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Metric Based Upscaling for Partial Differential Equations with a Continuum of Scales


Zhang, Lei (2007) Metric Based Upscaling for Partial Differential Equations with a Continuum of Scales. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/AZ06-4B54.


Numerical upscaling of problems with multiple scale structures have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation. Most existing methods are either based on the assumption of scale separation or heuristic arguments.

In this thesis, we present rigorous results on homogenization of partial differential equations with L coefficients which allow for a continuum of spatial and temporal scales. We propose a new type of compensation phenomena for elliptic, parabolic, and hyperbolic equations. The main idea is the use of the so-called "harmonic coordinates" ("caloric coordinates" in the parabolic case). Under these coordinates, the solutions of these differential equations have one more degree of differentiability. It has been deduced from this compensation phenomenon that numerical homogenization methods formulated as oscillating finite elements can converge in the presence of a continuum of scales, if one uses global caloric coordinates to obtain the test functions instead of using solutions of a local cell problem.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:continuum scales; harmonic coordinates; multiscale; numerical homogenization; partial differential equations; upscaling
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey and Co., Inc., Prize in Applied Mathematics, 2007.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Owhadi, Houman
Thesis Committee:
  • Owhadi, Houman (chair)
  • Candes, Emmanuel J.
  • Marsden, Jerrold E.
  • Hou, Thomas Y.
Defense Date:10 May 2007
Non-Caltech Author Email:mail4lei (AT)
Record Number:CaltechETD:etd-05162007-172755
Persistent URL:
Zhang, Lei0000-0002-2917-9652
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1841
Deposited By: Imported from ETD-db
Deposited On:18 May 2007
Last Modified:24 Mar 2020 23:01

Thesis Files

PDF (Thesis_LeiZhang.pdf) - Final Version
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