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This paper expands the analysis of randomized low-rank approximation beyond the Gaussian distribution to four classes of random matrices: (1) independent sub-Gaussian entries, (2) independent sub-Gaussian columns, (3) independent bounded columns, and (4) independent columns with bounded second moment. Using a novel interpretation of the low-rank approximation error involving sample covariance matrices, we provide insight into the requirements of a good random matrix for randomized low-rank approximations. Although our bounds involve unspecified absolute constants (a consequence of underlying nonasymptotic theory of random matrices), they allow for qualitative comparisons across distributions. The analysis offers some details on the minimal number of samples (the number of columns of the random matrix ) and the error in the resulting low-rank approximation. We illustrate our analysis in the context of the randomized subspace iteration method as a representative algorithm for low-rank approximation; however, all the results are broadly applicable to other low-rank approximation techniques. We conclude our discussion with numerical examples using both synthetic and real-world test matrices. 

Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyperdifferential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice-sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate).