expenses,
having a
global or strategic value, for the overall amount of about 41000 euros.
They
are: research contracts and research fellowships for 33.000 euros which
will
strengthen units U1 and U2; the contribution of about 8000 euros (U1)
for
organizing the fifth edition of the International Workshop "Matrix
Analytic Method in Stochastic Models" MAM5, to be held in Pisa, June
2005
(http://www.dm.unipi.it/~mam5). The conference, which covers topics in
applied
probability, engineering and numerical analysis, is an international
forum
where the state of the art in the field of numerical methods in applied
probability will be traced and discussed.
Concerning the research activity, we will pursue the goals reported in
section
2.1 of this form, to which we refer the reader for what follows. Here,
rather
than giving an exhaustive description of all the parts (which can be
found in
the "modello B" of each unit), we prefer to point out the synergies
between the different parts of the program.
PART A:
More specifically, concerning part A1 on Toeplitz and locally Toeplitz
matrices, the picture is quite complete so far concerning the
asymptotical
spectral analysis (distribution, localization, extremal behaviour),
even for
locally Toeplitz matrices (including PDEs with variable coefficients or
over
geometrically more complicated domains). Topics which still deserve
more study
are
 the analysis of the spectral conditioning (extremal behaviour) for
multiindex matrices, by following the approach of Boettcher and
Grudsky;
 the analysis of the interplay between spectral properties and
convergence of
preconditioned iterative methods in the case where asymptotic
properties of the
sequence of linear systems are known independently of the structure. In
fact,
we believe that the spectral distribution function plays a fundamental
role to
this regard even for unstructured problems.
 Concerning multilevel (multiindex) matrices (A7), the analysis is
motivated
by the need of designing efficient preconditioners (B7) for the
iterative
solution of multilevel Toeplitz systems encountered in many
applications like
image processing, integral equations, inverse scattering (B12,C3,C9).
Special
attention is deserved to the following topics.
 Multigrid methods (B6), where special emphasis is addressed to the
choice of
projector/smoother, to the extension of the optimality [ADSC] to the
multilevel
case, applications to image restoration (C3) and to PDEs (B5).
 Preconditioning (B7), trying to generalize the optimality results of
[S.
Serra, MathComp. 66,1997] to the multilevel case; introducing
"enriched" matrix algebras of the kind UTU* (A4) as preconditioners,
where U is unitary (related to a fast transform) and T has some
sparsity
feature in order to find preconditioners which are superlinear or
optimal. Designing
preconditioners for the GMRES (B13) where the preconditioner is updated
at each
step of the iteration.
 Projection methods in weighted Wiener algebras (B14) [Boettcher,
Silbermann,
"Analysis of Toeplitz operators" Springer 1990, Gohberg, Feldman, vol
41 of Transl. Math. Monog. 1974] will be considered for possible
extension to
multilevel Toeplitz matrices in the nondefinite and in the positive
definite
block case.
 Generation of classes of test matrices (D1) by generalizing the
results of
[MNS,MS].
 Using the computational methods of [MRS1] for infinite multilevel
perturbed
Toeplitz systems for the numerical solution of integral equations with
structured kernel (B12).
 Applications to inverse scattering problems in acoustics, quantum
mechanics,
and remote sensing (C9).
Concerning part A2 on semiseparable matrices, we will investigate on
 the interplay between diagonal plus semiseparable matrices and
rational
Krylov matrices and orthogonal rational functions;
 semiseparability properties of generalized companion matrices (see
part A5)
and eigenvalues computation (B2); in particular on the properties of
the QR
iteration applied to generalized nxn companion matrices and on
algorithms for
the QR step in O(n) ops [BDG], with applications to polynomial
rootfinding
(B8);
 reduction to a general symmetric matrix into semiseparable form.
Sequences of parametric matrices of the kind Aj=A+vj Ej will be
investigated in
part A3. This is a generalization of shifted sequences where Ej is the
identity
matrix. Systems with this class of matrices are encountered in several
popular
techniques for eigenvalues computation. Here we investigate on
indefinite
incomplete updated factorization preconditioners, with Krylov
projection
methods, and multilevel incomplete factorizations. Application of these
techniques to solving total least squares problems (TLS) are considered.
In part A4 concerning new matrix algebras, besides the enriched
matrices cited
above, we analyze the use of matrix algebras in approximating the
Hessian matrix
in minimization problems (B11) in the line of [DFLZ]. Other matrix
algebras
will be analyzed in studying the role of boundary conditions in image
restoration problems (C3, B4).
