A. Arico': Numerical solution of the nonlinear Schrodinger equation

Linear algebra techniques can be used to efficiently solve the nonlinear Schrodinger equation
                                 i q_t = q_xx + 2 q |q|²,         where q=q(x,t)
following the approach of the inverse scattering theory. We present an overview of the procedure and discuss some numerical experiments.  
 

Z.J. Bai: Nonnegative Inverse Eigenvalue Problems with Partial Eigendata
(joint work with S. Serra Capizzano and Z. Zhao)

We consider the inverse problem of constructing an n-by-n real nonnegative matrix A from prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint and provide a solution which is closest to a pre-estimated analytical nonnegative matrix. To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and then propose a nonsmooth Newton-type method for solving the nonsmooth equation related to the monotone complementarity problem. Under some very mild assumptions, we show that our method has simultaneously a global and quadratic convergence. We also specialize our method to the symmetric nonnegative inverse problem, and to the cases of a prescribed lower bound and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.  
 

D. Bertaccini: Adaptive updating preconditioners for image restoration with PDE models

We consider a semi implicit time step integration of a couple of nonlinear generalized Alvarez–Lions–Morel-like partial differential equation models where the discretized equations were solved by Krylov iterative solvers accelerated by an adaptive preconditioning paradigm based on the update of incomplete factorizations which presents a global computational cost slightly more than linear in the number of the image pixels, robust, able to manage spatially variant blurring operators and competitive with multigrid techniques but with a lower storage requirement and a greater flexibility. We demonstrated the efficiency of the strategy by denoising and deblurring some images.  
 

P. Brianzi: Ricostruzione di un segnale con formule di sovracampionamento
(in collaborazione con V. Del Prete)

Si affronta il problema della ricostruzione di un segnale a banda limitata quando un numero finito di suoi campioni e delle sue derivate sono persi. Si studiano le matrici che intervengono nei sistemi lineari di soluzione nella procedura di recupero e si affronta il particolare il caso mal-condizionato in cui i campioni mancanti sono consecutivi regolarizzando con il metodo di Tikhonov.  
 

D. Fasino: Ricostruzione di conduttività nella magneto-quasistatica con metodi level-set

Si presenta un metodo numerico per la risoluzione di un problema inverso nella magneto-quasistatica (una approssimazione della magnetodinamica per basse frequenze). Il problema consiste nella identificazione del coefficiente di conduttività in un dominio tridimensionale omogeneo in cui possono essere presenti piccole inclusioni, a partire da misure di distorsione di un campo magnetico esterno. Computazionalmente, il problema richiede l'inversione regolarizzata di una forward map fortemente non lineare; la soluzione cercata è una funzione costante a tratti con un supporto piuttosto limitato, che può assumere solo un numero molto piccolo di valori possibili, noti a priori. La regolarizzazione introdotta può essere vista come un vincolo di sparsità sulla discretizzazione del gradiente di una funzione parametrizzata mediante la tecnica dei metodi level-set.  
 

C. Estatico: Sviluppi recenti nella teoria della regolarizzazione, con un’applicazione in imaging
(in collaborazione con P. Brianzi, F. Di Benedetto, M. Pastorino, A. Randazzo e G. Rodriguez)

Formally, any inverse problem can be modeled by an operator equation
                                              A(x) = y,
where A : X ® Y is the so-called “forward” operator between two functional spaces X and Y, and xÎX is the “cause” of some “effects” yÎY . By means of the knowledge of (some approximation of) the effects y, the aim is to recover the unknown cause x. Many well known solving schemes approximate the solution x by means of the minimization of a cost functional like
                                       F(x) = ||A(x) − y||_Y + l||x||_X  
where X and Y are both Hilbert spaces. Iterative regularization algorithms for the minimization of F give rise in general to over-smoothed solutions and the discontinuities present in real solutions x are not well restored. More recently, it has been investigated the behavior of iterative methods based on more general Banach spaces X and Y . The new “geometry” of the Banach spaces can substantially reduce the over-smoothing of the iterative restoration process. In this talk, we discuss the regularization approach in Banach spaces. A non-linear iterative method is applied to a non-linear inverse scattering problem where the dielectric distributions x of a 2D domain must be recovered by means of its scattered microwave field y outside the domain. We will show how the new computational results well outperform classical “Hilbertian regularization”.  
 

