Graduation Year

2019

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Weiqing Gu

Reader 2

Nicholas Pippenger

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Rights Information

© 2019 William Conner DiPaolo

Abstract

The task of choosing a preconditioner M to use when solving a linear system Ax=b with iterative methods is often tedious and most methods remain ad-hoc. This thesis presents a randomized algorithm to make this chore less painful through use of randomized algorithms for estimating traces. In particular, we show that the preconditioner stability || I - M-1A ||F, known to forecast preconditioner quality, can be computed in the time it takes to run a constant number of iterations of conjugate gradients through use of sketching methods. This is in spite of folklore which suggests the quantity is impractical to compute, and a proof we give that ensures the quantity could not possibly be approximated in a useful amount of time by a deterministic algorithm. Using our estimator, we provide a method which can provably select a quality preconditioner among n candidates using floating operations commensurate with running about n log(n) steps of the conjugate gradients algorithm. In the absence of such a preconditioner among the candidates, our method can advise the practitioner to use no preconditioner at all. The algorithm is extremely easy to implement and trivially parallelizable, and along the way we provide theoretical improvements to the literature on trace estimation. In empirical experiments, we show the selection method can be quite helpful. For example, it allows us to create to the best of our knowledge the first preconditioning method for kernel regression which never uses more iterations over the non-preconditioned analog in standard settings.

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