Matrices and tensors in computational mathematics are so often well-approximated by low-rank objects. In the first part of the talk, we will use the ADI method, a classic partial differential equation (PDE) solver, to understand the prevalence of compressible matrices and tensors, and resolve a long-standing problem of finding an optimal complexity spectrally-accurate Poisson solver. In the second part of the talk, we will use low-rank techniques for PDE learning where one is given input-output training data from an unknown uniformly elliptic PDE and would like to recover the PDE operator. By exploiting the hierarchical low-rank structure of Green’s functions and randomized linear algebra, we will describe a rigorous scheme for PDE learning with a provable “learning rate.”
Department of Mathematics