Interpolation Operators as Optimal Recovery Schemes for Classes of Analytic Functions

Golomb, Michael

doi:10.1007/978-1-4684-2388-4_4

Cited by 11 publications

(3 citation statements)

References 11 publications

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“…Problems of optimal recovery of operators based on exact information were studied in [27,5,9,20], and on approximate information in [15,19,21,17,16,4]. We also refer the reader to the discussion of closely related questions in [28,20,12,30,24,1,2].…”

Section: 2mentioning

confidence: 99%

Optimal recovery of integral operators and its applications

Бабенко

Babenko

Parfinovych

et al. 2016

Journal of Complexity

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Abstract. In this paper we present the solution to the problem of recovering rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE's.Key words. optimal recovery, approximation, information with error, integral operators, integral equations, initial and boundary value problems AMS subject classifications. 41A35, 45P05, 35G151. Introduction. Solutions to boundary (or initial) value problems for various partial differential equations require knowledge of a boundary (or initial) function. However, often time, those functions are not fully known and only partial information about them can be measured, e.g. values at some finite set of points, average values over small measurement intervals, values of N first consecutive Fourier coefficients, etc. Thus, it is very important to find an approximate solution based on available information on the boundary (or initial) function. Furthermore, it is also natural and important to develop methods that provide an optimal (in some sense) approximation to the true solution. These research questions have been explored under the theory of optimal recovery of functions and operators, which is an area of Approximation Theory that started to develop in 1970s. More information on the development of the area can be found, for instance, in [20,28,21,12,17,24,25,11].As for specific applications to recovering solutions of boundary and initial value problems, Magaril-Ill'yaev, Osipenko, and co-authors (see, for instance, [16,22,18]) have considered the problem of optimal L 2 -approximation of the solution to the Dirichlet problem for Laplace's and Possion's equations in simple domains (disk, ball, annulus) based on the first N consecutive Fourier coefficients of the boundary function (possibly given with an error). In order to solve this problem they have used methods of Harmonic Analysis and general results from Optimization Theory.In this paper we address related questions of optimal approximation of the solution to several types of integral equations, boundary and initial value problems for PDE's. We begin by solving a more general problem of recovering a rather arbitrary integral operator and sum of operators. We then present the optimal method of recovery as well as the optimal error. Next, we apply this general result to recover solutions to various boundary and initial value problems. Moreover, we present optimal methods of recovery of the solution to boundary-value problems based on this incomplete information with error. Naturally, the solution to the problem when information with error is used will also lead to the solution to the problem with exact information. In this paper we focus on considering the Volterra's and Fredholm's linear integral equations as well as boundary value problems for wave, heat, and Poisson's equa-

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Section: 2mentioning

confidence: 99%

Optimal recovery of integral operators and its applications

Бабенко

Babenko

Parfinovych

et al. 2016

Journal of Complexity

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“…[13,18,20]), that explicitly provide an interpolant Rf for f . For instance, the optimal algorithm of [13] for p = 2 is given by…”

Section: Introductionmentioning

confidence: 99%

“…interesting remedy is proposed in [13], where a filtering is used in (2)). The optimal algorithm for p = +∞ (see [20]) behaves in a similar way with respect to perturbations δw.…”

Section: Introductionmentioning

confidence: 99%

On the numerical recovery of a holomorphic mapping from a finite set of approximate values

Prieto¹,

Palencia

2002

Numerische Mathematik

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The least squares approach for the recovery of a holomorphic mapping from given perturbed nodal values is considered. The mapping is assumed to be a priori bounded by a known quantity, so that the recovery problem is well-posed. The present analysis shows that for nodes that are the zeroes of Jacobi polynomials a quasioptimal algorithm results with a fairly moderate number of nodes. The analysis also takes into account the effect of the numerical linear algebra involved. Some numerical experiments are presented.

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Optimal nodes for interpolation in Hardy spaces

Schaback¹

1982

Math Z

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Interpolation Operators as Optimal Recovery Schemes for Classes of Analytic Functions

Cited by 11 publications

References 11 publications

Optimal recovery of integral operators and its applications

Optimal recovery of integral operators and its applications

On the numerical recovery of a holomorphic mapping from a finite set of approximate values

Optimal nodes for interpolation in Hardy spaces

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