Numerically the reconstructability of unknown parameters in inverse problems heavily relies on the chosen data. Therefore, it is crucial to design an experiment that yields data that is sensitive to the parameters. We approach this problem from the perspective of a least squares optimization, and examine the positivity of the Gauss-Newton Hessian at the global minimum point of the objective function. We propose a general framework that provides an efficient down-sampling strategy that can select data that preserves the strict positivity of the Hessian. Matrix sketching techniques from randomized linear algebra is heavily leaned on to achieve this goal. The method requires drawing samples from a certain distribution, and gradient free sampling methods are integrated to execute the data selection. Numerical experiments demonstrate the effectiveness of this method in selecting sensor locations for Schr\"odinger potential reconstruction.

翻译：在数值计算中，逆问题中未知参数的可重构性很大程度上依赖于所选数据。因此，设计能够产生对参数敏感数据的实验至关重要。我们从最小二乘优化的角度处理该问题，并考察目标函数全局极小点处高斯-牛顿海森矩阵的正定性。我们提出一个通用框架，该框架提供了一种高效的降采样策略，能够选择保持海森矩阵严格正定性的数据。为实现这一目标，我们大量借鉴了随机线性代数中的矩阵素描技术。该方法需要从特定分布中抽取样本，并集成无梯度采样方法来执行数据选择。数值实验证明了该方法在薛定谔势重构问题中选择传感器位置的有效性。