Computational Refinement and Existence Proofs for Singularities

9701540 R. Baker Kearfott 在过去的二十年里，人们对奇点、分岔和相关现象以及与物理世界的相关性的理解有所增加。 同时，边界离散化和舍入误差工具也得到了改进，以允许基于经典不动点理论的自动结果验证。 传统上，这种验证通过线性化进行，然后用区间算术进行限制。 然而，在迄今为止开发的方法中，当系统的解是奇异的时，存在性和唯一性验证必然会失败。 该项目的目标是（1）对现有验证技术进行新的、简单且自然的修改，以证明非线性系统奇异时的存在性和唯一性； (2) 研究所得工具的威力，首先针对有限维问题（代数系统），然后针对无限维问题（微分方程和积分方程）； (3)增加对奇点的理解。 整个科学和工程领域的许多复杂系统，从承重土木工程结构到生态系统再到人类心脏节律，都是通过表现出奇点和分岔的非线性方程组来建模的。 这种分叉对应于条件（负载、温度等）变化时行为的变化。 近年来，对此类现象的理解不断深入，但分析问题仍然存在。 这项研究将提供工具来实现高效、自动化且数学上无懈可击的分析。 这反过来又会导致物理世界更具可预测性。

9701540 R. Baker Kearfott During the past two decades, understanding of singularities, bifurcations and related phenomena, and relevance to the physical world have increased. At the same time, tools for bounding discretization and roundoff error have improved, to allow automatic result verification based on classical fixed point theory. Traditionally, such verification proceeds by linearization, then bounding with interval arithmetic. However, in methods developed to date, existence and uniqueness verification necessarily must fail when the system is singular at solutions. Goals of this project are to (1) develop new, simple and natural modifications to existing verification techniques, to prove existence and uniqueness when the nonlinear system is singular; (2) investigate the power of the resulting tools, first for finite-dimensional problems (algebraic systems), then for infinite-dimensional problems (differential and integral equations); (3) increase understanding of singularities. Many complex systems throughout the sciences and engineering, ranging from load-bearing civil engineering structures to ecological systems to human heart rythms, are modeled by sets of nonlinear equations that exhibit singularities and bifurcations. Such bifurcations correspond to changes in behavior as conditions (the load, a temperature, etc.) change. Understanding of such phenomena has blossomed in recent years, but analytical problems remain. This study will provide tools to enable an efficient and automated, yet mathematically airtight analysis. This in turn will lead to more predictability in the physical world.