Inverse Obstacle Scattering with Non-Over-Determined Scattering Data

Reconstructing an obstacle from scattered waves is an inverse problem with a deceptively sharp question: how little data can still determine the unknown surface? In Inverse Obstacle Scattering with Non-Over-Determined Scattering Data, Alexander G. Ramm develops a uniqueness result for the minimal-data setting, connecting scattering amplitude measurements to both the obstacle’s shape and its boundary condition.

When the data are just sufficient

Here, “non-over-determined” describes scattering data whose dimensionality matches that of the unknown object. Ramm considers data known over an open subset of the unit sphere and proves that these measurements uniquely determine the obstacle’s surface. The result also identifies its boundary condition as Dirichlet, Neumann, or impedance type.

From direct scattering to the inverse problem

The treatment addresses the direct scattering problem before turning to inverse obstacle scattering. This progression places the reconstruction question in its mathematical setting and clarifies why uniqueness from minimal data matters: it establishes that the available measurements can identify the unknown surface, rather than merely supplying an excess of observations.

A rigorous result for rough surfaces

Alongside the uniqueness theorem, the book gives a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces. Together, these results make the work especially relevant to readers interested in the foundations of scattering analysis and the mathematical limits of inverse reconstruction.

For readers in analysis and scattering theory

This focused research monograph is suited to mathematicians, graduate-level researchers, and advanced students working in inverse problems, scattering theory, mathematical analysis, or applied mathematics. It is a specialized treatment of a precise theoretical question, not a general introduction to wave physics.

For a closer look at uniqueness in inverse obstacle scattering—and at what can be recovered from minimally dimensioned scattering data—Ramm’s work offers a concentrated, proof-oriented study.