  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.
