Li-Tien Cheng, Ph.D. - US grants

The Scientific Computation Group (Scicomp) in the Mathematics
Department at UCSD will purchase a 16-processor MPI-based parallel
computer and four Linux-based graphics workstations for visualization.
The equipment will be dedicated to the support of research in the
mathematical sciences. The equipment will be used for several research
projects, including parallel adaptive multilevel finite element methods,
parallel algorithms for optimal control with PDE constraints, parallel
level set algorithm development, and applications in biochemistry and
physics.

Supported by this computing equipment, our research program will
provide graduate students and postdocs a broad educational experience
in an environment in which different aspects of scientific computation
are integrated. This produces scientists who are not only expert in
their own area of study, but are also able to communicate across
disciplinary boundaries.

As part of this proposal, SDSC (San Diego Supercomputer Center)
will provide fiber-optic infrastructure to connect the 16-processor
parallel computer to a similar large cluster located at SDSC. The
coupled clusters will be used in large production calculations
requiring substantial resources, and the individual Scicomp cluster
will be used for parallel algorithm development and for smaller
production calculations. The Scicomp satellite with fiber link to
SDSC will complement similar computational satellites recently
constructed by the biochemistry group on campus and by the UCSD
Medical School.

The investigator and his colleagues focus on several
fundamental computational issues involved in the parallel
implicit solution of optimization problems with partial
differential equation The goal of the research is to advance the
state of the art in four fundamental topics: (1) the formulation
and analysis of algorithms for large-scale optimization; (2)
adaptive mesh generation for partial differential equations; (3)
multilevel partial differential equation solvers; and (4)
parallel computation. Although each of these topics can be
investigated in isolation, the exploitation of their interactions
is crucial for the creation of effective global algorithms. The
investigators are members of a Scientific Computation Group that
offers a program of instruction and research emphasizing the role
of scientific computation in the formulation, modeling, and
solution of problems from diverse and changing areas. A major
part of the project involves the development of software and its
dissemination within the manufacturing, engineering, and
scientific community. Software developed as part of the project
provides an effective method of technology transfer and extends
the scope and effectiveness of the existing codes PLTMG, MC, and
SNOPT developed by the investigators.
Because partial differential equations conveniently
characterize the physical laws of many complex systems occurring
in science and engineering, they also lie at the heart of the
mathematical models used to simulate and predict the behavior of
these systems. The need to optimize the performance of such
systems is the common feature of practical applications that
range over such diverse areas as the optimal design of the hull
of an America's Cup yacht, the cleanup of toxic waste sites, the
construction of bioartificial arteries in tissue engineering, and
the management of stock portfolios and hedge funds. Software
developed in this project provides engineers and scientists with
instant access to state-of-the-art methods for the modeling and
optimization of complex systems involving partial differential
equation constraints. The resulting improvements in the
efficiency, accuracy and robustness of these models have a
substantial impact in areas of manufacturing and engineering that
are vital to US global competitiveness.

This project involves a three-year program of research on several
fundamental computational issues involved in the parallel implicit
solution of optimization problems with partial differential equation
(PDE) constraints. Such problems arise in many contexts in
engineering and scientific computation, since physical reality is
often expressed through models involving PDEs. Accurate
discretizations of PDEs lead to very large sparse constrained
optimization problems, where at least part of the structure reflects
the discretization. The goals of the research on methods include, but
are not limited to, advancing the state-of-the-art in four fundamental
topics: (1) the formulation and analysis of algorithms for large-scale
nonlinear optimization; (2) adaptive mesh generation for PDEs; (3)
multilevel PDE solvers; and (4) parallel computation. Although each
of these topics can be investigated in isolation, the investigators
believe that the exploitation of their interactions is crucial for the
creation of effective global algorithms. The research is motivated
and guided by three particularly challenging applications: (i)
geophysical inverse problems; (ii) projection methods for evolution
PDEs with constraints; and (iii) constrained level-set methods. These
topics cover a number of important applications of computational
science, including off-shore petroleum exploration, the numerical
modeling of black holes, the modeling of crystal growth and
biomembranes, capturing diffraction effects of waves and path
planning. Two features common to all these applications are that the
PDE constraints must be handled using modern adaptive multi-level
techniques and that the underlying optimization problem is highly
nonlinear and hence nonconvex.


