We introduce a time-dimensional reduction method for the inverse source problem in linear elasticity, where the goal is to reconstruct the initial displacement and velocity fields from partial boundary measurements of elastic wave propagation. The key idea is to employ a novel spectral representation in time, using an orthonormal basis composed of Legendre polynomials weighted by exponential functions. This Legendre polynomial-exponential basis enables a stable and accurate decomposition in the time variable, effectively reducing the original space-time inverse problem to a sequence of coupled spatial elasticity systems that no longer depend on time. These resulting systems are solved using the quasi-reversibility method. On the theoretical side, we establish a convergence theorem ensuring the stability and consistency of the regularized solution obtained by the quasi-reversibility method as the noise level tends to zero. On the computational side, two-dimensional numerical experiments confirm the theory and demonstrate the method's ability to accurately reconstruct both the geometry and amplitude of the initial data, even in the presence of substantial measurement noise. The results highlight the effectiveness of the proposed framework as a robust and computationally efficient strategy for inverse elastic source problems.

翻译：本文针对线性弹性力学中的反源问题，提出了一种时间维度约简方法，其目标是根据弹性波传播的部分边界测量数据重构初始位移场和速度场。该方法的核心思想是采用一种新颖的时间谱表示，其正交基由指数函数加权的勒让德多项式构成。该勒让德多项式-指数基实现了时间变量上的稳定且精确的分解，从而将原始时空反问题有效地约简为一系列不再依赖于时间的耦合空间弹性系统。这些约简后的系统通过拟可逆性方法求解。在理论方面，我们建立了一个收敛性定理，保证了当噪声水平趋于零时，由拟可逆性方法得到的正则化解的稳定性和一致性。在计算方面，二维数值实验验证了理论结果，并证明了该方法即使在存在显著测量噪声的情况下，仍能精确重构初始数据的几何形状和振幅。这些结果凸显了所提框架作为解决弹性反源问题的一种鲁棒且计算高效策略的有效性。