Numerical multiscale methods: from homogenization to milestoning

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dc.contributor.advisorEngquist, Björn, 1945-
dc.contributor.committeeMemberMartinsson, Per Gunnar
dc.contributor.committeeMemberArbogast, Todd J.
dc.contributor.committeeMemberTsai, Richard Yen-Hsi
dc.contributor.committeeMemberBajaj, Chandrajit
dc.creatorChen, Ziheng
dc.creator.orcid0000-0001-9671-3977
dc.date.accessioned2024-07-19T21:34:04Z
dc.date.available2024-07-19T21:34:04Z
dc.date.created2024-05
dc.date.issued2024-05
dc.date.submittedMay 2024
dc.date.updated2024-07-19T21:34:04Z
dc.description.abstractThe dissertation focuses on addressing the challenges posed by multiscale problems in applied mathematics, which stem from the intricate interplay between microscales and the computational demands of resolving fine details. To alleviate this burden, numerical homogenization and averaging methods are favored. This study explores three interconnected topics related to numerical techniques for handling multiscale problems in both spatial and temporal domains. In the first part, we establish the equivalence principle between time averaging and space homogenization. This principle facilitates the application of various numerical averaging techniques, such as FLAVORS, Seamless, and HMM, to boundary value problems. Moreover, we introduce the dilation operator as a decomposition-free approach for numerical homogenization in higher dimensions. Additionally, we utilize the Synchrosqueezing transform as a preprocessing step to extract oscillatory components, crucial for the structure-aware dilation method. The second part extends the Deep Ritz method to multiscale problems. We delve into the scale convergence theory to derive the [Gamma]-limit of energy functionals exhibiting oscillatory behavior. The resulting limit object, formulated as a minimization problem, captures spatial oscillations and can be tackled using existing neural network architectures. In the third part, we lay the groundwork for the milestoning algorithm, a successful tool in computational chemistry for molecular dynamics simulations. We adapt this algorithm to a domain-decomposition-based framework for coarse-grained descriptions and establish the well-posedness of primal and dual PDEs. Additionally, we investigate the convergence rate and optimal milestone placements. We illustrate this framework through the understanding of the Forward Flux algorithm as a specific example.
dc.description.departmentMathematics
dc.format.mimetypeapplication/pdf
dc.identifier.uri
dc.identifier.urihttps://hdl.handle.net/2152/126131
dc.identifier.urihttps://doi.org/10.26153/tsw/52669
dc.subjectApplied mathematics
dc.subjectMultiscale methods
dc.subjectHomogenization
dc.subjectNumerical methods
dc.subjectElliptic problems
dc.subjectAveraging
dc.subjectMolecular dynamics
dc.subjectMilestoning algorithm
dc.titleNumerical multiscale methods: from homogenization to milestoning
dc.typeThesis
dc.type.materialtext
local.embargo.lift2025-05-01
local.embargo.terms2025-05-01
thesis.degree.collegeNatural Sciences
thesis.degree.departmentMathematics
thesis.degree.disciplineMathematics
thesis.degree.grantorThe University of Texas at Austin
thesis.degree.nameDoctor of Philosophy

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