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)
PROGETTO DI UNA UNITÀ DI RICERCA - MODELLO B
Anno 2004 - prot. 2004015437_003
PARTE I
1.1 Tipologia del programma di ricerca
Aree scientifico disciplinari
Area 01:
Scienze matematiche e informatiche
(100%) |
|
|
1.2 Durata del Programma di Ricerca
24 Mesi
1.3 Coordinatore Scientifico del Programma di Ricerca
BINI |
DARIO ANDREA |
bini@dm.unipi.it |
MAT/08
- Analisi numerica |
Università
di PISA |
Facoltà
di SCIENZE MATEMATICHE
FISICHE e NATURALI |
Dipartimento
di MATEMATICA "Leonida Tonelli" |
1.4 Responsabile Scientifico dell'Unità di
Ricerca
SEATZU |
SEBASTIANO |
Professore
Ordinario |
10/04/1941 |
STZSST41D10F667J |
MAT/08
- Analisi numerica |
Università
degli Studi di CAGLIARI |
Facoltà
di INGEGNERIA |
Dipartimento
di MATEMATICA |
070/6755619
(Prefisso e telefono) |
070/6755601
(Numero fax) |
seatzu@unica.it
(Email) |
1.5 Curriculum scientifico del Responsabile
Scientifico dell'Unità di Ricerca
Sebastiano Seatzu has been full
professor of numerical analysis at the University of Cagliari (Italy)
since 1980 and professor of operational research from 1981-1995. He
served as chairman of the Department of Mathematics from 1986-1989 and,
for 12 years, as chairman of the Electrical Engineering Degree Council
from 1980-1994. He participated in the organization of various national
and international conferences (such as the XIV International Workshop
on Operator Theory and Applications, held in Cagliari in the period of
June 24-27, 2003). After his graduation in physics in 1965, he got
interested in numerical analysis, in particular in approximation
theory, the numerical solution of integral equations of the first kind,
and numerical linear algebra. He has published 65 papers. He has also
occupied himself with applied mathematics in various fields of physics
and chemistry. His recent work includes approximation theory, sampling
theory, spectral factorization of infinite Toeplitz matrices, and the
numerical solution of block structured and multi-index linear systems.
He is a coeditor of Calcolo, a quarterly on numerical analysis
supported by the Italian National Research Council, and an associate
editor of Advances in Computational Mathematics.
1.6 Pubblicazioni scientifiche più
significative del Responsabile Scientifico dell'Unità di Ricerca
1. |
VAN DER MEE
C.V.M.; RODRIGUEZ G.; SEATZU S.
(2003). Semi-infinite multi-index perturbed block Toeplitz systems
LINEAR ALGEBRA AND ITS APPLICATIONS. (vol. 366(C) pp. 459-482) |
2. |
BREZINSKI C.;
REDIVOZAGLIA M.; RODRIGUEZ G.;
SEATZU S. (2003). Multiparameter regularization techniques for
ill-conditioned linear systems NUMERISCHE MATHEMATIK. (vol. 94 pp.
203-228) |
3. |
VAN DER MEE
C.V.M.; NASHED M.Z.; SEATZU S.
(2003). Sampling expansions in unitarily translation invariant
reproducing kernel Hilbert spaces. ADVANCES IN COMPUTATIONAL
MATHEMATICS. (vol. 19(4) pp. 355-372) |
4. |
VAN DER MEE
C.V.M.; RODRIGUEZ G.; SEATZU S.
(2002). Spectral factorization of bi-infinite multi-index block
Toeplitz matrices LINEAR ALGEBRA AND ITS APPLICATIONS. (vol.
