Preserving positivity of Gauss-Newton Hessian through random sampling

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.