The Investigators are members of the Computational and Applied
Mathematics (CAM) Group within the Department of Mathematics at UC San
Diego. They have a combined expertise in applied mathematics,
numerical optimization, numerical partial differential equations and
parallel computation. An important goal is the development of
software embodying the above algorithms. Software developed as part
of the project will provide an effective method of technology transfer
and will extend the scope and effectiveness of existing codes that
have been developed by the investigators at UC San Diego. The
software component of the project will have a substantial impact on
research involving the modeling of complex systems as it will provide
scientists and engineers with instant access to state-of-the-art
methods. Within the Computational and Applied Mathematics Group, the
Investigators offer a program of instruction and research that
emphasizes the role of computational science in the formulation,
modeling, and solution of problems from diverse and changing areas.
The activities associated with this project will help attract advanced
graduate students into the area of computational science, which plays
a vital role in the study of systems arising in manufacturing,
engineering and the natural sciences.

The Computational and Applied Mathematics (CAM) group in the Department
of Mathematics at the University of California, San Diego is proposing
the purchase of a 64-bit, 33-node blade rack system, using NPACI Rocks
Linux-based software, to improve and upgrade their current parallel
computing capabilities. The proposed equipment will support fundamental
research and education in computational science. The research of the
group includes, but is not restricted to, five core projects:
(1) New algorithms for parallel computation with adaptive multilevel
finite-element methods.
(2) The level-set method in computationally intensive environments.
(3) Optimization with partial differential equation constraints.
(4) Finite element modeling applications in biochemistry and physics.
(5) Hybrid finite-element level-set methods for stress-driven
interface dynamics in materials science.
Each of these projects is computationally intensive and requires the
extensive use of parallel computation. The Investigators have combined
expertise in applied mathematics, numerical optimization, numerical
partial differential equations and parallel computation. Access to
state-of-the-art parallel computers has played a vital and formative
role in the research and educational program offered by the CAM group.
The proposed equipment will provide an order-of-magnitude improvement
in the computing capabilities of the group and will allow them to
consider even more challenging computational problems and applications.

The Investigators are participating in a wide range of outside research
projects that include applications of computational science in off-shore
petroleum exploration, protein modeling, the numerical modeling of
black holes, the modeling of crystal growth and biomembranes, the
capture of diffraction effects of waves, and path planning. An important
goal of the CAM group is the development of software for computational
science and its dissemination within the manufacturing, engineering and
scientific community. Software developed as part of the project will
provide an effective method of technology transfer and will extend the
scope and effectiveness of the existing codes PLTMG, FETK, SNOPT and
IOTR that have been developed by the investigators at UC San Diego.
The software component of the project will have a substantial impact on
research involving the modeling of complex systems as it will provide
scientists and engineers with instant access to state-of-the-art methods.
The members of the CAM group offer a program of instruction and research
that emphasizes the role of computational science in the formulation,
modeling, and solution of problems from diverse and changing areas. The
research and educational activities supported by the proposed equipment
will help attract advanced graduate students into the area of computational
science, which plays a vital role in the study of systems arising in
manufacturing, engineering and the natural sciences.

This project develops hybrid computational models and robust numerical methods for electrostatic interactions in biomolecular systems. The computational models are constructed at different levels. They include variational mean-field models with atomistic details, particularly ionic size effects, and Monte Carlo simulation models for treating individual ions. These models are coupled with an advanced, variational approach to the solvation of biomolecules. A robust numerical method for solving the related elliptic interface problem and calculating the dielectric boundary force is designed and analyzed. Special interface algebraic multigrid methods and the GPU (Graphics Processing Unit) implementation are developed to accelerate the related large-scale computations. Numerical analysis focuses on the accuracy of the proposed schemes, particularly that of the boundary force approximation.