343-344 pp. 355-380) Special Issue on Structured and Infinite Systems
of Linear Equations.. |
5. |
GOODMAN
T.N.T.; MICCHELLI C.A.; RODRIGUEZ G.;
SEATZU S. (2000). On the Cholesky factorization of the Gram matrix
of multivariate functions SIAM JOURNAL ON MATRIX ANALYSIS AND
APPLICATIONS. (vol. 22(2) pp. 501-526) |
1.7 Risorse umane impegnabili nel Programma
dell'Unità di Ricerca
1.7.1 Personale universitario dell'Università
sede dell'Unità di Ricerca
Personale docente
nº |
Cognome |
Nome |
Dipartimento |
Qualifica |
Settore Disc. |
Mesi Uomo |
1° anno |
2° anno |
1. |
SEATZU |
Sebastiano |
Dip. MATEMATICA |
Prof. Ordinario |
MAT/08 |
6 |
6 |
2. |
VAN DER MEE |
Cornelis
Victor Maria |
Dip. MATEMATICA |
Prof. Ordinario |
MAT/07 |
6 |
6 |
3. |
RODRIGUEZ |
Giuseppe |
Dip. MATEMATICA |
Prof. Associato |
MAT/08 |
6 |
6 |
|
TOTALE |
|
|
|
|
18 |
18 |
Altro personale
Nessuno
1.7.2 Personale universitario di altre
Università
Personale docente
nº |
Cognome |
Nome |
Università |
Dipartimento |
Qualifica |
Settore Disc. |
Mesi Uomo |
1° anno |
2° anno |
1. |
REDIVO ZAGLIA |
Michela |
PADOVA |
Dip.
MATEMATICA PURA ED APPLICATA |
PA |
MAT/08 |
6 |
6 |
|
TOTALE |
|
|
|
|
|
6 |
6 |
Altro personale
Nessuno
1.7.3 Titolari di assegni di ricerca
Nessuno
1.7.4 Titolari di borse
Nessuno
1.7.5 Personale a contratto da destinare a questo
specifico programma
Nessuno
1.7.6 Personale extrauniversitario indipendente o
dipendente da altri Enti
nº |
Cognome |
Nome |
Nome dell'ente |
Qualifica |
Mesi Uomo |
1° anno |
2° anno |
1. |
Theis |
Daniela |
Indipendente |
Dottore di
ricerca |
3 |
3 |
|
TOTALE |
|
|
|
3 |
3 |
PARTE II
2.1 Titolo specifico del programma svolto
dall'Unità di Ricerca
Multiindex structured matrices
with applications
2.2 Settori scientifico-disciplinari interessati dal
Programma di Ricerca
MAT/08 -
Analisi numerica |
MAT/07 -
Fisica matematica |
MAT/05 -
Analisi matematica |
2.3 Parole chiave
REGULARIZATION METHODS ;
EXTRAPOLATION METHODS ; STRUCTURED MATRICES ; SPECTRAL FACTORIZATION ;
INTEGRAL EQUATIONS WITH STRUCTURED KERNEL ; INVERSE SCATTERING
2.4 Base di partenza scientifica nazionale o
internazionale
On any of the subjects
indicated below there exists extensive literature represented by
numerous publications in international journals, specialized book
series and proceedings of international conferences. The major part of
the relevant papers regards, apart from numerical analysis, also
functional analysis and linear operator theory. Moreover, there exist
problems in various applied fields whose solution depends on the
research on the subjects indicated in the project. A large amount of
research on inverse scattering in, 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 and
in particular in [12,17,18,19,37]. The most interesting applied
results, including the most efficient methods for solving many inverse
scattering problems, are to be found in [9,14,27]. Significant
contributions of the local research unit can be found in [1] and [29].
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 [25]. Very interesting
results on linear operator theory, numerical methods for solving linear
systems with structured matrices, integral equations with structured
kernel and their applications have been presented at the Fourteenth
IWOTA conference (International Workshop on Operator Theory and its
Applications) held in Cagliari in the period of June 24-27, 2003, whose
proceedings will be published in the Birkhauser OT series under the
editorship of Kaashoek, van der Mee and Seatzu. On the solution
techniques for ill-conditioned linear systems there exists documented
scientific activity by the local research unit [5,6,38]. As to research
on the generalization of projection methods to the solution of
multi-index systems, the monographs [2,3,17] are of the utmost
interest. In this respect the most significant contributions of the
local unit are contained in the papers [22,30,33,34]. Many results of
specific interest to integral equations are contained in [17,18,19].
