Qiang Du - US grants

A centroidal Voronoi tessellation (CVT) is a Voronoi tessellation of a
given set such that the associated generating points are centers of
mass of the corresponding Voronoi regions. Applications of CVT's
range from problems in image compression, vector quantization,
quadrature rules, grid generation and optimization, finite difference
schemes, distribution of resources, cellular biology, cluster analysis,
and the territorial behavior of animals. The goals are to develop highly
efficient parallel algorithms for the computation of CVT's, to gain
further understanding of interesting features related to the these
tessellations, to implement and test these algorithms, and to produce
a useful software suite that can serve as a design tool for many
problems in applications.

Among the theoretical questions to be studied are the complexity of
algorithms for the construction of CVT's, the effects of nonuniform
densities, CVT's in general metrics, constrained CVT's, and generalized
CVT's based on lines, curves and surfaces.

Parallel deterministic and probabilistic algorithms will be developed
and tested. In the former case, different averaging and communication
strategies will be examined; for the latter,domain decomposition
ideas will be exploited. This effort can lead to parallel grid generation
algorithms and other software useful in applications such as
clustering analysis.

Applications of CVT's will also be studied, with particular emphasis
on grid generation and optimization. The questions addressed in this
connection include the role of the density functions used to generate
the CVT's on the optimal placement of grid points, the construction of
grids with special properties and the the use of various generalized
CVT's. Another application that will be examined are the use of CVT's
in the clustering analysis of large data sets.

There has been an increasing trend to conduct scientific research using numerical simulations on modern high performance computers in recent years. Considerable progress has been made in the area of computational material sciences. Computational tools have been used in the design of new materials as well as in the study of their properties. The central objectives of this project are: 1) to develop or refine certain mesoscale and macroscale models, so to enlarge the range of physical problems for which such models are valid; 2) to analyze these models in order to gain further understanding of their properties and solutions; 3) to develop, analyze, and implement algorithms, in particular, parallel and adaptive algorithms, for the numerical simulation of these models; and 4) to use our algorithms and codes to study some interesting phenomena in material sciences.

In the proposed work, the principal investigator will study models and develop numerical algorithms for some interesting material sciences problems that involve multiscale (mesoscale and macroscale) and stochastic effects, such as problems related to vortices and other defects in superconductivity and magnetism. A major part of the project is aimed at increasing the range of applications for the mesoscale codes and allow more comparative studies between the mesoscale and macroscopic models through the use of domain and scale decomposition/integration and adaptive computation techniques. The codes for mesoscale models can be of use in gaining information and insight about the physical behavior and interaction of the fine structures (such as vortices) with, for example, boundaries, interfaces, impurities, currents, and thermal fluctuations. They can be of indirect use to device designers, in particular, when connections with macroscopic properties can be identified. Models based on the stochastic partial differential equations and their numerical simulations will also be given emphasis, so as to gain insight to the macroscopic effect of thermal fluctuations and impurities in the materials like superconductors and liquid crystals. The work will be aimed at making the computational codes robust, efficient, flexible, accurate, scalable and user-friendly. It is hoped that these codes can be used by physicists, material scientists, and engineers in laboratories, universities, and industrial organizations as a tool for studying some specific material properties and also a tool in designing devices.

NSF proposal DMS-0215392
PIs: Xu, Belmonte, Du, Li and Zikatanov

ABSTRACT

The Department of Mathematics at the Pennsylvania State
University will purchase a 64-node parallel PC cluster
to be dedicated to the support of research and teaching
in the mathematical sciences. In particular, the PC
cluster will be used to support the research projects
of faculty members in the areas of the numerical
solution of partial differential equations in fluid
dynamics and material sciences, and computational
finance and in the studies of general numerical
techniques such as parallel multigrid algorithms and
quasi-Monte Carlo methods. In particular, the projects
include studies of important issues concerning modeling
and simulations of non-Newtonian flows, liquid crystals,
quantized vortices, water waves and fuel cells. Much of
the research efforts rely critically on the
establishment of the proposed PC cluster.

The proposal involves an integrated collaboration
between the numerical work to be performed in the PC
cluster and the experimental work to be performed in
the W. G. Pritchard Laboratories of the Department of
Mathematics. The new equipment will make it possible to
numerically simulate the various complex physical
phenomena observed in the fluid lab and will greatly
enhance collaborations among researchers in computational
and applied mathematics at Penn State. The PC cluster
will also be the basis for the creation of a new
computational laboratory, which together with the
Pritchard fluid lab, will provide a unique environment
for multidisciplinary research as well as for (both
undergraduate and graduate) student training.

This award is made under the Information Technology Research initiative and is funded jointly by the Division of Materials Research and the Advanced Computational Infrastructure Research Division.