Concerning generalized companion and resultant matrices in part A5 we
consider
nxn diagonal plus rankone matrices, arrowhead matrices, commerade and
Frobenius matrices for which there is an explicit relation between
their
entries and their characteristic polynomial. Algorithms for performing
one step
of the QR iteration with O(n) cost will be designed and analyzed in
terms of
their complexity and stability. The problem of computing all the
eigenvalues of
generalized companion matrices is investigated (B2) with extension to
higher
rank corrections. This part is related with part A2 for the
semiseparable
structure, and with part B8 for the design of polynomial rootfinders.
We consider also resultant matrices whose characteristic polynomial is
a
multiple of the resultant of two polynomials, say Bezout matrices. Here
the
goal is to represent such matrices in different polynomial bases, say
Bernstein
polynomials, and analyze and exploit the matrix structures in order to
find
more efficient algorithms for certain polynomial computations (B8).
Block Toeplitz matrices of part A6 will be investigated in the attempt
to
extend to positive definite multilevel block Toeplitz matrices the
results of
Boettcher, Silbermann, and of Gohberg, Feldman on weighted Wiener
algebras.
Semiinfinite Toeplitz matrices (A8) and WienerHopf factorizations
(B10) will
be investigated with the aim of providing a better understanding of the
numerical solution of certain Markov chains encountered in queueing
models (C6)
and in order to provide still more efficient solution algorithms in the
line of
[BM,BM3,BM4]
Nonconventional structures in part A9 consist of any matrix structures
which do
not fall in the described classes. In fact, we expect to encounter new
structures from the analysis of applications, especially in web search
engines
(C1) queueing models (C6), problems in astronomy (C9).
Part B:
Structured linear systems in part B1 will be dealt in terms of
preconditioned
iterative methods and multigrid according to the structure involved.
Eigenvalue
problems will be similarly treated in terms of the underlying
structures.
Functional approximation issues in part B3 will pursue the previous
research on
interpolation and approximation problems with rational functions,
particularly
to the analysis of structured conditioning issues and optimal pole
placements.
Inverse problems analysis (B4) will concern the analysis of the number
of
iterations as a regularizing parameter, another direction concerns the
extension of regularizing families of preconditioners to the
reconstruction of
nonnegative images and applications to the LBT system. Another topic is
the
design and analysis of algorithms for systems obtained by Tikhonovlike
regularization.
Concerning PDEs (B5), convergence properties of GMRES in solving
systems
discretizing the convectiondiffusion equation with variable diffusion
coefficients are investigated, as well as timedependent
convectiondiffusion
equations in human metabolism models (C2).
Multigrid technique will be used for regularizing the solution of
illposed
problems (B6,B4).
Preconditioning (B7) is mainly concerned with multilevel Toeplitz
matrices
Concerning polynomial computations (B8), the properties of weak
WienerHopf
factorizations (B10) are analyzed, where both the factors in the
factorization
may have zeros of unit modulus; polynomial rootfinders are designed
based on
the QR algorithm applied to generalized companion matrices (A5) which
maintain
the semiseparable structure (A2). The representations of polynomials
in the
Bernstein basis together with their structured matrix counterparts are
studied
in order to design efficient algorithms for the intersection of Bezier
curves
and of surfaces with applications in CAGD (C7). We will relay on hybrid
techniques for the related computational problems with tools in
projective
geometry, computer algebra and numerical analysis. Finally, algorithms
for the
computation of the eigenvalues of tridiagonal matrices are considered.
Concerning matrix equations (B9) we continue the study of [BM],
[BGM1],[BGM2],[BG2].
 We investigate the generalization of the shift technique (introduced
in
[He,Meini,Rhee, SIMAX,23 2001/02] in the context of Markov chains) to
general
matrix equations of polynomial type.
 The cyclic reduction algorithm of G. Golub [BGN] adjusted to Markov
chains by
[BM3],[BM4] and analyzed in [BGM1],[BGM2], will be considered for
solving
algebraic Riccati equations. Also the computation of the principal pth
root of
a matrix will be faced in terms of WienerHopf factorizations (B10),
Laurent
series inversion and Newton's iteration and with the techniques of
[M4], [Ia]
 Matrix equations modeling queueing problems will be taken into
account.
For minimization problems (B11) the study of [D], [DFLZ], [DLZ] is
continued
with the analysis of sufficient conditions which guarantee the
convergence of
the quasiNewton minimization techniques where the Hessian matrix is
replaced
with a matrix in the Hartley algebra [BF] or with some structured
matrix. Here
the conditions will concern the boundedness of the operators appearing
in the
minimization process and on the condition number of the structured
matrix. Also
algorithms for total least squares problems for structured matrices
will be
analyzed with applications to blind image deblurring (A1,C3).
