CAREER: Adaptive multilevel finite element methods with applications to biomolecules and gravitation

研究者设计并分析了在数学生物学和万有引力领域中求解一类复杂域上的非线性椭圆型和抛物型偏微分方程的有效数值技术。工作重点是自适应多层次有限元方法，这是一种强大的数值逼近技术，用于以最优或接近最优的方式处理不规则域几何、退化系数和复杂非线性。这项工作的具体目标应用是在生物分子的连续体模型中出现的非线性泊松-玻尔兹曼方程，爱因斯坦方程中的哈密顿和动量约束，以及在几何表面的拓扑分类中出现的一些相关的椭圆和抛物方程。虽然这项工作有很大的理论成分，但要对生物学和物理学产生影响，需要大量的软件开发，以及与相关领域科学家的跨学科合作。所需的实现是由研究者和他的学生开发的，作为他的软件包MC（歧管代码）的类库扩展，并且由此产生的软件可供生物学和物理学社区的研究人员使用。本项目为美国国家科学基金会CAREER基金。本项目开发的计算方法对合理药物设计研究人员使用基于结构的生物分子模型的建模能力有直接影响。泊松-玻尔兹曼方程是生物分子静电相互作用的一个完善的模型，允许人们预测候选药物分子在通过静电场单元的洋葱状层的过程中与目标生物分子的活性位点结合的有效性。例如，通过使用这些基于结构的数值模型，可以预测候选HIV蛋白酶抑制剂与HIV蛋白酶活性位点结合的有效性。然而，泊松-玻尔兹曼方程在计算机上用数值方法求解存在严重的技术困难。通过开发改进的多层数值技术来减少解决泊松-玻尔兹曼方程所需的计算机时间，同时通过使用自适应有限元方法产生更准确的解决方案，研究者为药物设计人员提供了在更短时间内产生更准确结果的模型。此外，这个项目和相关项目成功的一个关键因素是下一代计算数学家的教育和培训。因此，在本科和研究生计算数学教育的全面参与是该项目的一个组成部分。特别是，研究者在加州大学圣地亚哥分校开发了两个新的以项目为导向的跨学科计算数学课程，模仿了他1997-1998年在加州大学欧文分校教授的一个成功的实验课程。本项目为美国国家科学基金会CAREER基金项目。美国国家科学基金会强烈鼓励教师作为教育工作者和研究人员的早期发展。教师早期职业发展（Career）计划是一个基金会范围内的计划，为初级教师的整体职业发展提供支持。它在一个项目中结合了对最广泛意义上的高质量研究和教育的支持，以及那些传统上代表性不足的科学和工程领域的充分参与。该项目加强并强调了基金会对全面、平衡的学术生涯发展的重要性。

Holst9875856 The investigator designs and analyzes effective numericaltechniques for solving certain classes of nonlinear elliptic andparabolic partial differential equations on complicated domains,arising in mathematical biology and gravitation. The workfocuses on adaptive multilevel finite element methods, a powerfulclass of numerical approximation techniques for dealing withirregular domain geometries, degenerate coefficients, andcomplicated nonlinearities, in an optimal or nearly optimal way.Specific target applications for this work are the nonlinearPoisson-Boltzmann equation arising in continuum models ofbiomolecules, the Hamiltonian and momentum constraints in theEinstein equations, and some related elliptic and parabolicequations that arise in the topological classification ofsurfaces in geometry. While this work has a large theoreticalcomponent, to have an impact in biology and physics requires asubstantial amount of software development, as well asinterdisciplinary collaboration with scientists in the relevantareas. The required implementations are developed by theinvestigator and his students as class library extensions to hissoftware package MC (Manifold Code), and the resulting softwareis made available to researchers in the biology and physicscommunities. This project is an NSF CAREER grant. The computational methods developed in this project have adirect impact on the modeling capabilities of rational drugdesign researchers using structure-based models of biomolecules.The Poisson-Boltzmann equation is a well-established model forelectrostatic interactions of biomolecules, allowing one topredict the effectiveness of a candidate drug molecule innegotiating the onion-like layers of the electrostatic field onits journey to bind to the active site in a target biomolecule.For example, through the use of these types of structure-basednumerical models, one can predict the effectiveness of acandidate HIV proteinase inhibitor in binding to the active sitein the HIV proteinase. However, the Poisson-Boltzmann equationpresents severe technical difficulties for solution by numericalmethods on computers. By developing improved multilevelnumerical techniques to reduce the computer time required tosolve the Poisson-Boltzmann equation, and at the same timeproducing more accurate solutions through the use of adaptivefinite element methods, the investigator gives drug designersaccess to models that produce more accurate results in less time.In addition, a critical ingredient for the success of this andrelated projects is the education and training of the nextgeneration of computational mathematicians. Therefore, a fullyintegrated involvement in undergraduate and graduatecomputational mathematics education is an integral part of theproject. In particular, the investigator develops two newproject-oriented, interdisciplinary, computational mathematicscourses at UC San Diego, modeled after a successful experimentalcourse he taught at UC Irvine in 1997-1998. This project is aNational Science Foundation CAREER grant. NSF stronglyencourages the early development of academic faculty as botheducators and researchers. The Faculty Early Career Development(CAREER) Program is a Foundation-wide program that provides forthe support of junior faculty within the context of their overallcareer development. It combines in a single program the supportof quality research and education in the broadest sense and thefull participation of those traditionally underrepresented insciences and engineering. This program enhances and emphasizesthe importance the Foundation places on the development of full,balanced academic careers.