This collaborative research project involves two materials scientists, a computer
scientist, a mathematician, and two physicists from academia, industry and a national laboratory. The project is a synergistic effort that leverages the overlapping and complimentary expertise of the researchers in the areas of scalable parallel scientific computing, first-principles and atomistic calculations, computational thermodynamics, mesoscale microstructure evolution, and macroscopic mechanical property modeling. The main objective of the proposal is to develop a set of integrated computational tools to predict the relationships among the chemical, microstructural, and mechanical properties of multicomponent materials using technologically important aluminum-based alloys as model materials. A prototype GRID-enabled software will be developed for multicomponent materials design with efficient information exchange between design stages. Each design stage will incorporate effective algorithms and parallel computing schemes. Four computational components will be integrated, these are: (1) first-principles calculations to determine thermodynamic properties, lattice parameters, and kinetic data of unary, binary and ternary compounds; (2) CALPHAD data optimization computation to extract thermodynamic properties, lattice parameters, and kinetic data of multicomponent systems combining results from first-principles calculations and experimental data; (3) multicomponent phase-field modeling to produce microstructure; and (4) finite element analysis to obtain the mechanical response from the simulated microstructure. The research involves a parallel effort in information technology with two main components: (1) advanced discretization and parallel algorithms, and (2) a software architecture for distributed computing system. The first component includes: (a) a coupling of spectral and finite element approximations, (b) local adaptivity and multi-scale resolution, (c) high order stable semi-implicit in time schemes, (d) parallelization through domain decomposition, and (e) scalable sparse system solvers. The second component involves computational GRID-enabled software for the overall design process; this software architecture enables the use of geographically distributed high performance parallel computing resources to reduce application turnaround time while providing a flexible client-server interface that allows multiple design cycles to proceed.

The research project will be integrated with education and training of graduate students in the broad area of computational science and engineering through the participation of students and the PIs in the "High Performance Computing Graduate Minor" offered through the Institute of High Performance Computing at The Pennsylvania State University. Existing programs at Penn State will be used to integrate undergraduates into the project.
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This award is made under the Information Technology Research initiative and is funded jointly by the Division of Materials Research and the Advanced Computational Infrastructure Research Division.

This collaborative research project involves two materials scientists, a computer
scientist, a mathematician, and two physicists from academia, industry and a national laboratory. The project is a synergistic effort that leverages the overlapping and complimentary expertise of the researchers in the areas of scalable parallel scientific computing, first-principles and atomistic calculations, computational thermodynamics, mesoscale microstructure evolution, and macroscopic mechanical property modeling. The main objective of the proposal is to develop a set of integrated computational tools to predict the relationships among the chemical, microstructural and mechanical properties of multicomponent materials using technologically important aluminum-based alloys as model materials. Prototype GRID-enabled software will be developed for multicomponent materials design. Effective algorithms and parallel computing schemes will be incorporated into the design. The GRID-enabled software allows geographically distributed high performance parallel computing resources to be harnessed bringing greater computational power to bear on a given problem and enabling practical application of these computational tools. The prototype software, with improved predictive power in multicomponent materials design, may enable scientists to develop new materials with unique properties and to tailor existing materials for better performance.

The research project will be integrated with education and training of graduate students in the broad area of computational science and engineering through the participation of students and the PIs in the "High Performance Computing Graduate Minor" offered through the Institute of High Performance Computing at The Pennsylvania State University. Existing programs at Penn State will be used to integrate undergraduates into the project.
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The proposed project is concerned with the development,
analysis and applications of numerical simulation tools
to a number of problems in computational sciences,
in particular, in computational physics and computational
material sciences. It is a continuation of PI's past research
work in this area that has contributed to the modeling, analysis,
and computation of various problems in superconductivity,
Bose-Einstein condensation, and phase transitions in binary and
multicomponent alloys.

In the proposed work, while building upon the past progress,
the PI will take on new initiatives in the study of some interesting
physical problems at various time and spatial scales
and in the design of new algorithms and efficient
solvers which can then be used to understand experimental phenomena
and the underlying physical and material properties.
The proposed works are to be carried out in the following aspects:
to develop or refine mathematical models for the underlying physical
problems, so to enlarge the range of validity of such models; to
analyze these models in order to gain further understanding of their
properties and of their solutions; to develop, analyze, and
implement algorithms, in particular, parallel and adaptive algorithms,
for the numerical simulation of these models; and to use our algorithms
and codes, together with physicists and material scientists,
to study some interesting phenomena in physics and material sciences,
including the further studies on the quantized vortices in
superconductors and Bose-Einstein condensates, on the effects of
fluctuation and on the validity and the extension of mezoscopic
phenomenological models. While we will focus on developing innovative
mathematical theory and computational algorithms, comparisons with the
physical experiments will also be made whenever possible.

