Approximation of a fractional power of an elliptic operator

Vabishchevich, Petr N.

doi:10.1002/nla.2287

Cited by 14 publications

(14 citation statements)

References 20 publications

(32 reference statements)

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“…with the parameter κ > 1. We used such an integral representation in the article [20]. The advantages of this representation are (i) we integrate on a finite interval, (ii) we avoid the singularity of the integrand, (iii) and we control the smoothness of the integrand by choosing the parameter κ of the integral representation.…”

Section: Rational Approximationmentioning

confidence: 99%

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Vabishchevich¹

2021

Preprint

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Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. In this paper, the possibility of using approximation by exponential products is noted. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation.

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Section: Rational Approximationmentioning

confidence: 99%

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Vabishchevich¹

2021

Preprint

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“…Moreover we show the results of the approximations (40), ( 43) and (44), for τ = τ * . Clearly the ideal situation would be to have τ * such that ϕ(1, τ * ) = ϕ(λ, τ * ), but notwithstanding Remembering that In Figure 4 we show the behavior of the method for the computation of L −α , with L defined in (17), together with the estimate (47). For comparison, in the same pictures we also plot the error of the SE approach.…”

Section: 41mentioning

confidence: 99%

“…Among the approaches recently introduced we quote here the methods based on the best uniform rational approximations of functions closely related to λ −α that have been studied in [6,7,8,9]. Another class of methods relies on quadrature rules arising from the Dunford-Taylor integral representation of λ −α [1,2,3,4,5,17,18]. Very recently, time stepping methods for a parabolic reformulation of fractional diffusion equations, proposed in [19], have been interpreted by Hofreither in [10] as rational approximations of λ −α .…”

Section: Introductionmentioning

confidence: 99%

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

Aceto¹,

Novati²

2021

Preprint

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In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a double-exponential transform of the integrand function. For the first approach our aim is only to review some theoretical aspects in order to refine the choice of the parameters that allow a faster convergence. As for the double exponential transform, in this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.

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“…The variety of approximation options (10) (see for example [12,15,16,18]) is associated with the choice of coefficients a i (α), b i (α), i = 1, . .…”

Section: Rational Approximation For Fractional Powers Of the Operatormentioning

confidence: 99%

“…Many works (see, for example, [14,15,16]) are based on the integral representation (Balakrishnan formula [17]) of an operator's fractional power using specific quadrature formulas. We also used other integral representations [18]. The solution to a fractional power operator's problem can be represented as the solution to some auxiliary problems of a larger dimension.…”

Section: Introductionmentioning

confidence: 99%

Splitting Schemes for Non-Stationary Problems with a Rational Approximation for Fractional Powers of the Operator

Vabishchevich¹

2020

Preprint

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Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint positive operator. In computational practice, rational approximations of the fractional power operator are widely used in various versions. The purpose of this work is to construct special approximations in time when the transition to a new level in time provided a set of standard problems for the operator and not for the fractional power operator. Stable splitting schemes with weights parameters are proposed for the additive representation of rational approximation for a fractional power operator. Possibilities of using similar time approximations for other problems are noted. The numerical solution of a two-dimensional non-stationary problem with a fractional power of the Laplace operator is also presented.

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Approximation of a fractional power of an elliptic operator

Cited by 14 publications

References 20 publications

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

Splitting Schemes for Non-Stationary Problems with a Rational Approximation for Fractional Powers of the Operator

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