本研究旨在探討並開發一套針對波動方程逆問題（InverseProblems）的穩健數值重建框架，重點在於估計空間中變動的位能係數q(x)的波動方程。傳統基於偏微分方程理論的直接反演算法，在面對離散觀測數據時，往往受限於數值微分的不穩定性，且極易受到測量雜訊的干擾。為了解決此一困境，本研究提出將逆問題重新定義為參數空間內的隨機優化問題，透過蒙地卡羅方法（MonteCarlomethod）進行大規模隨機抽樣，並輔以有限元素法（FiniteElementMethod, FEM）精確求解正向波動方程，藉此尋求使模擬波場與參考數據間相對誤差最小化的參數分佈。
數值實驗結果證實，對於低維度的網格劃分，蒙地卡羅方法表現出極佳的收斂特性與抗噪能力，能有效捕捉到位能係數在空間中的分佈趨勢與量值特徵。針對計算效能需求，本研究進一步整合MATLAB平行運算環境，利用多核心工作節點平行化執行大量的正向模擬任務，實驗顯示其運算效率隨核心數增加有顯著提升，有效縮短了求解複雜參數識別問題所需的時間。最後，本研究探討了參數空間維度對重建精度之影響，指出隨著網格精細化導致的「維度詛咒」挑戰。整體而言，本研究所建立的框架為波動方程逆問題提供了一種兼具理論可行性與實作彈性的數值方案。

This research focuses on the development and evaluation of a robust numerical reconstruction framework for the inverse problem of the wave equation, with a specific emphasis on estimating the spatially varying potential coefficient q(x). In many physical applications,such coefficients characterize the inhomogeneous properties of a medium, and identifying their distribution from observed wavefields remains a significant challenge. Traditional direct inversion algorithms derived from partial differential equation theory often suffer from numerical instabilities associated with high-order differentiation and exhibit extreme sensitivity to observational noise. To address these issues, this study reformulates the inverse problem as a stochastic optimization task within the parameter space. By employing the Monte Carlo method for large-scale random sampling in conjunction with the Finite Element Method(FEM) for precise forward modeling, the framework seeks to identify the optimal parameter distribution that minimizes the relative L2 error between simulated wavefields and reference data.
Numerical experiments demonstrate that for lower-dimensional grid configurations, the Monte Carlo approach exhibits superior convergence and robustness against noise, effectively capturing the spatial patterns and magnitudes of the potential coefficient q(x). To accommodate the heavy computational workload inherent in stochastic sampling, the study
integrates the MATLAB Parallel Computing Toolbox to parallelize the execution of forward simulations across multiple local workers. The results indicate that computational efficiency scales significantly with the number of processing cores, substantially reducing the time required for complex parameter identification. Furthermore, this work investigates the impact of parameter dimensionality on reconstruction precision, highlighting the ”curse of dimensionality” encountered as the grid resolution increases. In conclusion, the framework established in this research provides a versatile and reliable numerical solution for inverse coefficient problems in wave equations, balancing theoretical rigor with practical computational performance.

[1] L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society,2010.

[2] N. Metropolis and S. Ulam, ”The Monte Carlo Method,” Journal of the American Statistical Association, 44(247), 335–341, 1949.

[3] J. S. Rosenthal, ”Parallel computing and Monte Carlo algorithms,” Far East Journal of Theoretical Statistics, 4, 207–236, 2000.

[4] The MathWorks, Inc., Parallel Computing Toolbox User’s Guide (R2025b), The Math Works, Inc., Natick, MA, USA, 2025. Available: https://www.mathworks.com/help/pdf_doc/parallel-computing/parallel-computing.pdf [Accessed: Mar.29, 2026].

[5] The MathWorks. parfeval: Execute function asynchronously on parallel pool worker.MATLAB Parallel Computing Toolbox Available: https://ww2.mathworks.cn/help/parallel-computing/parallel.pool.parfeval.html [Accessed: Mar. 29,2026].

[6] The MathWorks. tic, toc: Measure elapsed time. MATLAB Documentation. Available: https://ww2.mathworks.cn/help/matlab/ref/tic.html [Accessed: Mar.
29, 2026].