Inverse and ill-posed sources problems
type: Item
instance of: scholarly article
title: Inverse and ill-posed sources problems
zbMATH Open document ID: 0905.35095
author: Yu. E. Anikonov
author: B. A. Bubnov
author: G. N. Erokhin
published in: Inverse and Ill-Posed Problems Series
publication date: 1998-09-17T00:00:00Z
review text: In this monograph, different settings of inverse problems of mathematical physics, both in linear and nonlinear cases are considered. Most attention is concentrated on unique solvability and on the search for constructive methods. As a rule, the problems considered here are multidimensional. Effective methods for source recovery are described for sources of different nature. In Chapter 1 the authors describe various methods for solving inverse problems of source determination on the basis of integral transforms, the structure method and optimization methods. The main goal is creation of constructive solution methods. In the first paragraph the Meyer transformation is introduced into the inverse problem with axial symmetry. The need to consider more and more complicated processes has led to the structure scheme method for distributed systems. By means of this method, the complicated interactions of various fields are represented in paragraph 1.2. The basic notion of this method is that of a distributed block, where the entrance and exit are selected; and the input signal uniquely transforms into an output signal. The collection of such blocks with a description of its combining rules (parallel, sequential, with closing by an inverse relation and so on) gives a visual scheme of solving the direct problem step by step. The inverse problem for a linear differential equation, from the formal point of view, is either to determine the transmitted block function (determining differential operator), or to determine the input signal (the right side of the equation) by a certain output function (solution). The authors act analogously to the structure method for a linear inverse problem to determine the right-hand side of the differential equation (the structure block input) by certain information on its exit. The block representation of solutions to direct and inverse problems allows not only to construct new settings to determine the right-hand side of the equations but to re…
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 86A10
zbMATH DE Number: 1200113
zbMATH Keywords: integral transformation
zbMATH Keywords: ill-posed source problem
zbMATH Keywords: Meyer transformation
zbMATH Keywords: pseudodifferential equation
zbMATH Keywords: integro-differential equation
zbMATH Keywords: seismic moment tensor
zbMATH Keywords: regularization
zbMATH Keywords: ultrahyperbolic equation
zbMATH Keywords: multidimensional hyperbolic equation
reviewed by: G. Dimitriu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1200113