Journal
APPLIED NUMERICAL MATHEMATICS
Volume 61, Issue 7, Pages 803-820Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.apnum.2011.01.009
Keywords
Interval analysis; Convex relaxations; McCormick relaxations; Ordinary differential equations; Dynamic systems; Global optimization
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Funding
- Natural Sciences and Engineering Research Council of Canada (NSERC) [372078-2009]
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This paper presents a discretize-then-relax methodology to compute convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval methods for ODEs and uses the McCormick relaxation technique to propagate convex/concave bounds. At each integration step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire step are calculated in the first phase: then, pointwise-in-time convex/concave bounds at the end of the step are obtained in the second phase. An approach that refines the interval state bounds by considering subgradients and affine relaxations at a number of reference parameter values is also presented. The discretize-then-relax method is implemented in an object-oriented manner and is demonstrated using several numerical examples. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.
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