Math Department Invited Talks

[Math Department Invited Talk] Multidimentional Gelfand-Levitan-Krein-Marchenko Equations and Big Data


Hyperbolic equations describing the wave processes are of great concern in many areas of applied mathematics. Waves come through object and deliver information about its structure to the surface of measurements. Solutions of hyperbolic equations can contain non-smooth and singular components. This leads to more effective inversion of the operator compared with elliptic and parabolic equations. Usually inverse problems for hyperbolic equations are solved by minimizing the residual functional. Iterative method of minimizing the functional requires the solution of the direct (and, perhaps, adjoint) problem for every iteration of the method. In multidimensional case iterative methods for inverse problems are very time-consuming. The Gelfand-Levitan-Krein-Marchenko (GLKM) approach overcomes nonlinearity of the problems – the nonlinear inverse problem reduces to a system of linear integral equations.  GLKM method in some sense is the direct method – there is no need to solve the forward problem (no iteration process). In numerical solution of multidimentional GLKM equations we use fast Toeplitz matrix inversion (Levinson, Durbin, Trench, Tyrtyshnikov, Voevodin) and Monte Carlo approach.