[Paper Review] Stability of solutions to inverse scattering problems with fixed-energy data

А. Г. Рамм|ArXiv.org|Aug 3, 2000

Numerical methods in inverse problems20 references22 citations

TL;DR

This paper establishes stability estimates for 3D inverse scattering problems with fixed-energy data, deriving inversion formulas and error bounds for both exact and noisy potential and obstacle scattering data. It proves uniqueness and stability for inverse potential and obstacle scattering, constructs the Dirichlet-to-Neumann map from scattering data, and provides an analytical example of non-uniqueness in geophysical inverse problems, with rigorous error estimates for reconstruction under noisy conditions.

ABSTRACT

A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse potential scattering problem is discussed in detail, inversion formulas are derived and error estimates are obtained. Inverse obstacle scattering problem with data at a fixed frequency is studied. Uniqueness theorems and stability estimates are obtained. Inverse geophysical scattering problem is discussed. An algorithm for computing the Dirichlet-to-Neumann map from the scattering amplitude and vicxe versa is obtained. An analytical example of non-uniqueness of the solution to a 3D inverse geophysical problem is constructed. An inverse problem for a parabolic equation is discussed.

Motivation & Objective

• To establish stability estimates for inverse scattering problems in three dimensions with fixed-energy data.
• To derive inversion formulas and error estimates for both exact and noisy scattering data in potential scattering.
• To prove uniqueness and stability for inverse obstacle scattering with non-smooth and $C^{2,ar{\lambda}}$ boundaries.
• To construct the Dirichlet-to-Neumann map from scattering amplitude and vice versa.
• To analyze non-uniqueness in geophysical inverse scattering and prove a uniqueness result for parabolic inverse problems.

Proposed method

• Derives a fundamental equation for scattering solutions using the Lippmann-Schwinger integral equation framework.
• Applies the property $C$ for the pair $\{L_1 - k^2, L_2 - k^2\}$ to establish completeness and uniqueness.
• Uses $L^2$ and $L^∞$ estimates on compactly supported potentials $q(x) \in Q_a \cap L^∞(\mathbb{R}^3)$ to derive stability bounds.
• Constructs the Dirichlet-to-Neumann map via inversion of the scattering amplitude, using integral equation techniques.
• Applies Laplace transforms to reduce the time-dependent parabolic inverse problem to an elliptic one.
• Employs analytic continuation and series expansions in $t$ to show analyticity of solutions and extend data beyond $t > 0$.

Experimental results

Research questions

• RQ1 Can stable reconstruction be achieved for 3D inverse potential scattering with fixed energy and noisy data?
• RQ2 What are the stability estimates for inverse obstacle scattering when the boundary is only $C^{2,\lambda}$ with $0 < \lambda < 1$?
• RQ3 Is it possible to uniquely reconstruct the Dirichlet-to-Neumann map from scattering amplitude data, and vice versa?
• RQ4 Does non-uniqueness occur in 3D geophysical inverse scattering problems, and if so, under what conditions?
• RQ5 Can a uniqueness result be established for inverse boundary value problems of parabolic type using Laplace transform and elliptic regularity?

Key findings

• Stability estimates are derived for the inversion of exact data in 3D inverse potential scattering, with error bounds depending on the $L^2$-norm of the potential.
• For noisy data, error estimates are obtained that quantify the stability of the reconstruction under perturbations in the scattering amplitude.
• Uniqueness and stability are proven for inverse obstacle scattering in the class of $C^{2,\lambda}$ domains, $0 < \lambda < 1$, with explicit stability bounds.
• An analytical example of non-uniqueness is constructed for a 3D geophysical inverse problem, demonstrating that scattering data may not uniquely determine the velocity profile.
• A uniqueness theorem is proven for the inverse problem of parabolic equations by reducing it to an elliptic problem via Laplace transform and using Ramm’s uniqueness result.
• The Dirichlet-to-Neumann map can be reconstructed from the scattering amplitude and vice versa, establishing a duality between boundary and scattering data.