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

机译：限定参数弱耦合二阶半线性抛物面偏微分方程的解

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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. Problems within the important class of reaction-diffusion systems may be optimized with these tools. (c) 2016 The Authors. Optimal Control Applications and Methods published by John Wiley & Sons, Ltd.
机译：在本文中，开发了两种用于限定参数弱耦合二阶半线性抛物面偏微分方程的方法的三种新颖技术。第一个提供一个定理以构建间隔边界，而第二个提供定理以构建参数中的下限凸凸和上限凹陷。由于动态系统中的间隔分析所遭受的包装效果，凸起/凹形界限可以明显更严重。这两种类型的界限都是计算地的构造，需要求解辅助系统两次，分别比原始系统大的四倍。呈现了在单一使用间隔算术和McCormick放松技术的简单串行分支算法内实现了简单串行分支和绑定算法内的结合结构的说明性数值例和用于确定性的全局优化。可以用这些工具优化重要的反应扩散系统内的问题。 （c）2016年作者。 John Wiley＆Sons，Ltd.公布的最佳控制应用和方法

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