Abstract

Molecular Dynamics simulations are powerful
tools to study problems of materials science,
nanoscience, and biology. It naturally provides
ample opportunities for interdisciplinary research
that requires knowledge in mathematics, statistics,
computer science, physics, materials and biology.
The focus of this project is on developing learning-based
computational and statistical methods for potential
energy landscape modeling to accelerate ab-initio
molecular dynamics simulations. The set of tools
developed will substantially expand the limits of time
and system size without compromising the precision and
quality of the ab-initio simulation results.

Hongyuan Zha, Qiang Du, Runze Li and Jorge Sofo will
investigate learning and
computational methods 1) to characterize both the local and global
structures of the low-dimensional manifold in which the simulation
really occurs through manifold learning from the trajectories of the
ab-initio simulation; 2) to identify and extract suitable clusters in
the reduced dimension spaces corresponding to regions in the
configuration space that naturally emerge from the ab-initio
simulation and are visited frequently by the particles throughout the
simulation; 3) to conduct efficient energy and force interpolation
using Gaussian Kriging models with penalized likelihood. In this
learning and computational
framework, the interpolated potential energy surface will be
evaluated and it will replace the costly ab-initio evaluation when its
precision is good enough. As the simulation evolves, the interpolated
potential energy surface will be retested to detect the
eventual need of a retraining in case the simulation is exploring
new regions of the configuration space.

A group of researchers in the Department of Mathematics at Penn State
will purchase a symmetrical multiprocessor (SMP) cluster, and a
visualization server. The new equipment will be used to support
various ongoing mathematical research projects including number
theory, numerical solution of partial differential equations,
mathematical modeling and numerical simulations for hydrogen fuel
cells, modeling in fluid dynamics and material sciences, mathematical
modeling and numerical simulations in biological sciences, analysis
and implementation of new cutting-edge numerical techniques and
parallel algorithms. Most of the models and applications from these
projects are described by systems of partial or ordinary differential
equations and their numerical solutions rely on intensive computations
that require high memory capacity and high memory bandwidth. The
shared memory computer to be purchased will serve well for these
requirements and will significantly enhance the proposers' research
capabilities.


The research projects that benefit from the new equipment range from
the very basic mathematical research such as number theory to very
practical applications such as mathematical modeling and numerical
simulations of hydrogen fuel cells that have vital national interests
(for security and environment). The proposers communicate and
collaborate on a various research subjects within the department as
well as across the university. Such collaborations provide a
stimulating environment for the training of graduate students,
postdoctoral fellows, as well as junior faculty members. The proposed
equipment will also aid existing, and provide solid computational base
for new joint projects with researchers from other sciences and
engineering

This project is concerned with the study of interfaces and defects which are ubiquitous in physical and biological systems and are essential to the materials properties and biological functions. Building on the previous research work, the PI will develop and apply analytical and numerical simulation tools to study both deterministic and stochastic effects associated with various material interfaces and defects with particular emphasis on defects in superconductors and Bose-Einstein condensates as characterized by quantized vortices, and soft interfaces as characterized by model biomimetic cell membranes. The investigation will be largely following the general Ginzburg-Landau (diffuse interface, phase field) formalism with connections to other multiscale and stochastic modeling approaches. Systematic model derivations, analysis and simplifications will be conducted. Adaptive algorithms and statistics retrieval algorithms will be designed and analyzed. These research activities will enhance the simulation capability of complex systems and the predictive power of large scale numerical computation. While focusing on specific applications, the central algorithmic development work in this project will be in tune with the modern theme of integrated, adaptive and intelligent scientific computation.

This project lies at the interface of computational mathematics, physics, materials and biological sciences. The physical and biological objects to be studied through mathematical analysis and numerical simulations include quantized vortices which are well-known signatures of superfluidity, and biological membranes which are soft interfaces designed by nature as the fundamental building blocks of life. The PI will develop new analytical theory on the complex nonlinear models and new tools for extracting useful statistics and exploring hidden structures from massive simulations. The research activities will bring new advances to mathematics and provide better understanding of various basic physical and biological processes. They in turn may aid the efforts in discovering new technology based on the superconductivity and superfluidity, and new design of drug and drug delivery vehicles based on biomimetic membranes, both are of significant economic value and national priority. In addition, this project will provide valuable training environment and interdisciplinary research experience to the future generation of workforce and young researchers that showcases the TEAMS (Training in Experiments, Analysis, Modeling and Simulations) spirit.

