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

这一建议研究了一类新的系统，它是由时间相关的定常偏微分方程组表示的。系统具有固有的开关性质，导致它们被解释为自由边界问题。将它们定义为新的是它们作为系统而不是标量方程的特征，强调了相互关联、复杂的依赖关系。这项工作的一部分是与齐格弗里德·卡尔共同完成的，它扩展了许多数学家的工作，他们在过去的四十年里研究了标量方程的自由边界问题。自由边界问题系统涉及到不同的模型，如相变、半经典带隙理论、阈值切换和燃烧。以重要模型为例，我们采用了多层次分析，允许开发用于分析和计算的基础设施。首席研究员迈克尔·罗斯在1982年介绍了两相斯特凡问题的最初的全面有限元研究。在该方案中，迭代和离散化被作为伴生实体进行研究。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。Monica Torres得到了Monica Torres的合作，用齐次化理论和完全非线性偏微分方程组理论研究了这些系统的界面。自由边界问题的数学模型建立在丰富的传统基础上。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冰线的建设，这需要对永久冻土融化的复杂过程有所了解。自由边界指的是固体和液体之间的所谓糊状区。为此，采用了一个温度的数学模型，称为两相斯特凡问题。这项工作的主要贡献者之一是埃克森美孚公司的约翰·惠勒。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，由于系统内部的源或汇引起的不连续切换，可能会出现相关的自由边界问题。环境方面的一个例子是将一种污染物排放到河流中，对排放进行监管，以及随后控制机制的反应，以改善排放。利用了复杂的偏微分方程组，结合了描述污染物密度和河流温度的物理原理，并包括开关现象。将量子原理结合到纳米结构的模型中也会导致开关和自由边界问题，研究某些燃烧过程也是如此。这些想法的另一个应用是某些金融工具的定价，这一点在本提案中没有涉及。关于赞助计算电子学研讨会的国际社会，已经确定了该提案的一个更广泛的影响领域。首席研究员自1990年以来一直是顾问小组成员，其中一些工作的细节将在定期安排的研讨会和《计算电子学》杂志上向社区传播，这是使量子计算成为可能的努力的一部分。论文也将在正规的数学期刊上发表。

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.