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Title: Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations

Abstract

Here in this paper, two novel techniques for bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations are developed. The first provides a theorem to construct interval bounds, while the second provides a theorem to construct lower bounds convex and upper bounds concave in the parameter. The convex/concave bounds can be significantly tighter than the interval bounds because of the wrapping effect suffered by interval analysis in dynamical systems. Both types of bounds are computationally cheap to construct, requiring solving auxiliary systems twice and four times larger than the original system, respectively. An illustrative numerical example of bound construction and use for deterministic global optimization within a simple serial branch-and-bound algorithm, implemented numerically using interval arithmetic and a generalization of McCormick's relaxation technique, is presented. Finally, problems within the important class of reaction-diffusion systems may be optimized with these tools.

Authors:
 [1]
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Dept. of Electrical Engineering and Computer Science
Publication Date:
Research Org.:
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
OSTI Identifier:
1425976
Grant/Contract Number:  
SC0001088
Resource Type:
Accepted Manuscript
Journal Name:
Optimal Control Applications and Methods
Additional Journal Information:
Journal Volume: 38; Journal Issue: 4; Journal ID: ISSN 0143-2087
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 42 ENGINEERING; deterministic global optimization; semilinear parabolic PDEs; bounds

Citation Formats

Azunre, P. Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations. United States: N. p., 2016. Web. doi:10.1002/oca.2275.
Azunre, P. Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations. United States. https://doi.org/10.1002/oca.2275
Azunre, P. Wed . "Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations". United States. https://doi.org/10.1002/oca.2275. https://www.osti.gov/servlets/purl/1425976.
@article{osti_1425976,
title = {Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations},
author = {Azunre, P.},
abstractNote = {Here in this paper, two novel techniques for bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations are developed. The first provides a theorem to construct interval bounds, while the second provides a theorem to construct lower bounds convex and upper bounds concave in the parameter. The convex/concave bounds can be significantly tighter than the interval bounds because of the wrapping effect suffered by interval analysis in dynamical systems. Both types of bounds are computationally cheap to construct, requiring solving auxiliary systems twice and four times larger than the original system, respectively. An illustrative numerical example of bound construction and use for deterministic global optimization within a simple serial branch-and-bound algorithm, implemented numerically using interval arithmetic and a generalization of McCormick's relaxation technique, is presented. Finally, problems within the important class of reaction-diffusion systems may be optimized with these tools.},
doi = {10.1002/oca.2275},
journal = {Optimal Control Applications and Methods},
number = 4,
volume = 38,
place = {United States},
year = {Wed Sep 21 00:00:00 EDT 2016},
month = {Wed Sep 21 00:00:00 EDT 2016}
}