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

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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.