An Inverse Source Problem for Semilinear Stochastic Hyperbolic Equations

TL;DR

This work addresses the inverse source problem for general semilinear stochastic hyperbolic equations, focusing on reconstructing an unknown initial source from partial lateral boundary data. It introduces a globally convergent iterative regularization that fuses Carleman estimates with fixed-point iterations, and proves convergence in weighted spaces along with Hölder-type stability under data noise. A new Carleman estimate for linear stochastic hyperbolic equations underpins this approach, enabling an iterative scheme that does not require a close initial guess. Numerically, the authors solve the forward problem via finite differences and Euler–Maruyama, then minimize a discretized Tikhonov functional using the Adam optimizer with automatic differentiation, avoiding backward SPDEs and handling stochastic data efficiently. The method demonstrates robust source recovery across Lipschitz and non-Lipschitz nonlinearities and varying stochastic data, highlighting practical applicability to inverse problems in random media.

Abstract

This paper investigates an inverse source problem for general semilinear stochastic hyperbolic equations. Motivated by the challenges arising from both randomness and nonlinearity, we develop a globally convergent iterative regularization method that combines Carleman estimate with fixed-point iteration. Our approach enables the reconstruction of the unknown source function from partial lateral Cauchy data, without requiring a good initial guess. We establish a new Carleman estimate for stochastic hyperbolic equations and prove the convergence of the proposed method in weighted spaces. Furthermore, we design an efficient numerical algorithm that avoids solving backward stochastic partial differential equations and is robust to randomness in both the model and the data. Numerical experiments are provided to demonstrate the effectiveness of the method.

Figures (6)

• Figure 1: We adopt the setting of \ref{['ex4']} with $N_{S}=1$. The figure compares the performance of the Adam optimizer and the CGM in solving the inverse problem, demonstrating that Adam achieves faster convergence and more stable optimization when handling stochastic gradients and noisy data.
• Figure 2: We adopt the setting of \ref{['ex1']} with $N_{U}=3000$, $N_{I}=3$, and $N_{S}=100$. The figure shows how the $L^2$ relative error $\frac{|u_{0} - u_{0}^{*}|_{L^{2}(G)}}{|u_{0}^{*}|_{L^{2}(G)}}$ decreases with the number of sample paths $N_{S}$. In subsequent numerical tests, we set $N_{S}=30$.
• Figure 3: The numerical results for \ref{['ex1']}. The computed source function $u_{0}$ is very close to the true source function $u_{0}^{*}$. The relative difference is small, and the $L^{2}$ relative error is 2.1%.
• Figure 4: The numerical results for \ref{['ex2']}. The computed source function $u_{0}$ approximates the true source function $u_{0}^{*}$ well, despite the non-Lipschitz continuity of the nonlinearity $F$. The relative difference is small, and the $L^{2}$ relative error is 6%.
• Figure 5: The numerical results for \ref{['ex3']}. The computed source function $u_{0}$ closely approximates the true source function $u_{0}^{*}$. Despite both $f$ and $g$ being stochastic processes, which further complicates the problem, the relative difference remains minimal and the $L^{2}$ relative error is 17.9%.

Theorems & Definitions (5)

• Theorem 2.1
• proof : Proof of \ref{['thmCarlemanEstimate']}
• proof
• Theorem 3.2
• proof