Data-sparse algorithms for structured matrices
- In the first part of the dissertation, we present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is polylogarithmic in the total number of unknowns and linear in the number of perturbed unknowns. In the second part, we consider the application of hierarchical factorizations to the problem of spatial Gaussian process maximum likelihood estimation, i.e., parameter fitting for kriging. We present a framework for scattered (quasi-)two-dimensional observations using skeletonization factorizations to quickly evaluate the Gaussian process log-likelihood and its gradient, which we use in the context of black-box numerical optimization for parameter fitting of low-dimensional Gaussian processes. Finally, we introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral equations associated with elliptic partial differential equations in two and three dimensions (and other matrices with similar hierarchical rank structure). RS-S uses a simple modification of skeletonization, strong skeletonization, which compresses only far-field interactions. We further combine the strong skeletonization procedure with alternating near-field compression to obtain the hybrid recursive skeletonization factorization (RS-WS), a modification of RS-S that exhibits reduced storage cost in many settings.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Minden, Victor Lawrence
|Stanford University, Institute for Computational and Mathematical Engineering.
|Statement of responsibility
|Victor Lawrence Minden.
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2017.
- © 2017 by Victor Lawrence Minden
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