The methods of the field continuation from the part of the boundary
The field continuation problems from the part of the boundary (or the Cauchy Problem) for hyperbolic equations are ill-posed problems. The similar problems can be found, for instance, in geophysics and tomography when the field continuation allows to detect the parameters of the medium outside the investigated domain.
We reduce the ill-posed problem to the inverse problem and reformulate it in the operator equation Aq=f. For numerical solution of the continuation problem we apply singular value decomposition method and gradient methods.
Theory and numerical methods are developed for the continuation problem. The formulae to calculate the singular values of the continuation problem operator was obtained in case of the simple geometry. The numerical results are presented.
This work is partially supported by the Ministry of Education and Science of the Russian Federation and RFBR grant 14-01-00208.