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UBC Theses and Dissertations
Efficient algorithms for diffusion-generated motion by mean curvature Ruuth, Steven J.
Abstract
This thesis considers the problem of simulating the motion of evolving surfaces with a normal velocity equal to mean curvature plus a constant. Such motions arise in a variety of applications. A general method for this purpose was proposed by Merriman, Bence and Osher, and consists of alternately diffusing and sharpening the front in a certain manner. This method (referred to as the MBO-method) naturally handles complicated topological changes with junctions in several dimensions. However, the usual finite dif-ference discretization of the method is often exceedingly slow when accurate results are sought, especially in three spatial dimensions. We propose a new, spectral discretization of the MBO-method which obtains greatly improved efficiency over the usual finite difference approach. These efficiency gains are obtained, in part, through the use of a quadrature-based refinement technique, by in-tegrating Fourier modes exactly, and by neglecting the contribution of rapidly decaying solution transients. The resulting method provides a practical tool, not available hitherto, for accurately treating the motion by mean curvature of complicated surfaces with junc-tions. Indeed, we present numerical studies which demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization. New analytic and experimental results are also developed to explain important prop-erties of the MBO-method such as the order of the approximation error. Extrapolated algorithms, not possible when using the usual finite difference discretization, are proposed and demonstrated to achieve more accurate results. We apply our new, spectral method to simulate the motion of a number of three dimensional surfaces with junctions, and we visualize the results. We also propose and study a simple extension of our method to a nonlocal curvature model which is impractical to treat using the previously available finite difference discretization.
Item Metadata
Title |
Efficient algorithms for diffusion-generated motion by mean curvature
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1996
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Description |
This thesis considers the problem of simulating the motion of evolving surfaces with a
normal velocity equal to mean curvature plus a constant. Such motions arise in a variety
of applications. A general method for this purpose was proposed by Merriman, Bence
and Osher, and consists of alternately diffusing and sharpening the front in a certain
manner. This method (referred to as the MBO-method) naturally handles complicated
topological changes with junctions in several dimensions. However, the usual finite dif-ference
discretization of the method is often exceedingly slow when accurate results are
sought, especially in three spatial dimensions.
We propose a new, spectral discretization of the MBO-method which obtains greatly
improved efficiency over the usual finite difference approach. These efficiency gains are
obtained, in part, through the use of a quadrature-based refinement technique, by in-tegrating
Fourier modes exactly, and by neglecting the contribution of rapidly decaying
solution transients. The resulting method provides a practical tool, not available hitherto,
for accurately treating the motion by mean curvature of complicated surfaces with junc-tions.
Indeed, we present numerical studies which demonstrate that the new algorithm
is often more than 1000 times faster than the usual finite difference discretization.
New analytic and experimental results are also developed to explain important prop-erties
of the MBO-method such as the order of the approximation error. Extrapolated
algorithms, not possible when using the usual finite difference discretization, are proposed
and demonstrated to achieve more accurate results.
We apply our new, spectral method to simulate the motion of a number of three
dimensional surfaces with junctions, and we visualize the results. We also propose and
study a simple extension of our method to a nonlocal curvature model which is impractical
to treat using the previously available finite difference discretization.
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Extent |
8198422 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2009-03-17
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0079751
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
1996-11
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.