Inverse problem theory is often studied in the ideal infinite-dimensional setting. Through the lens of the PDE-constrained optimization, the well-posedness PDE theory suggests unique reconstruction of the parameter function that attain the zero-loss property of the mismatch function, when infinite amount of data is provided. Unfortunately, this is not the case in practice, when we are limited to finite amount of measurements due to experimental or economical reasons. Consequently, one must compromise the inference goal to a discrete approximation of the unknown smooth function. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size $(r)$ and the number of parameters to be uniquely identified $(m)$. The technical pillar is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing randomly sub-sampled Hessian matrix, we attain well-conditioned reconstruction problem with high probability. Our main theory is finally validated in numerical experiments. We set tests on both synthetic and the data from an elliptic inverse problem. The empirical performance shows that given suitable sampling quality, the well-conditioning of the sketched Hessian is certified with high probability.

翻译：逆问题理论通常在理想无穷维框架下研究。通过偏微分方程约束优化的视角，适定性理论表明，在提供无限量数据时，参数函数可达到失效函数零损失特性的唯一重建。然而在实际中，因实验或经济条件限制，我们只能获取有限数量的测量数据，这导致该理想情况难以实现。因此，必须将推断目标妥协为未知光滑函数的离散近似。固定数量的数据观测具有怎样的重建能力？我们能重建多少个参数？本文提出一种概率方法，系统阐明观测规模（r）与需唯一辨识的参数数量（m）之间的相互作用关系。其技术核心是随机素描策略，主要运用矩阵浓度不等式与采样理论。通过分析随机子采样Hessian矩阵，我们以高概率获得良态的重建问题。最终通过数值实验验证了核心理论，在合成数据及椭圆逆问题数据上进行了测试。实证结果表明，在采样质量适当的情况下，经素描处理的Hessian矩阵能以高概率保证良态性。