Biomolecules such as proteins and DNA are assemblies of atoms of which a significant portion are charged. Charged biomolecules polarize the solvent (water or salted water) and produce ions that are mobile charged particles in the solution. The electrostatic or charge-charge interaction gives rise to strong forces that determine the structure, dynamics, and function of underlying biological systems. For instance, the electrostatic interaction affects how a drug molecule binds to a target molecule, which in turn determines how effective the drug is in the process of curing a disease. Through the development of modern mathematical theories and computational tools, this project aims at understanding the fundamental principles of biological systems at the molecular level and advancing the research of computational mathematics. The success of this project can potentially help reduce the high cost often needed for experiments and speed up the process of drug discovery. In addition, this highly interdisciplinary research brings opportunities for students at different levels to receive training at the interface of computational mathematics and molecular biological science. Such training is critical to keeping our strength in scientific research in an competitive international environment.

Interfacial fluctuations are common in many physical and biological systems. Understanding the principles that underlie such fluctuations has far-reaching scientific and technological consequences. For instance, the manipulation of interfacial fluctuations in the so-called molecular beam epitaxy of growing nanometer-scale semiconductor materials can largely improve the quality and functionality of such materials that are widely used for high-technology electronic and military sensor devices. Effective treatment of some fatal diseases relies critically on our knowledge of anomalous water-protein interfacial structures that result from fluctuations and biological mutations and that characterize such diseases. This project develops a state-of-the-art computer program to investigate how the fluctuation affects the structures and long-time dynamics of interfaces, with a particular application to the binding of a drug molecule to a target protein that is a crucial step in the computer-aided drug design. The success of this project can therefore potentially help reduce the high cost often needed for laboratory experiments and speed up the process of drug discovery. In addition, this highly interdisciplinary research brings unique opportunities for students at different levels, particularly those from under represented groups, to receive training at the interface of mathematical and biological sciences. Such training is critical to keeping our nation's strength in scientific research in a highly competitive international environment.

Computationally tracking the motion of fluctuating interface is in general rather challenging, as such motion involves multiple but correlated spatial and temporal scales, high energy barriers between one stable interfacial structure to another, and the coupling of interface and bulk processes. The PIs construct two methods to overcome some of these difficulties. One is the stochastic level-set method that describes the fluctuating interface by solving a stochastic differential equation. The noise in the equation is spatially localized on or near the interface. Rigorous stochastic analysis is carried out to reformulate such an equation for accurate and efficient computations. The other is a stochastic lattice-phase method that treats the coupling of both interfacial and bulk fluctuations. This method describes the interface geometry by assigning a binary value on each of the discrete sites, and minimizes a Hamiltonian of all possible discrete binary fields using a Monte Carlo simulation method. This Hamiltonian mimics the continuum one with spatial gradient-square term and a double-well potential. The mathematical analysis using the notion of Gamma-convergence reveals the interplay between the numerical grid size and the interfacial width, and directly guides the design of fast algorithms. The PIs also develop a parallel computational algorithm with the GPU implementation to speed up their computations. They combine their new techniques with a molecular solvation theory to study molecular recognition, particularly the binding of a small drug molecule to a target protein. The computational models, numerical algorithms, and computer codes developed in the project can be incorporated into existing software that are used on a daily basis to study biomolecular interactions, and in particular, for computer-aided drug design

This project develops rigorous scientific theories and powerful computational tools to investigate the principal mechanisms by which drug and protein molecules associate and dissociate. Often, a drug molecule moves around in a crowded environment, and finds a spot of the surface of a protein to bind to, stays there, and can also leave, unbinding from the protein. During such binding and unbinding events, often repeated, both molecules constantly change their internal atomic positions. They also interact with other molecules, particularly the water molecules, in the surrounding environment. There are two key scientific questions on such complex processes that are characterized by multiple spatiotemporal scales and many-body effects. One is how stable the drug-protein bound unit is. Such thermodynamic stability severs as a criterion for searching drug molecules capable of binding to targeted proteins. The other is how fast or slow the binding and unbinding can occur. Such kinetics has been found recently in experiments and computer simulations to be critical to the drug effectiveness and efficacy. For decades, the scientific communities have made an enormous amount of effect, searching the quantitative answers to these questions to guide the computer-aided drug design and discovery. A recent assessment by the National Institutes of Health of the existing such computer programs, however, has concluded that advanced scientific theories are needed urgently to improve the practice. The success of this project can therefore provide a solid theoretical foundation as well as computational algorithms for drug design and discovery, potentially helping reduce the very high cost often needed for laboratory experiments and speed up the process of drug discovery. In addition, this highly interdisciplinary research project provides unique opportunities for students at different levels to receive training at the interface of mathematical, computational, and biological sciences, keeping our nation's strength in scientific research in a highly competitive international environment.

