Factorization and monotonicity methods for reconstructing impenetrable obstacles in inverse biharmonic scattering

TL;DR

This paper develops new factorization and monotonicity methods for reconstructing impenetrable obstacles in inverse biharmonic scattering, addressing analytical challenges unique to fourth-order operators.

Contribution

It introduces novel factorizations of the far-field operator and establishes range and monotonicity characterizations for obstacle reconstruction across all wavenumbers.

Findings

Proved range identities for the far-field operator.

Derived a monotonicity relation for eigenvalues.

Numerical experiments confirm robustness of the methods.

Abstract

The inverse scattering problem for biharmonic waves, governing flexural vibrations of elastic plates, presents fundamental analytical challenges distinct from acoustic inverse problems due to the fourth-order differential operator and higher-order boundary conditions. This paper addresses the reconstruction of impenetrable obstacles with Dirichlet or Neumann boundary conditions from far-field measurements. We establish new factorizations of the far-field operator by considering structures of the biharmonic fundamental solution and the boundary conditions. We rigorously prove that the factorizations satisfy the range identities and derive characterizations of the obstacle's support by the factorization methods, valid for all wavenumbers except the associated transmission eigenvalues. Furthermore, we establish a monotonicity relation for the eigenvalues of the far-field operator, which yields an alternative characterization of the obstacle's support that remains applicable for all wavenumbers. Numerical experiments for the Dirichlet obstacles with various shapes are presented to demonstrate the effectiveness and robustness of the proposed reconstruction schemes.