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DOI: 10.46698/z9615-4666-9912-l On Reconstry the Solution to the Dirichlet Problem Using Information Known with a Random Error
Burikova, S. V.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3.
Abstract:
This paper considers the problem of optimal recovery of a solution to the Dirichlet problem for the unit ball in \(\mathbb R^d\) centered at zero. The solution is to be recovered on a sphere of radius \(\rho <1\) centered at zero, given the following information: it is known that the boundary function \(f\), defined on the unit sphere \(\mathbb S^{d-1}\), belongs to a certain class of functions contained in the space \(L_2(\mathbb S^{d-1})\), and for each element of this class, a finite set of its first Fourier coefficients is known with random error. The class of functions is the set of those functions \(f\in L_2(\mathbb S^{d-1})\) for which the norm in \(L_2(\mathbb S^{d-1})\) of a given fractional power of the spherical Laplace-Beltrami operator of this function does not exceed unity. In this paper, explicit expressions are constructed for optimal recovery methods that depend on the variance of the random error, and exact values of the corresponding optimal recovery errors are found. These methods are linear and, generally speaking, do not use all the information available for measurement. Explicit expressions for optimal methods allow their use in constructing efficient numerical algorithms for solving similar problems, where the initial information is specified inaccurately. The problem solved here belongs to a branch of approximation theory that emerged in the 1970s and is called the theory of optimal recovery of linear functionals and operators on classes of sets whose elements are known approximately. This theory has been actively developed and generalized in various directions. Many specific problems have been solved, but the primary focus has been on problems with deterministic error. The present is one of the first to provide an exact solution to the optimal recovery problem when information is known with random error.
Keywords: Dirichlet problem, optimal recovery, Fourier coefficients, extreme problem
Language: Russian
For citation: Burikova, S. V. On Reconstry the Solution to the Dirichlet Problem Using Information Known with a Random Error, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp. 51-59 (in Russian). DOI 10.46698/z9615-4666-9912-l
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue |
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