M. Donatelli: Matrici di regolarizzazione quadrate per problemi mal-posti discreti di grandi dimensioni
(in collaborazione con L. Reichel)

Si considera la soluzione di problemi mal-posti discreti lineari di grandi dimensioni mediante regolarizzazione con il metodo di Tikhonov. Ricorrendo a metodi iterativi basati sul procedimento di Arnoldi si ha la necessita' di utilizzare matrici di regolarizzazione quadrate. Tali matrici solitamente sono approssimazioni alle differenze finite di operatori differenziali. In questo seminario si discutera' come costruire matrici di regolarizzazione quadrate per preservare particolari componenti della soluzione con un costo computazionale contenuto. Una tecnica generale ed efficace per ottenere tali matrici e' l'imposizione di opportune condizioni al contorno.  
 

F. Ferri: Importance of 3D deconvolution in Confocal Microscopy of biopolymer gels
(in collaborazione con D. Magatti e M. Molteni)

Biopolymer gels are three‑dimensional (3D) networks of fibers grown from the polymerization of biological macromolecules, usually proteins, that polymerize under the action of some specific enzyme. The bio-functional properties of the network is directly linked to its mechanical and physical structure, which, in turn, depends on the physical parameters characterizing the gel, such as fibers diameter and stiffness, distribution of fiber length and branching/cross-linking points, mesh size, fractal morphology.
Among non invasive techniques suitable for studying these systems, Confocal Microscopy (CM) is one of the most prominent, having the distinctive advantages of allowing the direct examination of the samples in real space and as a function of time. The quantitative analysis of the 3D confocal images, however, requires a non-trivial 3D deconvolution of the raw data, which might be quite critical because the Point Spread Function (PSF) of the microscope is often larger than the fibers diameter, it can be non isotropic and non uniform, and its experimental determination is not straightforward as well. Assessing and testing the performances of the deconvolution method used in the data analysis is, therefore, of crucial importance.
In this talk we present a novel method for the generation of in silico 3D-filamentous networks that exhibit the same structural and morphological properties of fibrin gels (the gels involved in the process of blood coagulation) and can be easily exploited for validating the deconvolution method. The network is made of cylindrical straight fibers of known diameter joined together at some randomly distributed nodal points. The resulting 3D network exhibits a fractal morphology similar to that of the fibrin gel (mass fractal dimension ~1.2-1.5) and, after convolution with the point spread function of our confocal set-up, appears strikingly similar to the confocal images taken on fibrin gels. Furthermore, based on these in silico gels, we have developed a simple statistical method for the determination of some of the gel physical parameters mentioned above (fibers diameter, gel mesh size, fractal dimension). This method of analysis, based on the 3D correlation function of entire confocal stack, is rather simple, fast and reliable, but if applied to the same in silico gels convolved with the PSF of the microscope, produces highly inaccurate results. Thus, in the case of real data, it is of crucial importance to deconvolve them before any quantitative analysis.  
 

N. Mastronardi: An algorithm for computing a new factorization of indefinite symmetric matrices
(in collaborazione con P. Van Dooren)

Let AÎR^nxn be a symmetric indefinite matrix and let {I₁,I₂,I₃} be its inertia, i.e., I₁, I₂ and I₃ are the number of eigenvalues of A greater, equal or less than zero. Without loss of generality, we assume I₁ ³ I₂+I₃. Then, the following factorization of A exists,
                                                      A = QLQ^T
where Q is an orthogonal matrix and L is an lower anti--triangular one, i.e., a matrix with zero entries above the anti--diagonal, with a bulge matrix of size I₁-I₂-I₃ in the main diagonal.
In this talk a stable and efficient algorithm for computing the latter factorization is described. Moreover, the computation of the inertia with the proposed algorithm is straightforward. Comparisons with algorithms available in the literature are shown.  
 

