Inverse and ill-posed sources problems
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 reduce the complicated problem to simpler ones. Paragraph 1.2 also contains formulas for problems of source determination applied to ten specific examples. The problem of determining the index and the right-hand side of the pseudodifferential equation is treated in paragraph 1.3. Investigation of inverse problems connected with determining the source characteristics of various physical fields leads to group analysis methods for initial information. Under some conditions, a certain ``symmetry'' in the measured information corresponds to a certain ``symmetry'' of the desired source. The method described in section 1.4 is based on the two physical suppositions: a) the source event is bounded and its domain does not intersect the domain of measurements, and b) the medium has a certain symmetry. Section 1.5 contains inversion formulas for the wave equation. Section 1.6 entitled ``Conjugated equations in control problems of nonstationary geophysical fields'' treats with the conjugate equations method for problems of geophysical fields control on the basis of variation of radiation source parameters. The inverse problem of source determination as the problem of statistical evaluation of parameters of the mathematical model is treated in section 1.7. The last paragraph of Chapter 1 contains formulas in multidimensional inversion problems for evolution equations. Chapter 2 treats different multidimensional inverse problems for the determination of sources, both in linear and in nonlinear cases. Here, the authors cite many results concerning the newly-posed problems, their solvability and uniqueness. When one solves inverse problems for hyperbolic and parabolic equations, one often meets the Cauchy problem for elliptic and other types of equations. The Cauchy problem in the class of analytic functions has been investigated quite thoroughly, but in applications the problem arises of its solvability in classes of finite smoothness. As follows from the well-known Petrovskii theorem, the Cauchy problem is ill-posed for elliptic equations in classes of finite smoothness. In the first paragraph of Chapter 2 the authors give necessary and sufficient conditions of solvability for elliptic and ultrahyperbolic equations in classes of functions of finite smoothness, and also show how these conditions are effectively applied for inverse problem research. The next section analyses inverse problems for multidimensional hyperbolic equations. In section 2.3 some problems of control through the nature of the source are interpreted as inverse problems for differential equations. The control considered here is described by two coefficients entering into the evolutionary equations and not depending on time. If control data are given, the inverse problems are reduced to the boundary-value problems for nonlinear integro-differential equations or to boundary-value problems for infinite systems of nonlinear differential equations. The basic mathematical difficulties are connected with ill-posedness study of such settings, especially, the problem of existence. After solving these problems, the control functions are obtained constructively. Using a portion of the control data, the authors write out nonlinear integro-differential equations. Then, on the basis of the obtained equation and taking into account the boundary-value information, results of existence, uniqueness and stability are formulated. The paragraphs 2.4 and 2.5 analyse inverse problems for nonlinear equations and inverse problems of transfer theory, respectively. The results are related to both the uniqueness and existence of solutions to these problems. The last paragraph, 2.6, entitled ``Geometric problems'' considers the integral equation of the first kind, which arises when we try to determine the convex domain by functionals of its orthogonal projections (shadows) and illuminated parts. There are given some theorems on unique definiteness of a closed convex surface, if the functionals of its orthogonal projections and illuminated parts are known. In Chapter 3 the authors consider the numerical solution of mathematical simulation of seismic and acoustic fields from complicated sources on the basis of measured information. The peculiarity of these sources is their possible motion. As for solution of inverse problems, because of their instability, the main focus of attention is the construction of optimal regularization methods. The first paragraph of this chapter deals with the problems of simulation of a seismic vibration field from sources which are characterized by a wide spectrum of radiation, by complicated spatial directness, and by their spatial motion. The second paragraph contains the problems of numerical solution of the inverse problem of determining the seismic moment tensor of the seismic centre and the coordinates of the source hypocentre, by the measured volume waves in some points of elastic space. Constructive algorithms of inversion for certain right-hand sides of differential equations are also suggested. The next section presents the numerical realization of the suggested algorithms. The stability of the regularized solution is theoretically and numerically discussed. It is pointed out the importance of using inhomogeneous waves in the inversion algorithms. The problem of distance quality control of landing an airplane on a flight-landing strip on the basis of analysis of seismic records of the wave process which arises during the landing is analyzed in the last paragraph of the third chapter. The authors give an analytic description of the source which simulates the airplane landing and justify the chosen mathematical model which describes the process of disturbance propagation in the medium. This monograph, together with previous monographs of Anikonov Yu. E., ``Multidimensional Inverse and Ill-Posed Problems for Differential Equations'' and ``Formulas in Inverse and Ill-Posed Problems'', propose a wide constructive collection of investigation methods in this area. The main results of this book are due to the authors.
scientific article; zbMATH DE number 1200113