Weinan E - US grants

Nanoscale magnetic devices are of critical technological importance. This project will advance our understanding of their properties through a coordinated program of modeling, analysis, simulation and experiment. Topics to be addressed include (a) development of improved numerical methods for the simulation of magnetic materials and devices; (b) exploration of the micromagnetic "energy landscape" and the role of noise in thermally activated switching; and (c) investigation of specific nanoscale effects such as configurational anisotropy and geometrically constrained walls. Mathematics has much to contribute and much to gain. The study of appropriate limits leads to challenging problems of analysis whose solution will shed light on the essential physics. The analysis of noise and switching leads to the study of the energy landscape and to physically relevant examples of stochastic partial differential equations. Modeling coordinated with laboratory experiments will refine our understanding of the relevant phenomena. This Focused Research Group activity will draw expertise from a multidisciplinary group of mathematicians, physicists and computational scientists. The project includes a collaboration with IBM and training of postdoctoral scientists and graduate students.

Magnetic storage devices lie at the foundation of modern computing. Their modeling, simulation, analysis and design raise fundamental questions of physics and mathematics, many still unanswered. As device size decreases, the relevant science changes: defects, spatial disorder and thermal fluctuations become crucial in the nanoscale regime. Mathematics has much to contribute and much to gain. The study of appropriate limits leads to challenging problems of analysis whose solution will shed light on the essential physics. The analysis of noise and switching will be studied in a three-pronged approach: by mathematical analysis, numerical modeling and experimental investigation. Modeling coordinated with laboratory experiments will refine our understanding of the relevant phenomena. This Focused Research Group activity will draw expertise from a multidisciplinary group of mathematicians, physicists and computational scientists. The project includes a collaboration with IBM and training of postdoctoral scientists and graduate students.

A workshop would be organized and held at Princeton to celebrate the introduction of quasiconvexity by C.B. Morrey 50 years earlier by surveying recent developments and stimulating future applications. Quasiconvixity was originally introduced as a condition to understand the closely related questions of lower semicontinuity and existence in the calculus of variations. It is now clear that quasiconvexity is fundamental in many other applications.

E
0708026

The investigator continues work on developing a systematic
mathematical theory of crystalline solids starting from first
principles, with emphasis on understanding the connection between
atomistic/electronic structure models and continuum models. He
and his collaborators study the continuum limit of electronic
structure models such as models in Kohn-Sham density functional
theory. They also study numerical algorithms that combine
Kohn-Sham density functional theory and continuum models. In
particular, they study the accuracy of these algorithms and find
strategies for achieving uniform accuracy. These issues are
critical for understanding the electronic structure models and
making them powerful and practical tools for material science.
The project helps to provide the microscopic foundation of the
continuum mechanics of solids.

From a broader perspective, the project serves as an example
of the kind of work that needs to be done in order to establish
the foundation for multi-scale, multi-physics modeling. It is
now generally recognized across a wide spectrum of scientific
disciplines that the next major advances will come from
multi-scale, multi-physics modeling that integrates models from
quantum mechanics with models from classical or continuum
mechanics. However, due to the lack of a solid foundation, much
of the current activities in multi-scale, multi-physics modeling
are ad hoc in nature. Such a practice defeats the original
purpose of multi-scale, multi-physics modeling. This project is
aimed at establishing the necessary foundation for the case of
crystalline solids. In addition, interest in nanotechnology and
first-principle-based material design demands a much better
understanding of the connection between the mechanical properties
of solids and their electronic structure. Traditional concepts
that are often used in continuum theory will have to be modified
in order to take into account the effects at the electronic
structure level. Establishing such a connection is a major focus
of the proposed project.

The Department of Mathematics and Program in Applied and Computational Mathematics proposes to purchase a computational cluster which will be dedicated to support computational research in mathematical sciences. This research will be concentrated in particular in the following areas: (1) new algorithms for the separation of independent components in magnetic imaging data to study brain function, (2) the dynamics of how bio-molecules acquire and move between different conformations, (3) study of the analytic rank in connection with the Goldfeld conjecture on elliptic curves (a well-known conjecture in number theory), and (4) computer-assisted enumeration of cusped hyperbolic 3-manifolds.

Each of these projects require dedicated facilities that Fine Hall does not have currently. From a scientific and educational point of view, the proposed projects lie at the interfaces between different disciplines: signal processing, statistics and psychology for the fMRI project; applied mathematics and biochemistry for the second project; topology and geometry for the 3-manifold project. Progress on these projects will impact several communities. In addition, these projects will be part of the PIs' continuing effort on bringing frontier research to science education in the classrooms, as well as attracting women and other under-represented groups to the area of mathematics. New courses and discussion sessions will be organized so that a much larger part of the mathematical and scientific community will have access to these research projects through the proposed facility.

The main objective of the present proposal is to develop efficient algorithms for electronic structure analysis that are applicable for both metals and insulators. This will be done by developing multipole representation of the Fermi operator, which is a fundamental object in electronic structure analysis and more generally quantum theories of matter. In addition, the PI also proposes to develop efficient algorithms for representing and computing the Green's functions that arise in this context. A second component of the project is the numerical analysis of the algorithms in electronic structure analysis. Topics to be studied include the accuracy of linear scaling algorithms, convergence and convergence rates of self-consistent iterations. This will be done by developing and using simple but canonical model problems that capture the essential aspects of the problem but allow explicit analytical calculations.


