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Multi-parameter determination in the semilinear Helmholtz equation

Long-Ling Du, Zejun Sun, Li-Li Wang, Guang-Hui Zheng

Abstract

This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ ($n\ge2$). 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 $C^γ(\overlineΩ)$ regularity assumptions with $0<γ<1$, while in the two-dimensional case uniqueness is obtained under Sobolev regularity $W^{1,p}(Ω)$ 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.

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 (). 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 , uniqueness holds under regularity assumptions with , while in the two-dimensional case uniqueness is obtained under Sobolev regularity with . 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.
Paper Structure (10 sections, 11 theorems, 94 equations, 12 figures, 2 tables)

This paper contains 10 sections, 11 theorems, 94 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Well-posedness of the forward problem
  3. Uniqueness of the linear inverse problem
  4. Multiparameter determination in the semilinear He-lmholtz equation
  5. Numerical reconstruction
  6. Reconstruction algorithm
  7. Numerical examples
  8. Example 1
  9. Example 2
  10. Conclusion

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$ ($n \geq 3$) be a connected bounded open domain with $C^\infty$ boundary. Suppose $\alpha_1, \alpha_2, \beta_1, \beta_2 \in C^\gamma(\overline{\Omega})$. If $N_{\alpha_1,\beta_1} = N_{\alpha_2,\beta_2}$, then $\alpha_1 = \alpha_2$ and $\beta_1 = \beta_2$ in $\Omega$

Figures (12)

  • Figure 5.1: region ABCD
  • Figure 5.2: Divided ABCD
  • Figure 5.3: Sample traces of all coefficients in Example \ref{['exa1']}.
  • Figure 5.4: Sample autocorrelation functions of all coefficients in Example \ref{['exa1']}.
  • Figure 5.5: Sample mean and standard deviation ($\sigma$) for reconstructing $\alpha$ and $\beta$ in Example \ref{['exa1']}
  • ...and 7 more figures

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 7 more