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We develop an approach to recover the underlying properties of fluid-dynamical processes from sparse measurements. We are motivated by the task of imaging the stochastically evolving environment surrounding black holes, and demonstrate how flow parameters can be estimated from sparse interferometric measurements used in radio astronomical imaging. To model the stochastic flow we use spatio-temporal Gaussian Random Fields (GRFs). The high dimensionality of the underlying source video makes direct representation via a GRF’s full covariance matrix intractable. In contrast, stochastic partial differential equations are able to capture correlations at multiple scales by specifying only local interaction coefficients. Our approach estimates the coefficients of a space-time diffusion equation that dictates the stationary statistics of the dynamical process. We analyze our approach on realistic simulations of black hole evolution and demonstrate its advantage over state-of-the-art dynamic black hole imaging techniques. 

This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues. Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.