Electronic structure analysis is at the foundation of chemistry and material science, as well as some aspects of biology. Our ability to understand chemical reactions and fundamental aspect of materials relies heavily on efficient numerical algorithms for solving models from quantum chemistry or density functional theory. Existing algorithms are much more effective for insulators than for metals. This is particularly true for the recently developed linear scaling algorithms which relies heavily on the exponential decay property of the wave functions or density matrices, a property that holds for insulators but not for metals. The present project is aimed at bringing powerful mathematical tools to bear on the problem of electronic structure analysis. The proposed work will explore the mathematical features of the electronic structure problem in a way that has never been done before. By doing so, new insights and new algorithms will result that greatly advance our ability to analyze the electronic structure of matter.

The quantum physics of many interacting electrons lies at the foundation of chemistry and condensed matter physics. A direct treatment of the many-electron problem is impossible due to its shear complexity: dealing with N interacting electrons requires solving partial differential equations in 3N dimensions. Equilibrium and non-equilibrium Density Functional Theories (DFT) are rigorous and formally exact theories which map the interacting N-electron problem into a non-interacting N-electron problem. The non-interacting electrons move in an effective potential that has a universal functional dependence on the total electron density. As a result, the problem is reduced to a problem in dimension 3, amenable for computation. In this proposal the PIs propose to study a number of dynamical problems in many-body quantum mechanics within an interdisciplinary environment of mathematicians and physicists. In particular, the PIs propose to develop further the mathematical foundations of density-functional theory, for equilibrium as well as the time-dependent case. The mathematical structure of the theory and its solutions will be further investigated and the insight from this analysis will be used to develop efficient numerical simulations. Particular emphasis will be given to the treatment of the spin-orbit interaction, within the full relativistic formulations and in non-relativistic formulations that include relativistic corrections. The PIs also plan to establish the foundations of the Dissipative Time-Dependent Density Functional Theory, and to apply the theory to the problem of charge and spin transport in materials.

The present technological progress is in great part based on design and discovery of new materials. Nowadays, the design of advanced materials involves laboratory work and computer simulations. Enhancing the accuracy and efficiency of computer simulations will reduce the costs, broaden the array of interesting and potentially useful materials, and speed up the process of testing and characterization. This is the target of the proposed research. The plan is to combine rigorous mathematical analysis, the insights from physics, chemistry and computer simulations in order to push the boundaries of theoretical simulations of advanced materials such as nano-structured materials, topological insulators and molecular electronic devices. The proposed research could have significant technological impact in applications such as nano-science and other areas of interest such as solar cell devices and energy conversion and storage. The PIs propose to integrate research and education by involving undergraduate and graduate students, and post-doctoral associates, in an interdisciplinary environment. Special attention will be paid to the recruitment of women and students from other underrepresented groups through the utilization of a diverse number of programs at the participating institutions.

This MRI award will serve to purchase a 256 Processor/1536 core SGI Altix UV1000 with 9.2 TB of RAM and 14.4 TB of raw scratch disk space. The instrument will provide scientists at Princeton University, the Institute for Advanced Studies, and partner institutions with the computational resources needed to model multi-scale phenomena in the sciences and engineering. The flexibility of the Altix architecture, which supports both shared and distributed memory applications, along with an outstanding bus architecture to support the addition of extra processing units such as GPGPUs is an ideal platform for developing algorithms for multi-scale problems. The setting of the instrument in the University?s High Performance Computing Research Center will facilitate a cross-disciplinary approach combining expertise in applied mathematics, computer science and domain-specific disciplines enabling innovative approaches for memory intensive applications. The new instrument will play an essential role in educating a new generation of scientists and training students across many disciplines in the use of advanced modeling tools on modern computer platforms, contributing to new graduate student certificate programs offered by PACM, the Program in Applied and Computational Mathematics, and PICSciE, the Princeton Institute for Computational Science and Engineering. Finally, the instrument will provide a necessary link between local and national facilities, preparing the Princeton scientific community to the emerging multicore and massively parallel architectures of the future.

The instrument will enhance international scientific cooperation by contributing to projects like the Munich-Princeton collaboration in cosmological computational science, and will contribute to science education of the general public through collaboration with the American Museum of Natural History in New York City, with planned new visualizations for use in the Cosmos series and in conjunction with the Museum?s ongoing public education work on earthquakes and geologic movement. Women?s participation in computational projects enabled by the instrument will set examples to encourage greater access of women to science. Finally, access provided to partners at
California State University-Northridge will contribute to training minority scientists and engineers.

This research concerns improved methods for the modeling of material systems. Many materials modeling problems rely heavily on quantum mechanical models. However, for quantum mechanics to be really useful, one has to couple it with either continuum models or molecular mechanics models. The success of such an approach has been well documented in chemistry applications. This project explores approaches based on quantum mechanics (density functional theory) coupled with molecular mechanics or continuum mechanics in modeling materials. Although the coupled quantum mechanics/molecular mechanics approach has enjoyed a considerable amount of success, winning the 2013 Nobel Prize in chemistry, its application to general material systems, particularly metallic systems, has been at issue for a long time. The problem comes from the non-local effect of the errors made at the interface between the molecular mechanics and quantum mechanics regions. This problem is particularly severe for metallic systems. This project tackles these important challenges in the modeling of material systems.

There are many technical hurdles that one needs to overcome. First of all, one needs to improve the efficiency of density functional theory (DFT) algorithms to be able to handle inhomogeneous systems, such as systems with point defects or a small portion of extended defects. The next hurdle is to derive classical models, based on either molecular mechanics or continuum theory, that are consistent with DFT. This means that the DFT models reduce to classical models in some limit. A third problem is to design coupling schemes that move smoothly from DFT to classical models. The current project will focus on the second and third problems, beginning with one-dimensional model problems. The investigation's starting point is the Fermi operator formulation of DFT. By making successive approximations on the Fermi operator formalism, the PI aim to arrive at a consistent DFT/molecular mechanics or DFT/continuum model coupling scheme.