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Abstract Computational inverse problems utilize a finite number of measurements to infer a discrete approximation of the unknown parameter function. With motivation from the setting of PDE-based optimization, we study the unique reconstruction of discretized inverse problems by examining the positivity of the Hessian matrix. 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 here is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing a randomly subsampled Hessian matrix, we attain a well-conditioned reconstruction problem with high probability. Our main theory is validated in numerical experiments, using an elliptic inverse problem as an example.