Band 16, Heft 7

In this paper we construct and investigate localization algorithms for isolated singularities of a function which is a solution to a linear convolution equation of the first kind whose right-hand side is given with an error. We consider two types of singularities: δ -functions and discontinuities of the first kind. A problem for singularities localization is an ill-posed problem having perturbation. We use averaging methods defined by an averaging functional to obtain the singularities. We formulate conditions that the averaging functional must satisfy. For convolution equations, using the Fourier transform, we obtained upper estimates of precision of the singularities localization, separation threshold and other important characteristics of the supposed methods.

In this paper the method of obtaining unimprovable (in certain sense) estimates of solutions of some integral inequalities with the operators of Volterra type is stated. The basis of this method is the theory of monotone operators in partially ordered Banach spaces. This theory allows us to reduce obtaining these estimates to solving corresponding equations. The paper consists of two parts. The first part is devoted to unimprovable estimates of solutions for linear multidimensional inequalities. In the second part the author states nonlinear inequalities which arise while researching multilinear Volterra equations of the first kind connected with modelling nonlinear dynamic systems of black body type by Volterra polynomials.

The estimates for the number of operations needed to implement two different iteratively regularized Gauss–Newton methods as well as the iteratively regularized gradient scheme are given. The operation count is illustrated by simulations for a two dimensional version of the exponentially ill-posed optical tomography inverse problem for the diffusion ( D ) and absorption ( μ a ) coefficient spatial distributions.

The inverse problem for a semilinear system of partial differential equations is studied and the theorem of uniqueness of solution to the inverse problem is proved.

In this paper we apply the notion of quasi-solution to nonlinear inverse coefficient problems. Instead of a compact set M we use the ball B (0, r ) in which the radius r occurred to be sometimes a regularization parameter. Moreover this constant allows one to estimate the convergence rate for many well-known algorithms for solving inverse coefficient problems and to decrease crucially the number of iterations.

Inverse nodal problems are studied for second-order differential operators on star-type graphs with standard matching conditions in the internal vertex. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.