标题： 关于从边界数据驱动的Gramian矩阵生成的内部解的最优性与界的研究 显示英文标题

标题： On optimality and bounds for internal solutions generated from boundary data-driven Gramians

摘要： 我们考虑了时间域等离子体波动方程内部解的计算问题，该方程带有未知系数，数据来源于对其边界上的传递函数进行采样得到的结果。 计算过程通过利用数据驱动的 Gramian 的 Cholesky 分解来变换已知背景快照完成。 我们证明，这种近似值在渐近意义上接近于真实内部解在背景快照子空间上的投影。 这使我们能够推导出一个适用于时间域等离子体波动方程内部场从边界数据近似时的一般误差界，其中方程具有未知势能$q$。 对于一般情况$q\in L^\infty$，我们证明了一维下两个例子的数据生成内部场的收敛性。 第一个例子是分段常数初始数据和采样间隔$\tau$等于脉冲宽度。 第二个例子是分段线性初始数据和以脉冲宽度一半进行采样。 我们表明在这两种情况下，数据生成的解在$L^2$中以阶数$\sqrt{\tau}$收敛。 我们还给出了数值实验来验证这一结果以及收敛率的尖锐性。 显示英文摘要

摘要： We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the true internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data for a time domain plasma wave equation with an unknown potential $q$. For general $q\in L^\infty$, we prove convergence of these data generated internal fields in one dimension for two examples. The first is for piecewise constant initial data and sampling $\tau$ equal to the pulse width. The second is piecewise linear initial data and sampling at half the pulse width. We show that in both cases the data generated solutions converge in $L^2$ at order $\sqrt{\tau}$. We present numerical experiments validating the result and the sharpness of this convergence rate.