Operator shifting and model reduction techniques for efficient and accurate computational physics
- This thesis is split into two distinct parts. The first (and principal) part of the thesis will discuss the problem of more accurately estimating the solution of a linear system corrupted by noise. The second part will discuss a handful of gaps in the literature on model reduction and introduce novel techniques to address these shortcomings. In part I, we note that many problems in computational physics require knowledge of quantities subject to uncertainty or noise. For example, one may be interested in simulating the scattering pattern of light passing through a scattering medium, but only have a noisy estimate of the scattering background. Since most computational physics problems reduce to linear systems, we want to build more accurate estimates of linear systems corrupted by noise. While many techniques exist for rectifying this in solution space (i.e., Tikhonov regularization, etc.), we instead examine this problem as one of producing a better estimate of a matrix inverse from a noisy matrix sample. We introduce the operator shifting technique, a loose adaptation of James-Stein estimation to matrices, and examine theoretical guarantees about shift direction and magnitude in both positive definite and general settings, as well as discuss efficient Monte Carlo estimation of shift magnitude via monotonic polynomial truncations of power series. In part II, we will pivot to model reduction techniques. While operator shifting helps provide more accurate solutions to physics problems, model reduction helps provide efficiently computable surrogate models for computationally expensive high-fidelity physics models. We will discuss novel model reduction techniques for integral equations and dense linear systems, as well as new adaptive refinement and compression schemes for linear reduced models. Finally, we will demonstrate new work in realtime model reduction for soft-body physics.
|Type of resource
|electronic resource; remote; computer; online resource
|1 online resource.
|Etter, Philip Allen
|Degree committee member
|Degree committee member
|Stanford University, Institute for Computational and Mathematical Engineering
|Statement of responsibility
|Philip A. Etter.
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis Ph.D. Stanford University 2022.
- © 2022 by Philip Allen Etter
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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