Multi-parameter determination in the semilinear Helmholtz equation

Abstract

This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n\ge2$). The objective is to recover the unknown linear and nonlinear coefficients from the associated Neumann-to-Dirichlet (NtD) map. Using a higher-order linearization approach, we establish the unique determination of both coefficients from boundary measurements. For spatial dimensions $n\ge3$, uniqueness holds under $C^\gamma(\overline{\Omega})$ regularity assumptions with $0<\gamma<1$, while in the two-dimensional case uniqueness is obtained under Sobolev regularity $W^{1,p}(\Omega)$ with $p>2$. The analysis relies on the well-posedness of the forward problem together with techniques from linear inverse problems, including Runge-type approximation arguments and Fourier analysis. In addition, we develop a numerical reconstruction framework for recovering the coefficients from boundary data. The forward problem is discretized using a finite difference scheme combined with a quasi-Newton iteration, and the inverse problem is formulated within a Bayesian inference framework. Posterior distributions of the coefficients are explored using the preconditioned Crank-Nicolson (pCN) Markov chain Monte Carlo algorithm, which provides both point estimates and uncertainty quantification. Numerical experiments demonstrate the effectiveness of the proposed reconstruction method and illustrate the theoretical uniqueness results.