Fast Algorithms for Solution of Some Nonlinear Coefficient Inverse Problems for Differential Equations and Applications

Two classes of nonlinear coefficient Inverse Problems (IP) for differential equations are considered. The first class of the IP corresponds to the case of a stationary process, is described by an elliptic equation with some boundary conditions. The problem of the coefficient reconstruction, using measurements on a boundary only, corresponds to the Electrical Tomography. Known methods for the electrical tomography imaging are nonlinear. The new GR-principle is proposed for mathematical modeling the measurement scheme and the basic differential equation. It leads to the new fast linear algorithm, based on the inversion of the classic Radon transformation. The second class is presented as an ordinary differential equation of second order that describes a model of a stationary process of the cerebral cortex dynamics. Two types of approximations of the non lineal member of this equation are used: 1) by a sigmoid function of exponential type; 2) by a lineal spline. The inverse problem consists in reconstruction of parameters in used approximations on the discreet values (measurements) of the solution. Algorithms for the solution of the direct and inverse problems are constructed corresponding to both approximations. For the spline approximation of nonlinear term can be applied the Local Approach that leads to explicit algorithms, which are simpler and faster then algorithms for exponential type approximation. The regularization of the numerical solution is realized for considered classes of problems by the Full Spline Approximation Method. The theoretical justification of constructed algorithms and results of numerical experiments are given.