Integral equation B12 will be considered in connection with multilevel
semiinfinite Toeplitz systems (A7,A8) with applications in inverse
scattering,
acoustics and remote sensing (C4) and in astronomy problems (C9).
PART C:
 Concerning C1, the most recent web search engines compute the
dominant
eigenvalue of a sparse block matrix with a rankone correction [KHMG].
Here our
goal is to design faster algorithms for computing the PageRank vector.
 The twolevel quasiNewton approach implemented for remote sensing
problems,
will be used together with the structured matrix technology for solving
nonlinear inverse scattering problems in electric engineering (C4).
 Nonclassical inverse eigenvalue problems for tridiagonal matrices
and other
closely related matrix structures (also including
diagonalplussemiseparable
matrices) have been used as models in identification problems (C5) in
civil
engineering. Our aim is solving both theoretical and computational
issues
arising in this framework.
 The most part of queueing models encountered in engineering
(telecommunications,
internet, metropolitan networks) are modeled by Markov chains described
by
infinite block Toeplitz matrices. We aim to continue applying the
Toeplitz
tools to solving the main computational problems encountered in this
framework,
among which solving matrix equations and solving infinite systems, in
the line
of [Anastasi, Lenzini, Meini, Perf.Eval. 29,1997].
Problems of CAGD (C7) involve computing intersection of curves and
surfaces
represented in terms of Bernstein polynomials. Many computational
problems can
be reduced to performing operations with structured matrices (Bezout,
Sylvester
etc). The aim is to apply structured matrix technology to the problems
in this
area together with tools in computer algebra and projective geometry by
integrating symbolic and numerical computations.
Neural networks and automatic learning (C8) are applications of
algorithms for
minimization problems (B11).
PART D:
Concerning part D, we already mentioned test matrices (D1). Here we
want to
continue the line of the previous project 20022003. We propose to
develop a
Matlab toolbox specialized to structured matrices with the following
features:
user friendliness, integration with the Matlab environment and its
intrinsic
numerical routines, and optimization of computing time and memory
requirements.
As a first goal we intend to pay attention to certain types of
structured
matrices and basic algorithms for solving linear systems, both by using
preconditioned iterative methods and by using direct methods based on
displacement
structure. The final goal is to construct software that is flexible and
can
easily be extended to new computational algorithms, also in the
multiindex
case. In this part we intend to better organize the experimental work
made by
the different units and the different researchers in the previous
project, by
providing a common framework based on Matlab.
In the project we planned the following collaborations partially
started in the
previous project 20022003:
D. Calvetti and G. Saidel (Ohio),
G. Golub (Stanford)
A. Morassi (Udine)
D. Noutsos and P. Vassalos (Ioannina)
M. Pastorino (DIBE, Genova)
M. Van Barel, S. Van Huffel (Leuven)
G. Latouche (Bruxelles)
N. Higham, F. Tisseur (Manchester)
J. Winkler (Sheffield)
V. Pan (New York)
N. Rhee e K.Sorhaby (Kansas City)
L. Rodman and I.M. Spitkovsky, H.J. Woerdeman (College of William and
Mary)
C. Brezinski (Lille)
M. Tasche (Rostock, Germany)
Subjects treated by each unit:
U1: A1 A2 __ A4 A5 A6 __ A8 A9
U2: A1 A2 A3 A4 A5 A6 A7 __ A9
U3: A1 ___________ A6 A7 A8 A9
U1: B1 B2 ______________ B8 B9 B10 B11 ___ ___ ___
U2: B1 B2 B3 B4 B5 B6 B7 _________________ B13 ___
U3: B1 _____ B4 _____ B7 _____ B10 ___ B12 B13 B14
U1: __ __ __ __ __ C6 C7 C8 __ __ __ D3
U2: C1 C2 C3 C4 C5 __ __ __ __ __ __ D3
U3: __ __ __ C4 __ __ __ __ C9 D1 D2 D3
Risultati parziali attesi:
We expect that the theoretical achievements of our research
(part A)
will allow us to obtain very effective computational and algorithmic
advances,
among which:
U1,U3: Better understanding of numerical methods for structured Markov
chains
(C6) in terms of weak WienerHopf factorizations (B10,A1,A6,A8). New
existence
results on WienerHopf factorization for certain multilevel block
Toeplitz
matrices.
