Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems

  title={Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems},
  author={Mauricio Bogoya and Sergei M. Grudsky and Stefano Serra Capizzano and Cristina Tablino Possio},
In the present article we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs). From one side they could look standard, since they are real, symmetric and positive definite. On the other hand they cause specific difficulties which prevent the successful use of classical tools. In particular the associated matrix-sequence, with respect to the matrix-size, is ill-conditioned and it is such that a… 
1 Citations

On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems

The analysis of the spectral features of a Toeplitz matrix-sequence { Tn(f) } n∈N, generated by a symbol f ∈ L([−π, π]), real-valued almost everywhere (a.e.), has been provided in great detail in the



Symbol-based preconditioning for Riesz distributed-order space-fractional diffusion equations

In this work, the numerical solution of a 1D distributed-order space-fractional diffusion equation is examined by means of an implicit finite difference scheme based on the shifted Grünwald-Letnikov formula, and the resulting linear systems show a Toeplitz structure.

New PCG based algorithms for the solution of Hermitian Toeplitz systems

  • S. Serra
  • Computer Science, Mathematics
  • 1995
It is proved that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and asymptotic information is obtained about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R).

Toeplitz Preconditioners Constructed from Linear Approximation Processes

  • S. Capizzano
  • Computer Science, Mathematics
    SIAM J. Matrix Anal. Appl.
  • 1998
It is proved that the necessary (and sufficient) condition, in order to devise a superlinear PCG method, is that the spectrum of the preconditioners is described by a sequence of approximation operators "converging" to f.

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

This paper extends the technique to the nonlinear, nonsymmetric case and proves some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear.

On the extreme eigenvalues of hermitian (block) toeplitz matrices

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Abstract. Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system $A_n\vec{x}=\vec{b}$ [59,9,20,5,34,62,6,10,28,45,44,46,49].

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

  • S. Capizzano
  • Computer Science, Mathematics
    Numerische Mathematik
  • 2002
By extending the latter approach, this work performs a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at $x^0=0$ of finite order.

A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative

This work considers the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter p, and derives (two-sided) Jacobi polyfracnomials as eigenfunctions of a Sturm--Liouville problem with weights uniquely determined by the parameter p.