Rational approximations to fractional powers of self-adjoint positive operators

Aceto, Lidia; Novati, Paolo

doi:10.1007/s00211-019-01048-4

Cited by 34 publications

(46 citation statements)

References 17 publications

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“…First of all we recall the following result given in [1,Proposition 2], and based on the representation of the error arising from the Padé approximation of the fractional power…”

Section: Error Analysismentioning

confidence: 99%

“…, N ) and L N = A p so that σ(L N ) ⊆ [1, N p ]. Taking N = 100, p = 7, and h = 10 −2 , in Figure 2, for α = 0.2, 0.4, 0.6, 0.8 we plot the error obtained using τ k taken as in (3.19) andτ k as defined in [1,Eq. (24)], that is,…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In particular, working with unbounded operator, in [1] it has been shown how to suitably define the parameter τ by looking for an approximation of the optimal value given by the solution of…”

Section: Introductionmentioning

confidence: 99%

“…We derive an approximate solution τ k of (1.5) that depends on k and h, and we are able to show that the error E k decays like O(k −4α ), that is, sublinearly. We experimentally show that using this new parameter sequence it is possible to improve the approximation attainable by taking τ k as in [1] for L −α and then using (1.1) to compute (I + hL α ) −1 . The latter approach has recently been used in [4].…”

Section: Introductionmentioning

confidence: 99%

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Padé-type Approximations to the Resolvent of Fractional Powers of Operators

Aceto

Novati

2020

J Sci Comput

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We study a reliable pole selection for the rational approximation of the resolvent of fractional powers of operators in both the finite and infinite dimensional setting. The analysis exploits the representation in terms of hypergeometric functions of the error of the Padé approximation of the fractional power. We provide quantitatively accurate error estimates that can be used fruitfully for practical computations. We present some numerical examples to corroborate the theoretical results. The behavior of the rational Krylov methods based on this theory is also presented.

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“…First of all we recall the following result given in [1,Proposition 2], and based on the representation of the error arising from the Padé approximation of the fractional power…”

Section: Error Analysismentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Padé-type Approximations to the Resolvent of Fractional Powers of Operators

Aceto

Novati

2020

J Sci Comput

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“…Our proposal for selecting the poles for the construction of the rational Krylov subspace (4) relies on the rational approximation of z −α/2 proposed in [7][8][9]. In particular, following [8, eq.…”

Section: Poles Selectionmentioning

confidence: 99%

Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

Aceto

Bertaccini

Durastante

et al. 2019

Journal of Computational Physics

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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 numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising.

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Approximation of a fractional power of an elliptic operator

Vabishchevich

2020

Numerical Linear Algebra App

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Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two-dimensional boundary value problem for fractional diffusion. K E Y W O R D S elliptic operator, finite difference approximation, fractional diffusion, fractional power of an operator 1 ∞ 0 − (A + I) −1 d .Numer Linear Algebra Appl. 2020;27:e2287. wileyonlinelibrary.com/journal/nla

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Rational approximations to fractional powers of self-adjoint positive operators

Cited by 34 publications

References 17 publications

Padé-type Approximations to the Resolvent of Fractional Powers of Operators

Padé-type Approximations to the Resolvent of Fractional Powers of Operators

Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

Approximation of a fractional power of an elliptic operator

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