Volume 17, Issue 7

A short review is given on inverse problems of spectral analysis for the Sturm–Liouville differential operator on a finite interval. Using this operator as a model we present the main ideas and methods in the inverse problem theory, and describe the main results for this class of inverse problems.

New constructive methods of investigation of multidimensional inverse problems for nonlinear equations are proposed.

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the equivalence between these two problems for an arbitrary Lipschitz domain in . Moreover, uniqueness of the solution is proved for the general case when Γ is a non-empty open subset of the Lipschitz boundary.

A backward heat conduction problem is reduced to an operator equation Aq = ƒ and is solved in two versions by minimization of functionals R ( q ) = 〈 Aq – 2ƒ, q 〉 and J ( q ) = 〈 Aq – ƒ, Aq – ƒ〉. The performance of the versions is compared both theoretically and numerically. An a priori convergence rate estimate has been obtained in the case of exact data and an a posteriori two-side estimate of the solution error norm has been derived in the case of noisy data. The estimates are illustrated by a model image deblurring problem. Numerical experiments demonstrated that minimization of J ( q ) is more effective for noisy data while R ( q ) works better for exact data.