| dc.contributor |
Shu-Cherng Fang, Committee Chair |
|
| dc.creator |
Wang, Yong |
|
| dc.date |
2010-04-02T19:17:09Z |
|
| dc.date |
2010-04-02T19:17:09Z |
|
| dc.date |
2005-04-12 |
|
| dc.date.accessioned |
2023-02-28T17:07:38Z |
|
| dc.date.available |
2023-02-28T17:07:38Z |
|
| dc.identifier |
etd-04082005-144213 |
|
| dc.identifier |
http://www.lib.ncsu.edu/resolver/1840.16/5651 |
|
| dc.identifier.uri |
http://localhost:8080/xmlui/handle/CUHPOERS/265561 |
|
| dc.description |
A major objective of modelling geophysical features, biological objects, financial processes and many other irregular surfaces and functions is to develop "shape-preserving" methodologies for smoothly interpolating bivariate data with sudden changes in magnitude or spacing. Shape preservation usually means the elimination of extraneous non-physical oscillations. Classical splines do not preserve shape well in this sense.
Empirical experiments have shown that the recently proposed cubic L₁ splines are cable of providing C₁-smooth, shape-preserving, multi-scale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for node adjustment or other user input. However, a theoretic treatment of the bivariate cubic L₁ splines is still lacking. The currently available approximation algorithms are not able to generate the exact coefficients of a bivariate cubic L₁ spline.
For theoretical treatment and the algorithm development, we propose to solve bivariate cubic L*#8321; spline problems in a generalized geometric programming framework. Our framework includes a primal problem, a geometric dual problem with a linear objective function and convex cubic constraints, and a linear system for dual-to-primal transformation. We show that bivariate cubic L₁ splines indeed preserve linearity under some mild conditions.
Since solving the dual geometric program involves heavy computation, to improve computational efficiency, we further develop three methods for generating bivariate cubic L₁ splines: a tensor-product approach that generates a good approximation for large scale bivariate cubic L₁ splines; a primal-dual interior point method that obtains discretized bivariate cubic L₁ splines robustly for small and medium size problems; and a compressed primal-dual method that efficiently and robustly generates discretized bivariate cubic L₁ splines of large size. |
|
| dc.rights |
I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
|
| dc.subject |
spline function |
|
| dc.subject |
tensor-product |
|
| dc.subject |
cubic L1 spline |
|
| dc.subject |
interpolation |
|
| dc.subject |
geometric programming |
|
| dc.subject |
primal-dual method |
|
| dc.title |
Theory and Algorithms for Shape-preserving Bivariate Cubic L1 Splines. |
|