B. Morini: Risoluzione di sistemi lineari in problemi ai minimi quadrati vincolati ed applicazioni nella ricostruzione di immagini

Si considerano procedimenti iterativi per la risoluzione di problemi ai minimi quadrati lineari, vincolati e di grandi dimensioni. L'efficienza dei metodi di ottimizzazione considerati dipende fortemente dalla formulazione dei sistemi lineari algebrici che nascono ad ogni iterazione e dalla loro risoluzione iterativa. Presentiamo due strategie per la risoluzione della fase di algebra lineare ed risultati numerici ottenuti su problemi di ricostruzione di immagini.  
 

P. Novati: Preconditioning linear systems via matrix function evaluation
(in collaborazione con M. Redivo-Zaglia, M. R. Russo)

For the solution of discrete ill-posed problems, a novel preconditioned iterative method based on the Arnoldi algorithm for matrix functions is presented. The method is also extended to work in connection with Tikhonov regularization. Numerical experiments arising from the solution of integral equations and image restoration are presented.
 

M. Piana: A linear inverse problem in X-ray solar astronomy
(in collaborazione con S. Allavena)

The scientific aim of the NASA mission Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is to collect X-ray observations with high spatial and spectral resolution. The purest data provided by RHESSI are Fourier components of the X-ray radiation sampled at discrete points of the spatial frequency plane, named visibilities. This talk will describe an interpolation/extrapolation method for the reconstruction of X-ray images from RHESSI visibilities.
 

M. Semplice: Simulazioni numeriche con PDE paraboliche degeneri per la conservazione ed il restauro monumentale
(in collaborazione con M. Donatelli, S. Serra-Capizzano)

Dopo un breve excursus su recenti modelli matematici di interesse nel campo della conservazione dei monumenti e dei manufatti marmorei, consideriamo la discretizzazione di equazioni paraboliche non lineari (e degeneri) con la tecnica delle differenze finite e la loro applicazione nelle simulazioni. In particolare si descrivono la struttura dei sistemi non lineari da risolvere ad ogni passo temporale e la scelta di opportuni precondizionatori per il metodo di Newton-Krylov impiegato. Si illustrano sia i risultati di convergenza ed ottimalità recentemente ottenuti, che i limiti della loro applicabilità, illustrando i possibili sviluppi futuri di questa ricerca.
 

S. Serra Capizzano: Canonical eigenvalue distribution of multilevel block Toeplitz sequences with non-Hermitian symbols
(in collaborazione con M. Donatelli, N. Maya)

Let f  be a bounded symbol from the linear space of the complex s x s matrices to I_k=(-p,p)^k . We consider the sequence of multilevel block Toeplitz matrices {T_n(f)}, n=(n₁, ..., n_k). When s=1, thanks to the work of Tilli, if R(f) has empty interior and does not disconnect the complex plane, then {T_n(f)} ~_l (f, I_k). Here we generalize the latter result for the case where the role of R(f) is played by U_j=1...s R(l_j(f)),
l_j(f) being the eigenvalues of the matrix-valued symbol f. The result is extended to the algebra generated by Toeplitz sequences with bounded symbols.
 

D. Sesana: Successioni di matrici g-circolanti e g-Toeplitz: analisi spettrale, precondizionatori regolarizzanti e applicazioni
(in collaborazione con C. Estatico, E. Ngondiep e S. Serra-Capizzano)

For a given nonnegative integer g, a matrix A_n of size n is called g-Toeplitz if its entries obey the rule A_n=[a_r-gs]_{r,s=0,...,n-1}. Analogously, a matrix A_n again of size n is called g-circulant if A_n=[a_(r-gs) mod n]_{r,s=0,...,n-1}. In this talk we describe the asymptotic properties, in term of spectral distribution, of both g-circulant and g-Toeplitz sequences in the case where a_k can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (-p,p). Moreover, we consider the preconditioning problem which is well understood and widely studied in the last three decades for g=1, and we show that, while a standard preconditioning cannot be achieved for g³2, the result has a positive implication since there exist choices of g-circulant sequences which can be used as basic regularizing preconditioning sequences for the corresponding g-Toeplitz structures. Few numerical experiments are presented and discussed.