U1: Efficient solution of fluid queues obtained (C6, B9) with cyclic
reduction
and Cayley transform. Computation of reward of M/G/1 Markov chains
(C6). Extension
and generalization of the shift technique to general Markov chains
(B9,C6)
U1: Algorithms for the pth root of a matrix (B9) and WienerHopf
factorizations (B10). Solution of algebraic Riccati (B9) equations by
cyclic
reduction
U1: Analysis of algorithms for polynomials represented in the Bernstein
basis
with applications to CAGD (B8, C7)
U1: Design and analysis of quasiNewton minimization methods based on
approximating the Hessian matrix in matrix algebras, applications to
neural
networks (A4, B11, C8).
U1, U2: Design analysis and implementation of polynomial rootfinders
based on
the computation of eigenvalues of diagonal plus rankone matrices (A2,
B8, B2).
Polynomial rootfinders for polynomials represented in terms of Szego
polynomials computed as eigenvalues of unitary Hessenberg matrices plus
rankone (A2, B8). Algorithms for eigenvalues of diagonal plus
semiseparable
matrices (A2,A5,B2). Reduction of simmetric matrices to semiseparable
form
(A2,B2).
U1,U2: Analysis of total least squares problems for structured matrices
(B1)
U2: Design of algorithms for parametric systems based on Krylov
projection
methods and multilevel preconditioning (A3, A7, B7, B13)
U2: Inverse problems in imaging, extension of regularizing families of
preconditioners, reconstruction of nonnegative images, design of
algorithms for
linear systems generated by Tikhonovlike regularization, applications
to the LBT
problem (A1,B4,C3)
U2: Denoising by wavelets in astronomical images, chopped and nodded
images
(near infrared), measured PSF used in classical restoration,
overiterated
reconstruction (B4,C3).
U2,U3: Preconditioning analysis for the GMRES, applications to PDEs in
convection diffusion problems in ddimensional domains (B13, B5, A7).
Use of
GMRES and Krylov subspaces for ill conditioned structured systems (B13,
A7,
B15).
U1,U2: Regularization by multigrid, design of algorithms based on
multigrid
techniques for the regularized solution of ill conditioned systems,
applications to image processing (B6, B15, C3).
U2: Analysis of boundary conditions in image restoration, analysis of
antireflective conditions, analysis of the associated matrix algebra,
analysis
of the interplay of boundary conditions and regularization
(A1,A4,B15,C3).
U2: Web search engines, analysis of the structures encountered in
problems of
information retrieval (Google) in order to improve the efficiency of
their
solution, by computing the dominant eigenvector of large sparse plus
rankone
matrices (C1)
U3: Design analysis and implementation of prototype software for
solving
(perturbed) semiinfinite matrices (A8,B1,D3)
U3: Analysis of nonconventional matrix structures arising from the
analysis of
polarized light propagation in astronomy problems (C9)
U3: Creation of a set of multilevel (multiindex) test matrices with
the
identification of their asymptotic condition number (A7, D1). Creation
of a
MatLab toolbox with the main software for structured matrices (D2).
U1,U2,U3: Implementation and validation of the algorithms designed in
the
project at a prototype level (D3).
Among the expected results we have the organization of the
international
conference "Matrix Analytic Methods in Stochastic Models", Pisa June
2005, and the organization of intermediate coordination workshops and
of a
final workshop concluding the project.
We also expect to create the web pages of the project where the
research
advances are reported with a continuous update, and where the most
recent
results, in the form of reports (postscript or pdf files), can be
freely
downloaded as well as the software developed during the project.
Unita' di ricerca impegnate: · Unità nº 1·
Unità
nº 2· Unità nº 3
2.5 Criteri
suggeriti per la valutazione globale e delle singole fasi
The evaluation of the project should be done in terms of the
quality of
the scientific results obtained. This evaluation can be performed as
follows:
1 by looking at the quality of the journals where the main results are
published;
2 by looking at the international visibility and dissemination of the
results;
this can be performed by looking at the number and the relevance of the
international conferences where the results are presented.
3 The quality of the algorithmic achievements can be also evaluated in
terms
of the reduction of the cpu time needed to execute the algorithms and
in terms
of the numerical performances.
A dynamical monitoring will be performed: as already written in part
2.4, the
coordinating committee, formed by the coordinators of the Research
Units, will
meet periodically. At the end of each meeting, a report on the state of
the
advancement of the project will be written and inserted in a web page
containing all the relevant information of the project.