32 research outputs found
Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications
In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively.
Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given.
All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator.
In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8
Efficient approximation of functions of some large matrices by partial fraction expansions
Some important applicative problems require the evaluation of functions
of large and sparse and/or \emph{localized} matrices . Popular and
interesting techniques for computing and , where
is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from by a complex multiple of the identity matrix for computing
or require inverting sequences of matrices with the same
characteristics for computing . Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests
Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications
In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively.
Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given.
All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator.
In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8
Efficient computation of the sinc matrix function for the integration of second-order differential equations
This work deals with the numerical solution of systems of oscillatory
second-order differential equations which often arise from the
semi-discretization in space of partial differential equations. Since these
differential equations exhibit pronounced or highly) oscillatory behavior,
standard numerical methods are known to perform poorly. Our approach consists
in directly discretizing the problem by means of Gautschi-type integrators
based on matrix functions. The novelty contained here is
that of using a suitable rational approximation formula for the
matrix function to apply a rational Krylov-like
approximation method with suitable choices of poles. In particular, we discuss
the application of the whole strategy to a finite element discretization of the
wave equation
EFFICIENT COMPUTATION OF THE WRIGHT FUNCTION AND ITS APPLICATIONS TO FRACTIONAL DIFFUSION-WAVE EQUATIONS
In this article, we deal with the efficient computation of the Wright function in the cases of
interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations
Theoretical error estimates for computing the matrix logarithm by Pad\'e-type approximants
In this article, we focus on the error that is committed when computing the
matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can
be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric
function. Empirical observation tells us that the convergence of these
quadratures becomes slow when the matrix is not close to the identity matrix,
thus suggesting the usage of an inverse scaling and squaring approach for
obtaining a matrix with this property. The novelty of this work is the
introduction of error estimates that can be used to select a priori both the
number of Legendre points needed to obtain a given accuracy and the number of
inverse scaling and squaring to be performed. We include some numerical
experiments to show the reliability of the estimates introduced
Nonlocal PageRank
In this work we introduce and study a nonlocal version of the PageRank. In
our approach, the random walker explores the graph using longer excursions than
just moving between neighboring nodes. As a result, the corresponding ranking
of the nodes, which takes into account a \textit{long-range interaction}
between them, does not exhibit concentration phenomena typical of spectral
rankings which take into account just local interactions. We show that the
predictive value of the rankings obtained using our proposals is considerably
improved on different real world problems
Rational Krylov methods for functions of matrices with applications to fractional partial differential equations
In this paper, we propose a new choice of poles to define reliable rational
Krylov methods. These methods are used for approximating function of positive
definite matrices. In particular, the fractional power and the fractional
resolvent are considered because of their importance in the numerical solution
of fractional partial differential equations. The results of the numerical
experiments we have carried out on some fractional models confirm that the
proposed approach is promising
Why diffusion-based preconditioning of Richards equation works: spectral analysis and computational experiments at very large scale
We consider here a cell-centered finite difference approximation of the
Richards equation in three dimensions, averaging for interface values the
hydraulic conductivity , a highly nonlinear function, by arithmetic,
upstream, and harmonic means. The nonlinearities in the equation can lead to
changes in soil conductivity over several orders of magnitude and
discretizations with respect to space variables often produce stiff systems of
differential equations. A fully implicit time discretization is provided by
\emph{backward Euler} one-step formula; the resulting nonlinear algebraic
system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the
solution of a sequence of linear systems involving Jacobian matrices. We prove
some new results concerning the distribution of the Jacobians eigenvalues and
the explicit expression of their entries. Moreover, we explore some connections
between the saturation of the soil and the ill-conditioning of the Jacobians.
The information on eigenvalues justifies the effectiveness of some
preconditioner approaches which are widely used in the solution of Richards
equation. We also propose a new software framework to experiment with scalable
and robust preconditioners suitable for efficient parallel simulations at very
large scales. Performance results on a literature test case show that our
framework is very promising in the advance towards realistic simulations at
extreme scale