AI 中文总结
本文研究二维时域等离子体波动方程的内部解，利用数据驱动格拉姆矩阵的Cholesky分解变换背景快照，证明其收敛性，数值实验验证其在方形域及高对比度介质中的表现，特征值下限正则化更鲁棒。
AI 中文摘要
我们考虑从边界响应数据计算含未知势q的时域等离子体波动方程的内部解。该内部解通过利用数据驱动格拉姆矩阵（或称质量矩阵）的Cholesky分解变换已知背景快照得到。近期研究表明，在一维情形下，当初始波选择适当时，这些数据生成的内部解在L²范数下以√τ阶收敛。本文研究二维情形下的内部解重构，该情形需要多输入多输出（MIMO）设置与块格拉姆矩阵，且边界源数量随时间采样的精细化而增加。我们证明通用误差界可推广至该情形：数据生成解与背景快照最优因果近似之间的距离由最优近似质量矩阵失配控制。在方形区域上的数值精细化研究测量了数据生成解及背景快照最优因果近似的收敛性，相对误差似乎以√τ阶趋于零，绝对误差在L¹范数下趋于零。我们还表明，该解重构在高对比度复合介质中仍保持准确。最后，由于从含噪响应数据组装的质量矩阵可能无法保持正定，需引入正则化；通过对比对角移位与特征值下限两种方法，我们发现特征值下限法更鲁棒，在噪声达数据均方根振幅10%的情况下，其重构结果仍比未受扰的背景场更准确。
英文摘要
We consider the computation of internal solutions for a time domain plasma wave equation with an unknown potential $q$ from boundary response data. The internal solutions are computed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian, or mass matrix. It was recently shown that in one dimension these data generated internal solutions converge in $L^2$ at order $\sqrtτ$ for well chosen initial waves. Here we study the internal solution reconstruction in two dimensions, where a multiple input/multiple output (MIMO) setup and a block Gramian are needed, with the number of boundary sources increasing as the time sampling is refined. We show that the general error bound carries over to this setting: the distance between the data generated solutions and the best causal approximation from background snapshots is controlled by the best approximation mass matrix mismatch. Numerical refinement studies on a square domain measure the convergence of the data generated solutions alongside the best causal approximation from the background snapshots, with the relative errors appearing to go to zero at rate $\sqrtτ$, and the absolute errors going to zero in $L^1$. We also show that the solution reconstructions remain accurate for high contrast composite media. Finally, since the mass matrix assembled from noisy response data can fail to be positive definite, regularization is needed; comparing a diagonal shift with an eigenvalue floor, we find the floor more robust, with reconstructions that remain more accurate than the unperturbed background field with noise that is up to ten percent of the root mean square data amplitude.
