Optimal recovery of integral operators and its applications

Abstract

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-

Optimal recovery of operator sequences

“…Remark that similar and related lower estimates were established in many papers (see, e.g., [10,5]).…”

On optimal recovery of integrals of random processes

“…In [14] the problem of optimal quadratures for set-valued functions was considered. [15] contains results concerning a very broad generalizations of the classes H ω .…”

“…Recall, that I(t) = and the statement of the corollary follows from (15). Let now the measurements be done by such device, that the first measurement is triggered by some random event (which occurs at the random time τ 1 ) and each of the rest n − 1 measurements are done at time τ k = τ 1 + t k , i. e. in t k time units after the first measurement, k = 2, .…”