Introduction to the Theory of Inverse Problems

Chapter 1 Inverse problems for difference operators: inverse problem of spectral analysis for Jacobi matrices; inverse problem for a difference equation with constant coefficients; the problems of determining a difference operator in non-stationary statement; remarks and references. Chapter 2 A priori estimates and the uniqueness of integro-differential equations with operator coefficients: estimates of the Carleman type and their connection with the uniqueness of solutions of inverse problems; estimates for the Schrodinger equation with operator coefficients; remarks and references. Chapter 3 Inverse problems for differential equations: one-dimensional inverse problem for the wave equation in linearized statement; the method of transformation operators; uniqueness in multidimensional inverse problems in nonstationary and spectral statements; remarks and references. Chapter 4 Volterra operator equations and their applications: Volterra operator equations in scales of Banach spaces; non-hyperbolic Cauchy problem for the wave equation; the problem of integral geometry in a strip; the inverse problem of variational calculus; remarks and references. Chapter 5 Foundations of the theory of conditionally well-posed problems: conditional well-posedness lh; well-posedness of difference schemes; variational methods of solution of lh-stable difference schemes; remarks and references. Chapter 6 Theory of stability of difference schemes: statement of the problem and the necessary conditions of finite stability; basic estimates; sufficient stability conditions; estimates of l-stability up to the boundary; convergence theorems; finite stability of two-layer schemes of the canonical form; conditions of stability in terms of the transition operator; remarks and references.