Computational Mathematics and Scientific Computing Seminar

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Can one recover a structured matrix efficiently from only matrix-vector products? If so, how many are needed? In this talk, we will describe algorithms to recover structured matrices, such as tridiagonal, Toeplitz-like, and hierarchical low-rank, from matrix-vector products. Then, we will use insights from matrix recovery to understand the data-efficiency of operator learning in the context of PDE learning. We will partially explain the success of neural operators, like Fourier Neural Operators and DeepONet, by understanding the data-efficiency of recovery of the solution operators associated with elliptic, parabolic, and hyperbolic PDEs.