Free Boundary Problems for Systems: Toward a Comprehensive Theory of Computation

本文研究了一类新的系统，它是用与时间相关的平稳偏微分方程来表示的。该系统具有固有的开关特性，因此可以将其解释为自由边界问题。它们之所以是新的，是因为它们的特征是系统而不是标量方程，强调了联锁和复杂的依赖关系。这项工作的一部分是与齐格弗里德·卡尔（Siegfried Carl）共同完成的，它扩展了许多数学家在过去四十年中研究标量方程自由边界问题的工作。自由边界问题系统与相变化、半经典带隙理论、阈值开关和燃烧等多种模型有关。使用重要的模型进行说明，我们采用多层分析，允许开发用于分析和计算的基础设施。首席研究员和Michael rose1982年首次介绍了两相Stefan问题的综合有限元研究。在该方案中，迭代和离散化作为并行实体进行研究。一个特别重要的方面是系统映射的识别及其通过平滑和线性化的有效逼近。利用均匀化理论和全非线性偏微分方程理论研究这些系统的界面，得到了Monica Torres的合作。自由边界问题的数学模型建立在丰富的传统之上。自由边界问题在20世纪70年代和80年代得到了深入的研究。美国的一个主要推动力是跨阿拉斯加管道的建设，这需要了解永久冻土融化的复杂过程。自由边界是指所谓的介于固相和液相之间的糊状区域。为此，采用了一种称为两相斯特芬问题的温度数学模型。埃克森公司的约翰·惠勒是这项工作的主要贡献者之一。在系统建模方面进展缓慢，其中可能存在几种影响。在许多应用领域中，由于系统内的源或汇引起的不连续开关，可能会出现相关的自由边界问题。环境方面的一个例子是污染物排放到河流中，对排放的调节，以及随后控制机制的响应，以改善排放。利用复杂的偏微分方程系统，结合描述污染物密度和河流温度的物理原理，并包括切换现象。将量子原理结合到纳米结构模型中也会引起开关和自由边界问题，对某些燃烧过程的研究也是如此。这些思想的另一个应用是某些金融工具的定价，这在本建议中没有涉及。关于国际社会主办的计算电子学讲习班，已经确定了该提案具有更广泛影响的领域。自1990年以来，首席研究员一直是顾问小组成员，这项工作的一些细节，使量子计算成为可能的一部分，将在定期召开的研讨会上向社区传播，并发表在《计算电子学》杂志上。论文也将发表在正规的数学期刊上。

This proposal investigates a new class of systems, formulated in terms of time dependent and stationary partial differential equations. The systemshave inherent switching properties, leading to their interpretation as freeboundary problems. What defines them as new is their characterization assystems rather than scalar equations, underscoring interlocking, complexdependencies. This work, part of which is joint with Siegfried Carl,extends the work of many mathematicians who have worked on free boundaryproblems for scalar equations during the last four decades. Systems of free boundary problems are associated with such diverse models aschange of phase, semi-classical band gap theories, threshold switching, and combustion. Using significant models for illustration, we employa multi-tiered analysis, permitting the development of an infrastructurefor analysis and computation. The principal investigator and Michael Roseintroduced the initial comprehensive finite element study of the two-phase Stefan problem in 1982. In this proposal, iteration and discretization are studied as companion entities. A particularly significant aspect is the identification of the system map and its efficient approximation via smoothing and linearization. The cooperation of Monica Torres has beensecured to study the interface of these systems with homogenizationtheory and the theory of fully nonlinear partial differential equations.Mathematical models of free boundary problems build upon a rich tradition.Free boundary problems were studied intensively during the 1970s and1980s. A major impetus in the U.S. was the construction of the trans-Alaskapipeline, which required knowledge of the complex process of meltingof permafrost. The free boundary refers to the so-called mushy regionbetween solid and liquid phases. A mathematical model of temperature,called the two-phase Stefan problem, was employed for this purpose. One ofthe principal contributors to this effort was John Wheeler of the Exxon Corporation. Progress was slow in coming in the modeling of systems,where several effects may be present. Associated free boundary problems mayarise in many application areas, due to discontinuous switching induced bysources or sinks within the system. An environmental example is thedischarge of a contaminant species into a river, the regulation of thedischarge, and the ensuing response of control mechanisms to ameliorate thedischarge. Sophisticated systems of partial differential equations,incorporating physical principles describing the contaminant density andthe river temperature, are utilized, and include the switching phenomena. The incorporation of quantum principles into models of nanostructures alsoinduces switching and free boundary problems, as does the study of certainprocesses of combustion. Another application of these ideas, which is not pursued in this proposal, is the pricing of certain financialinstruments. An area of broader impact of the proposal has been identified in relation to the international community sponsoring the Workshops onComputational Electronics. The principal investigator has been an advisorypanel member since 1990, and details of some of this work, part of the effort toward making quantum computing possible, will be disseminated tothe community at a regularly scheduled workshop, and in the journal,Computational Electronics. Publication will also occur in regularmathematics journals.