On increasing stability and minimal data in inverse problems

Abstract: We expose (with basic ideas of proofs) recent results about improving stability in the Cauchy problem for general elliptic partial differential equations of second order of Helmholtz type without any geometrical assumptions on domains and operators when the wave number is growing. The next topic is better stability in in the inverse source scattering problems with the boundary data at an interval of wave numbers when this interval is getting larger. We give rather complete theory for the Helmholtz equation (based on sharp bounds of analytic and exact observability for the wave equation), as well as convincing numerical examples. Similarly we discuss recovery of the Schroedinger potential from the Dirichlet-to Neumann map. Finally, we report on first results on the inverse problems where the wave number is zero (or small), showing that in the two dimensional case of inverse gravimetry in a realistic practical situation one can stably find only 5 real parameters of gravity force at the boundary and with this data uniquely determine an ellipse.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine