A Recursive Algorithm for Multi-Coefficient Inversion in Nonlinear Helmholtz Equations

Abstract

We present a recursive algorithm for multi-coefficient inversion in nonlinear Helmholtz equations with polynomial-type nonlinearities, utilizing the linearized Dirichlet-to-Neumann map as measurement data. To achieve effective recursive decoupling and simultaneous recovery of multiple coefficients, we develop a novel Fourier-based approach that combines the principle of inclusion-exclusion with carefully constructed complex exponential solutions. This methodology not only ensures unique identifiability but also yields progressively increasing stability with enhanced wavenumbers. Comprehensive numerical experiments demonstrate the algorithm's computational efficiency and excellent reconstruction accuracy.