In the multi-index case there is not a lot of literature because of the
comparative nonexistence of an algebraic theory of multi-index matrix
operators, in spite of the objective interest in the subject in many
applied fields. In the latter situation, but exclusively in the scalar
case, two spectral factorization methods in weighted Banach algebras
with respect to a fixed total order have been developed [15,22]. The
results obtained allow one to characterize the type of decay of both
the elements of the inverse of the matrix and of the factors, in terms
of the characteristics of the matrix. One of the two methods, which is
easily implemented by using the FFT, has been used to identify the
limiting profile in the asymptotic process of orthonormalization of
sequences of uniform translates of a fixed multivariate function by
means of the Gram-Schmidt method [22]. In the multi-index block
Toeplitz case there only exist partial results on spectral
factorization [33,34]. In particular, nobody has so far succeeded in
characterizing the type of decay of the elements of the Cholesky
factors of a positive definite multi-index block Toeplitz matrix in
terms of that of the elements of the matrix under consideration. An
interesting extension of the band extension method that seems promising
for solving multi-index systems has recently been obtained in [16].
The research planned on the study of integral equations is based in
part on results involving their discrete analogs [36]. Because many
applied problems typical of remote sensing, inverse scattering and
image processing can be reduced to the solution of integral equations
of convolution type in several variables, it appears essential to
discretize to obtain a pure or perturbed Toeplitz system and to solve
the corresponding linear system with regularization methods that
exploit the structure.
The regularization theory applied to structured matrices [23] as well
as the computational methods based on fast algorithms that exploit the
displacement structure [13,20,24,26] form a solid basis for studying
this problem. These comprise preconditioned iterative methods, among
which those formulated in Krylov spaces, and in particular the GMRES,
Lanczos and conjugate gradient methods for which there exists a lot of
consolidated literature, evoke particular interest. Locally there
exists substantial expertise in both iterative methods [4,6,7,8] and
structured matrices [30-34]. For this reason we expect to be able to
develop iterative algorithms specialized to the case of structured
matrices. On the basis of partial results that are as yet unpublished,
the local unit expects to be able to develop an algorithm for
generating preconditioners of structured multi-index matrices.
For the numerical validation of the existing methods in the field, it
is crucial to have at one's disposal an extensive class of test
matrices. The principal reference for generating such test matrices is
the article [35] in which an operational method for generating families
of multi-index test matrices is indicated.
For this reason it will be essential to collaborate with the other
units to choose the optimal algorithms for solving perturbed structured
linear systems of large order, as required when using the projection
method. In the other two research units there exists a very substantial
theoretical and numerical expertise that is by now established and
internationally recognized. As to the development of software, we can
base ourselves on many algorithms and programs for treating structured
matrices and solving ill-conditioned linear systems which have already
been developed and used by the local unit. In this field there exists
specialistic know-how guaranteeing the realizability of software of
public domain, as indicated in part 5) of the research project.
2.4.a Riferimenti bibliografici
[1] T. Aktosun, M. Klaus, and
C. van der Mee. Direct and inverse scattering for selfadjoint
Hamiltonian systems on the line. Integral Equations and Operator Theory
38, 129-178 (2000).
[2] A. Boettcher and B. Silbermann. Analysis of Toeplitz operators.
Springer, New York, 1990.
[3] A. Boettcher and B. Silbermann. Introduction to large truncated
Toeplitz matrices. Springer, New York, 1999.
[4] C. Brezinski, M. Redivo Zaglia. Block projection methods for linear
systems, Numerical Algorithms, 29 (2002) 33?43.
[5] C. Brezinski, M. Redivo-Zaglia, G. Rodriguez, and S. Seatzu.
Multiparameter regularization techniques for ill-conditioned linear
systems, Numer. Math. 94, 203-228 (2003).
[6] C. Brezinski, M. Redivo Zaglia, H. Sadok. New look-ahead
Lanczos-type algorithms for solving linear systems, Numer. Math., 83
(1999) 53-85.
[7] C. Brezinski, M. Redivo Zaglia, H. Sadok. The matrix and polynomial
approaches to Lanczos-type algorithms, J. Comput. Appl. Math., 123
(2000) 241-260.
[8] C. Brezinski, M. Redivo Zaglia, H. Sadok. A review of formal
orthogonality in Lanczos-based methods, J. Comput. Appl. Math., 140
(2002) 81-98.
[9] K. Chadan and P. C. Sabatier. Inverse Problems in Quantum
Scattering Theory, 2nd ed., Springer, New York, 1989.