To tackle the extreme complex problem of molecular association and dissociation, the investigators design, implement, and analyze a very fast binary level-set method for interface relaxation to capture the molecular interfacial structures in the framework of an advanced, variational molecular solvation theory. The new method combines the strength of the threshold dynamics and the binary level-set representation, and utilizes the locality of the underlying energy landscape, and new pixel-flipping techniques to achieve very high efficiency. They also develop a new and hybrid computational approach to the kinetics of interface stochastic dynamics, coupling the interfacial energy minimization by the fast algorithm, the string method for transition pathways, and a novel, multi-state Brownian dynamics simulations. All these are applied specifically to investigating the molecular binding and unbinding kinetics for which, some of the conventional methods such as the standard Brownian dynamics simulations may fail. It is expected that this project will advance significantly the basic research in scientific computing and numerical analysis, particularly those of the interface dynamics and stochastic modeling. If successful, this research can help resolve some of the bottle-neck issues in solving very complex scientific problems.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

This project develops rigorous scientific theories and powerful computational tools to investigate the principal mechanisms by which drug and protein molecules associate and dissociate. Often, a drug molecule moves around in a crowded environment, and finds a spot of the surface of a protein to bind to, stays there, and can also leave, unbinding from the protein. During such binding and unbinding events, often repeated, both molecules constantly change their internal atomic positions. They also interact with other molecules, particularly the water molecules, in the surrounding environment. There are two key scientific questions on such complex processes that are characterized by multiple spatiotemporal scales and many-body effects. One is how stable the drug-protein bound unit is. Such thermodynamic stability serves as a criterion for searching drug molecules capable of binding to targeted proteins. The other is how fast or slow the binding and unbinding can occur. Such kinetics has been found recently in experiments and computer simulations to be critical to the drug effectiveness and efficacy. For decades, the scientific communities have made an enormous amount of effect, searching the quantitative answers to these questions to guide the computer-aided drug design and discovery. A recent assessment by the National Institutes of Health of the existing such computer programs, however, has concluded that advanced scientific theories are needed urgently to improve the practice. The success of this project can therefore provide a solid theoretical foundation as well as computational algorithms for drug design and discovery, potentially helping reduce the very high cost often needed for laboratory experiments and speed up the process of drug discovery. In addition, this highly interdisciplinary research project provides unique opportunities for students at different levels to receive training at the interface of mathematical, computational, and biological sciences, keeping our nation's strength in scientific research in a highly competitive international environment.<br/><br/>To tackle the complex problem of molecular association and dissociation, the investigators will design, implement, and analyze a very fast binary level-set method for interface relaxation to capture the molecular interfacial structures in the framework of an advanced, variational molecular solvation theory. The new method combines the strength of the threshold dynamics and the binary level-set representation, and utilizes the locality of the underlying energy landscape, and new pixel-flipping techniques, to achieve very high efficiency. They also develop a new and hybrid computational approach to the kinetics of interface stochastic dynamics, coupling the interfacial energy minimization by the fast algorithm, the string method for transition pathways, and a novel, multi-state Brownian dynamics simulations. All these are applied specifically to investigating the molecular binding and unbinding kinetics for which some of the conventional methods, such as the standard Brownian dynamics simulations, may fail. It is expected that this project will advance significantly basic research in scientific computing and numerical analysis, particularly at the interface of dynamics and stochastic modeling. This research will help resolve some of the bottle-neck issues in solving very complex scientific problems.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.