Best Recovery of the Solution of the Dirichlet Problem in a Half-Space from Inaccurate Data
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L DIFFERENTIAL EQUATIONS
Best Recovery of the Solution of the Dirichlet Problem in a Half-Space from Inaccurate Data E. V. Abramovaa,*, G. G. Magaril-Il’yaevb,**, and E. O. Sivkovac,*** a National
Research University “Moscow Power Engineering Institute”, Moscow, 111250 Russia b Lomonosov Moscow State University, Moscow, 119991 Russia c Moscow State Pedagogical University, Moscow, 119991 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received February 20, 2020; revised February 20, 2020; accepted June 9, 2020
Abstract—A family of linear optimal methods for reconstructing the solution of the Dirichlet problem on a hyperplane from information about its approximate measurements on a finite number of other hyperplanes is constructed. In this case, optimal methods do not use all the available information, but only information about the measurements of the solution on at most two planes. Keywords: Dirichlet problem, optimal recovery, extremal problem DOI: 10.1134/S0965542520100036
INTRODUCTION In this paper, we consider the Dirichlet problem in a half-space {( x, y) ∈ d × + } and raise the question of the best recovery of the solution to this problem on a hyperplane “parallel” to d from inaccurate measurements of this solution on a finite number of other hyperplanes parallel to the original one. Explicit expressions are found for methods of optimal recovery and the corresponding reconstruction error is calculated. It should be noted that optimal methods are linear and do not use all available information about measurements of the solution of the Dirichlet problem, but only information about its measurements on at most two hyperplanes. The problem considered in this paper is related to issues concerning the optimal recovery of the values of linear functionals and operators on sets of elements that are specifies imprecisely. Such problems arose in the second half of the last century and was initiated by the works of K. Shannon and A.N. Kolmogorov. The first formulation of the problem of optimal recovery of a linear functional on a class of elements known approximately belongs to A.S. Smolyak [1]. This subject has been actively developed. The initial stage of its development is reflected in reviews [2–4]. Subsequently, the main attention was paid to problems of recovery of functions and their derivatives from an imprecisely specified spectrum and to problems of optimal recovery of solutions of equations of mathematical physics (see, e.g., [5–13]). This work deals with the second class of problems. 1. STATEMENT OF THE PROBLEM AND FORMULATION OF THE MAIN RESULT Consider the Dirichlet problem:
Δu = 0, u(⋅, 0) = f (⋅),
(1)
where Δ is the Laplace operator in d +1 and f (⋅) ∈ L2(d ), which consists in finding a harmonic function u(⋅,⋅) in the half-space ( x, y) ∈ d +1 : y > 0 , such that u(⋅, y) ∈ L2(d ) for any y > 0 ,
{
}
d sup y >0 u(⋅, y) L2(d ) < ∞ , and u(⋅, y) → f (⋅) as y → 0 in the metric of L2( ).
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BEST RECOVERY OF THE SOLUTION OF THE DIR
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