[10] R. Chan and G. Strang. Toeplitz equations by conjugate gradients
with circulant preconditioner. SIAM J. Sci. Stat. Comput.
10(1):104-119, 1989.
[11] T.F. Chan. An optimal preconditioner for Toeplitz systems. SIAM J.
Sci. Stat. Comput., 9(4):766-771, 1988.
[12] K. Clancey and I. Gohberg. Factorization of Matrix Functions and
Singular Integral Operators, volume 3 of Operator Theory: Advances and
Applications. Birkhauser, Basel-Boston, 1981.
[13] T. Constantinescu, A.H. Sayed, and T. Kailath. Displacement
structure and H-infinity problems. System theory: Modeling, analysis
and control. Kluwer, Boston, 2000.
[14] W. Eckhaus and A. van Harten. The inverse scattering
transformation and the theory of solitons. An introduction.
North-Holland Mathematics Studies, 50. North-Holland, Amsterdam, 1981.
[15] T. Ehrhardt and C. van der Mee. Canonical factorization of
continuous functions on the d-torus. Proc. Amer. Math. Soc.,131,
801-813 (2002).
[16] J.S. Geronimo and H. Woerdeman. Positive extensions and
Fejer-Riesz factorization in Autoregressive Filters in two variable,
Ann. Math. to appear.
[17] I.C. Gohberg and I.A. Feldman. Convolution Equations and
Projection Methods for their Solution, volume 41 of Translations of
Mathematical Monographs. Amer. Math. Soc., Providence, 1974.
[18] I. Gohberg, S. Goldberg, and M.A. Kaashoek. Classes of Linear
Operators, Vol. I, volume 49 of Operator Theory: Advances and
Applications. Birkhauser, Basel-Boston, 1990.
[19] I. Gohberg, S. Goldberg, and M.A. Kaashoek. Classes of Linear
Operators, Vol. II, volume 63 of Operator Theory: Advances and
Applications. Birkhauser, Basel-Boston, 1993.
[20] I. Gohberg, T. Kailath and V. Olshevsky. Fast gaussian elimination
with partial pivoting for matrices with displacement structure. Math.
Comp., 64(212), 1557-1576 (1995).
[21] T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, and S. Seatzu.
Spectral factorization of Laurent polynomials. Adv. in Comp. Math.,
7:429-454, 1997.
[22] T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, and S. Seatzu. On
the Cholesky factorisation of the Gram matrix of multivariate
functions. SIAM J. Matrix Anal. Appl., 22(2):501-526, 2000.
[23] P.C. Hansen. Rank-deficient and discrete ill-posed problems.
Numerical aspects of linear inversion. SIAM, Philadelphia, 1997.
[24] G. Heinig and K. Rost. Algebraic methods for Toeplitz-like
matrices and operators. Vol. 13 of Operator Theory: Advances and
Applications, Birkhauser, Basel-Boston, 1984.
[25] 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).
[26] T. Kailath and A.H. Sayed. Displacement structure: Theory and
applications. SIAM Review 37(3): 297-386 (1995).
[27] V. A. Marchenko. Sturm-Liouville operators and applications,
Birkhauser, Basel, 1986.
[28] C.V.M. van der Mee, M.Z. Nashed, and S. Seatzu, Sampling
expansions and interpolation in unitarily translation invariant
reproducing kernel Hilbert spaces, Adv. Comp. Math.,19(4), 455-472,
2003.
[29] C.V.M. van der Mee, V. Pivovarchik. Inverse scattering for
Schrödinger equation with energy dependent potential. J. Math.
Phys. 42 (2001) 1-24.
[30] C.V.M. van der Mee, G. Rodriguez, and S. Seatzu. Block Cholesky
factorization of infinite matrices, and orthonormalization of vectors
of functions. In Z. Chen, Y. Li, C.A. Micchelli, and Y. Xu, editors,
Advances in Computational Mathematics, volume 202 of Lecture Notes in
Pure and Applied Mathematics, pages 423-455, New York and Basel, 1998.
