Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.Comment: 17 pages, 9 figure

Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure

Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability

Equatorial Lensing in the Balasin-Grumiller Galaxy Model

The Balasin-Grumiller model has been the first model employed as an attempt towards providing a fully general relativistic description of the dynamics of a disc galaxy. In this paper, we compute the equatorial gravitational lensing observables of the model. Indeed, our purpose is to investigate the role that gravitational lensing plays as an observable in distinguishing between the state-of-the-art galaxy models and the fully general relativistic ones, with the latter stressing the role of frame-dragging and hence conceivably pointing to a possible re-weighting of the dark matter content of disc galaxies. We obtain for the Balasin-Grumiller model the exact formula for the bending angle of light and we provide a corresponding estimate for the time delay between images in the equatorial plane. For a reasonable choice for the values of the parameters of the solution (bulge and scale radiuses, and average rotational star speeds), the values that we obtain for the bending angle are in agreement with those observed for typical disc galaxies. On the other hand, the calculated time delay, which is directly tied to the frame-dragging generated by the angular momentum of the galaxy, turns out to be some orders of magnitude larger than the ones measured for the class of galaxies that the Balasin-Grumiller model would claim to describe. We believe this abnormal discrepancy to be due to the very nature of the Balasin-Grumiller model. Namely, it being rigidly rotating, hence providing an unphysical amount of frame-dragging. Therefore, we conclude that, in spite of its simplicity and its unquestionable didactical value, the Balasin-Grumiller model is far too crude to provide an instrument for a reliable general relativistic description of a disc galaxy and that further work in the fully general relativistic modelling of galaxies is required to reach a satisfactory stage.Comment: 19 pages and 6 figure

Cryphonectria nitschkei chrysovirus 1 with unique molecular features and a very narrow host range

Cryphonectria nitschkei chrysovirus 1 (CnCV1), was described earlier from an ascomycetous fungus, Cryphonectria nitschkei strain OB5/11, collected in Japan; its partial sequence was reported a decade ago. Complete sequencing of the four genomic dsRNA segments revealed molecular features similar to but distinct from previously reported members of the family Chrysoviridae. Unique features include the presence of a mini-cistron preceding the major large open reading frame in each genomic segment. Common features include the presence of CAA repeats in the 5′-untranslated regions and conserved terminal sequences. CnCV1-OB5/11 could be laterally transferred to C. nitschkei and its relatives C. radicalis and C. naterciae via coculturing, virion transfection and protoplast fusion, but not to fungal species other than the three species mentioned above, even within the genus Cryphonectria, suggesting a very narrow host range. Phenotypic comparison of a few sets of CnCV1-infected and -free isogenic strains showed symptomless infection in new hosts

Peptide nanofibres for drug delivery

It is possible that peptides and proteins may be used to treat a wide range of Central Nervous System diseases if these molecules were able to cross the blood brain barrier (BBB). Currently these molecules do not cross the BBB due to their hydrophilic nature and large size. The aim of this work was to investigate the therapeutic applicability of peptide nanofibres as a new peptide brain delivery system. We hypothesise that hydrophilic peptides, when made lipophilic by attaching an acyl chain via a cleavable ester linkage, are able to cross the blood brain barrier and that the resulting lipidic peptides nanofibres assist in the delivery of these molecules to the brain. Dalargin, a neuropeptide and hexapeptide analogue of Leu-enkephalin that is unable to cross the blood brain barrier, was chosen as model drug; on direct injection into the brain, dalargin acts on brain opioid receptors, resulting in analgesia. An amphiphilic derivative of dalargin, palmitoyl dalargin (pDal) was synthesized which self assembled into high-axial-ratio nanostructures in aqueous environments. We have investigated the physicochemical interactions that control to the formation of peptide nanofibres and found that hydrophobic interactions as well as the formation of amino acid ~-sheets are the main drivers of self-assembly. Brain peptide delivery was assessed following intravenous administration of formulations containing pDal nanofibres, nanofibres prepared from pDal and a chitosan amphiphile - quaternary ammonium palmitoyl glycol chitosan (GCPQA) and dalargin (in the absence and presence of GCPQA). While the administration of control samples of dalargin did not result in dalargin being detected in any tissues (only the primary metabolite was detected), pDal was clearly detected in the brain both in the presence and absence of GCPQA. Furthermore only animals administered with pDal experienced analgesia when assayed using the tail flick test. We conclude that peptide nanofibres offer a unique method for delivering hydrophilic peptides across the blood brain barrier.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

A rational preconditioner for multi-dimensional Riesz fractional diffusion equations

We propose a rational preconditioner for an efficient numerical solution of linear systems arising from the discretization of multi-dimensional Riesz fractional diffusion equations. In particular, the discrete problem is obtained by employing finite difference or finite element methods to approximate the fractional derivatives of order α with α∈(1,2]. The proposed preconditioner is then defined as a rational approximation of the Riesz operator expressed as the integral of the standard heat diffusion semigroup. We show that, being the sum of k inverses of shifted Laplacian matrices, the resulting preconditioner belongs to the generalized locally Toeplitz class. As a consequence, we are able to provide the asymptotic description of the spectrum of the preconditioned matrices and we show that, despite the lack of clustering just as for the Laplacian, our preconditioner for α close to 1 and k≠1 reasonably small, provides better results than the Laplacian itself, while sharing the same computational complexity