Volume 20, Issue 5-6

Abstract. The problem of a posteriori accuracy estimation for approximate solutions of ill-posed extremal problem is under investigation. We present a new approach to a posteriori estimation along with the study of principal properties of the used estimator. The numerical algorithm for the solution of central extremal problem of a posteriori accuracy estimation is formulated as well. We analyze the optimality in order of obtained a posteriori accuracy estimates and introduce a new important notion of extra-optimal regularizing algorithm as a method for the solution of ill-posed problems having optimal in order a posteriori accuracy estimate. Finally, we demonstrate the results of numerical experiments on applications of extra-optimal regularizing methods, constructed on the base of Tikhonov and Lavrent'ev regularization, for calculating a posteriori accuracy estimates.

Abstract. We prove regularization for a certain ill-posed parabolic evolution problem in a Banach space X by obtaining Hölder-continuous dependence of its solution on modeling. In particular, we consider the generally ill-posed problem , , with initial data , in X where is the infinitesimal generator of a bounded holomorphic semigroup of angle on X , and for each with for . Assuming there exists a solution of the problem adhering to certain stabilizing conditions, we approximate by the solution of an approximate well-posed problem. We then use this estimate to prove the existence of a family of regularizing operators for the given ill-posed problem. The theory has applications to the backwards heat equation and other ill-posed partial differential equations in , , with time-dependent coefficients.

Abstract. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention in the classical literature on inverse heat conduction problems. In this article, the Ritz–Galerkin method with satisfier function is utilized to solve the inverse problems of identifying the temperature and the unknown diffusion coefficient in the one-dimensional heat equation from various additional information. Numerical examples are presented and discussed.

Abstract. We solve an identification problem for source functions to one-dimensional second order parabolic-elliptic system. Considered the system of equations obtained from the original system, where the time derivative added to the elliptic equation, containing a small parameter , we prove the following: the solvability “in general” of the inverse problem for , uniqueness of classical solutions of the inverse problem, periodicity of the spatial variable solutions of approximating problems for , a priori (uniform in ) estimates of solutions of approximating problems, convergence on the basis of the a priori estimates of solutions approximating the inverse problems to solutions original for , the rate of convergence (order ) of solutions of approximating problems in classes of continuous functions. An identification problem of source functions in the system of composite type is treated by some authors. The case where the unknown component of the vector source function in the equation that does not contain the small parameter was studied by Belov (2010).

Abstract. The ill-posed backward parabolic equation with a densely defined linear operator A such that generates an analytic semigroup in a Banach space X and being given is stabilized by the Tikhonov regularization method and by the well-posed non-local boundary value problem A priori and a posteriori parameter choice rules for these regularization methods are suggested which yield the error estimate for all , where are computable constants, E is a bound for and is a defined harmonic function.

Abstract. A family of real functions defining a spectral regularization method with optimal qualification is considered. Sufficient condition on the family and on the optimal qualification guaranteeing the existence of saturation are established. Appropriate characterizations of both the saturation function and the saturation set are found and some examples are provided.

Abstract. The direct measurement of temperatures and heat fluxes may be difficult or impossible on boundaries that are inaccessible, such as internal cavities, or exposed to harsh environmental conditions that would destroy the thermal sensors. In such circumstances, one may inversely determine the temperature and heat fluxes on these unknown boundaries by using over-specified conditions on boundaries where such information can be readily collected. This assumes the geometry and material properties of the domain are known. Algorithms for solving these problems, such as those based on finite difference, finite element, and boundary element, are well known for the case where measured boundary conditions are not a function of time. In this work, we demonstrate an inverse finite element method that effectively solves this inverse heat conduction problem using over-specified temperatures and heat fluxes that are time varying. The material properties may be highly heterogeneous and non-linear. A boundary regularization method is used to stabilize the method for cases involving errors in temperature and heat flux measurements. Several three-dimensional examples are given using simulated measurements with and without measurement errors, to demonstrate the accuracy of the method.

Abstract. There is studied well-posedness of the Cauchy problem on torus to a magnetoelastic system. The mathematical model consists of three coupled partial differential equations. One of them is a hyperbolic equation describing the elastic medium and the other two equations form a parabolic system originated from diffusion Maxwell's equations. Experimental measurements suggest that the elastic medium has a periodic structure, moreover with finite number of discontinuities on the fundamental domain. Thus we have studied in this paper the problem which we have defined as periodically Cauchy diffraction problem. The basic result is to prove existence, uniqueness and stability of solutions of the formulated problem.

Abstract. We consider the Lavrentiev method for the regularization of linear ill-posed problems with noisy data. Classical rules for the choice of the regularization parameter that use the noise level work well for almost exact noise level information, they fail in the case of the underestimated noise level and give a much larger error than the error by the optimal parameter in the case of the overestimated noise level. Heuristic rules that do not use the noise level give often good results but can not guarantee the convergence. We propose a general family of parameter choice rules, which give good results also in the case of many times under- or overestimated noise level. This family combines heuristic rules and quasi-optimal rules that use the noise level. The quasi-optimality is proved for a sub-family of rules. The advantages of the new rules are demonstrated in extensive numerical experiments.

Abstract. Inverse spectral problems are studied for second-order differential pencils on a finite interval with arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function for the selfadjoint Sturm–Liouville operator. The connection with other types of spectral characteristics is investigated. By the method of spectral mappings a uniqueness theorem is proved and constructive procedures for solving the inverse problem are obtained.