M. Dekker Inc.
[31] C.V.M. van der Mee, G. Rodriguez, and S. Seatzu. Solution methods
for semi-infinite linear systems of block Toeplitz type and their
perturbations. in D. Bini, E. Tyrtyshnikov, and P. Yalamov, editors,
Structured Matrices: Recent Developments in Theory and Computations,
pages 93-109, New York, 2000. Nova Science Publisher Inc.
[32] C.V.M. van der Mee, G. Rodriguez, and S. Seatzu. Spectral
factorization of bi-infinite block Toeplitz matrices with applications.
In L. Brugnano and D. Trigiante, editors, Recent Trends in Numerical
Analysis, Nova Science Publ., New York, 2000, pp. 203-225.
[33] C.V.M. van der Mee, G. Rodriguez, and S. Seatzu. Spectral
factorization of bi-infinite multi-index block Toeplitz matrices.
Linear Algebra and its Applications 343-344: 355-380 (2001).
[34] C.V.M. van der Mee, G. Rodriguez, and S. Seatzu. Semi-infinite
multi-index perturbed block Toeplitz systems. Linear Algebra and its
Applications, 366 (2003) 459-482.
[35] C.V.M. van der Mee, S. Seatzu. A method for generating infinite
positive definite self-adjoint test matrices and Riesz bases. Preprint
(2004).
[36] S. Proessdorf and B. Silbermann. Numerical analysis for integral
and related operator equations. Vol. 52 of Operator Theory: Advances
and Applications, Birkhauser, Basel-Boston, 1991.
[37] L. Rodman. Operator Polynomials, Vol. 38 of Operator Theory:
Advances and Applications, Birkhauser, Basel-Boston, 1989.
[38] G. Rodriguez, S. Seatzu, and D. Theis. A new technique for
ill-conditioned linear systems. Numerical Algorithms, 33 (2003) 433-422.
[39] E.E. Tyrtyshnikov. Optimal and superoptimal circulant
preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992) 459-473.
2.5 Descrizione del programma e dei compiti
dell'Unità di Ricerca
As agreed upon with the other
units, the Cagliari unit will conduct research on the following
arguments:
1) development of innovative methods to solve linear systems with
ill-conditioned matrices and linear systems with multi-index matrices;
2) development of preconditioners to solve linear systems with
multi-index matrices;
3) generation of families of multi-index test matrices, identifying the
asymptotic behaviour of their condition numbers;
4) numerical solution of integral equations with multivariate
structured kernels and their application to acoustics and remote
sensing;
5) development of prototypical software specialized to structured
matrices.
1) As to the first research argument, the Cagliari unit proposes to
develop computational methods for ill-conditioned structured linear
systems by applying fast algorithms based on displacement structure to
Tikhonov regularization methods. In the very frequent cases in which
the regularization matrix itself is structured, these methods should
allow us to deal with linear systems characterized by various types of
displacement structure, such as for instance Toeplitz, Hankel and
Cauchy structures, which are of particular interest in the numerical
solution of integral equations with structured kernels. We also propose
to study the solution of ill-conditioned linear systems by means of
preconditioned iterative methods applied in Krylov spaces, such as for
instance GMRES, the conjugate gradient method and the Lanczos method.
More precisely, we are thinking of using the preconditioning methods
mentioned in 2) as preconditioners in various iterative methods and in
particular in the GMRES method.
A second part of the research will regard the extension of projection
methods in weighted Wiener algebras to the solution of linear systems
with structured multi-index matrices. This method, which is well-known
when solving linear Toeplitz systems in the Wiener algebra, is easily
extended to weighted Wiener algebras, provided one deals with
mono-index block Toeplitz systems or with positive definite multi-index
block Toeplitz systems with scalar blocks [17],[2]. To the contrary,
the extension to the multi-index case is not easy for non positive
definite Toeplitz systems or for positive definite block Toeplitz
systems. Such an extension would be very useful because it would allow
us to characterize the asymptotic behaviour of the error as a function
of the decay of the elements of the matrix away from the diagonal and
of the decay of the elements of the vector on the right-hand side.
The local unit also proposes to study the solution of linear systems
whose matrices have a nonconventional structure stemming from problems
in astronomy such as the transfer of polarized light in planetary media
where there exists considerable local expertise [25].
2) As is well-known [10,11,39], there exist various methods of optimal
computational complexity to solve mono-index linear systems. On the
other hand, there exist only partial results on the generation of
preconditioners of low computational complexity in the case of
multi-index matrices [39]. Research in progress by the Cagliari unit
appears able to generate in a uniform way preconditioners for mono- and
multi-index matrices. The research is presently evolving towards the
optimization of the computational complexity independently of the
number of indices. Here we propose to experimentally determine the
efficiency of the solution method for an extensive class of linear
systems by means of preconditioned iterative methods. The development
of a vast class of test matrices, as to be discussed in 3), should
allow us to validate the method in an appropriate way. We also plan
research on the recursive generation of preconditioners that are to be
applied iteratively to e.g. the GMRES method. More explicitly, the
basic idea is the following: once the preconditioner has been
constructed for the matrix at step k, the preconditioner at the next
step is obtained from its predecessor by simply evaluating two
additional elements.
3) In [28] a general method has been developed that can be applied in a
simple way to generate bi-infinite positive definite matrices. In [35]
this method has been improved considerably and generalized to the
multi-index case, but only to bi-infinite matrices. Here we propose to
adapt the method to generate semi-infinite matrices, aiming at
identifying the asymptotic behaviour of the condition numbers of an
extensive class of multi-index matrices as a function of their order.
Such a result should allow us to have at our disposal a sufficiently
large class of matrices for which the asymptotic behaviour of their
condition numbers is known. The purpose is to be able to compare the
efficiency of the newly proposed methods to those in present use. The
Cagliari unit is particularly interested in the evaluation of the
efficiency of the preconditioners proposed in connection with the
conjugate gradient method and the GMRES method.
4) Because linear systems with structured multi-index matrices
represent the discrete analogue of integral equations in several
variables with structured kernel, there exists a natural and deep
connection between the two research topics that is potentially very
profitable but has presently not been valued in an adequate way. This
depends fundamentally on the fact that operator theory has so far been
developed by functional analysts and structured matrix theory by
numerical analysts. Because either type of expertise is present in the
local unit, we could in fact develop innovative methods for the
numerical solution of multivariate integral equations with structured
kernel. From the theoretical point of view the computational method for
infinite multi-index systems proposed in [34] should be very useful. An
important applied field where there exist significant contributions of
the local unit [1],[29], is represented by inverse scattering in
acoustics and quantum mechanics. In collaboration with scientists at
the Universities of Cagliari and Napels that are experts on remote
sensing, the local unit proposes to apply the theoretical and numerical
results on integral equations and linear multi-index systems to solve
typical problems in this field.
5) In spite of a great interest, both from the scientific and the
applied side, in the arguments connected with structured matrices and
their treatment, there does not seem to exist a specialized software
library which implements the most recent algorithms that have
specifically been developed for this class of matrices. Because there
exist various algorithms and programs of applied interest developed by
the local unit as well as a real expertise in writing and evaluating
specialized software (*), we propose to develop a toolbox specialized
to structured matrices in MatLab, which is one of the most widely
disseminated software instruments for scientific computation and
visualization. The characteristics to which we will give particular
attention are the following: 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 multi-index case.
(*) G. Rodriguez has developed the codes for the numerical algorithms
proposed in [35-38] and M. Redivo Zaglia is software editor of the
journal Numerical Algorithms.
2.8 Mesi uomo complessivi dedicati al
programma
|
|
Numero |
Mesi uomo
1° anno |
Mesi uomo
2° anno |
Totale mesi uomo |
University
Personnel |
3 |
18 |
18 |
36 |
Other
University Personnel |
1 |
6 |
6 |
12 |
Work
contract (research grants, free lance
contracts) |
0 |
|
|
|
PHD
Fellows & PHD Students |
PHD Students |
0 |
|
|
|
Post-Doctoral
Fellows |
0 |
|
|
|
Specialization
School |
0 |
|
|
|
Personnel
to be hired |
Work contract
(research grants, free lance
contracts) |
0 |
|
|
|
PHD Fellows
& PHD Students |
0 |
|
|
|
PHD Students |
0 |
|
|
|
Other tipologies |
0 |
|
|
|
No
cost Non University Personnel |
1 |
3 |
3 |
6 |
TOTALE |
|
5 |
27 |
27 |
54 |
.