This project will focus on the development and analysis of numerical methods
to probe the complex energy landscapes related to various interface and defect
problems, such as microstructures in materials undergoing phase transitions,
quantized vortices in geometrically frustrated configurations, and so on.
The associated stochastic dynamics and hydrodynamics, as well as their
numerical approximations will also be investigated.
He will also study robust and adaptive algorithms that are useful in the
statistical analysis of the underlying structures and important features.
These research issues, on one hand, are driven by practical applications
through collaborations with other scientists, and on the other hand, also
motivate new studies of mathematical subjects ranging from geometry and
topology to numerical and stochastic analysis. The research to be carried out is of interdisciplinary nature, encompassing subjects like computational mathematics, physics, information, materials and biological sciences.
There are many mathematical and numerical challenges involved in the research
such as the understanding of the collective behaviors of families and paths of
solutions of nonlinear PDEs and stochastic dynamics, and the exploration of
the hidden structures and statistics in the simulated results.

The research on the algorithmic development and numerical simulations can help
enhancing the capability as well as the predictive power of scientific
computations, which has potentially significant scientific, social and
economic impact and is one of the top research priorities internationally.
Meanwhile, interfaces and defects are also ubiquitous in nature which play
fundamental roles in many aspects of physical and biological systems. A better
mathematical and computational understanding of interfaces and defects,
especially in a stochastic setting, will enrich the scientific knowledge base, which in turn may aid the efforts by physicists, materials scientists and engineers in discovering new materials with desirable properties and in
developing new scientific devices and commercial instruments. This project
will also provide valuable interdisciplinary research opportunities for the
future generation of workforce and researchers. With an emphasis on the TEAMS
(Training in Experiments, Analysis, Modeling and Simulations) spirit, young
students can be better prepared to conduct interdisciplinary research in their
career.

Center for Computational Materials Design (CCMD)

IIP-1034965 Pennsylvania State University (PSU)
IIP-1034968 Georgia Tech (GT)

This is a proposal to renew the Center for Computational Materials Design (CCMD), an I/UCRC center that was created in 2005. The lead institution is Pennsylvania State University, and the research partner is Georgia Tech. The main research mission of the CCMD is to develop simulation tools and methods to support materials design decisions and novel methods for collaborative, decision-based systems robust design of materials.

The intellectual merit of CCMD is based on the integration of multiscale, interdisciplinary computational expertise at PSU and GT. CCMD provides leadership in articulating the importance of integrated design of materials and products to industry and the broad profession of materials engineering; and is developing new methods and algorithms for concurrent design of components and materials.

CCMD has operated successfully in Phase I, and has helped develop a partnership amongst academe, industry and national laboratories. Based on feedback received from the various members, CCMD has outlined in the renewal proposal research thrusts and initiatives for Phase II; and has also identified gaps that will be addressed as research opportunities in Phase II.

CCMD will have a large impact on how industry addresses material selection and development. The expanded university/industry interaction of this multi-university center offers all participants a broader view of material design activities in all sectors. CCMD contributes to US competitiveness in computational materials design by educating new generations of students who have valuable perspectives on fundamental modeling and simulation methods, as well as industry-relevant design integration and materials development. CCMD participates in programs at PSU and GT that support K-12 STEM issues, women and underrepresented groups, undergraduate students, and high school teachers. CCMD plans to disseminate research results via papers, conferences and the CCMD website.

Mengesha
1312809

The principal investigator and his colleague study the mathematical properties of nonlocal models, particularly the peridynamic model of continuum mechanics. The project focuses on establishing the well-posedness of both linear and nonlinear models for dynamic and equilibrium problems involving nonlocal boundary (volume-constrained) value conditions; classifying emerging discontinuities and developing a regularity theory for solutions of nonlocal models as functions of given data; and studying conditions that enable nucleation and propagation of singularities in peridynamic equations and other nonlocal models. The investigators use analytical tools ranging from perturbation methods to calculus of variations. The project addresses technical challenges that are absent from the traditional local approach, leading to extensions of classical mathematical concepts and techniques to the nonlocal setting.

The project offers a mathematical framework for studying nonlocal operators and equations that may be useful to the study of many common nonlocal phenomena. It helps establish the necessary theoretical footing for nonlocal models. The project findings are integrated by the investigators into classroom teaching and other educational endeavors. The investigators' long-term goal is to work with mechanical engineers, physicists, biologists and materials scientists to make nonlocal models useful tools for studying complex systems in various scientific and engineering applications such as anomalous diffusion, nonlocal heat conduction, phase transition, and biological aggregation.

The project is concerned with mathematical and computational issues related to the simulation and analysis of equilibria, metastable and transition states and minimum energy paths for complex energy landscapes of practical interests, and associated stochastic dynamics. The research to be carried out is closely motivated by applications in a number of areas of federal strategic interests: the development of effective algorithms and codes is a crucial part of high-performance computing, and numerical methods and software tools to be developed may be potentially useful for effective computational materials and drug design.

The principal investigator will carry out interdisciplinary research that encompassing subjects like computational mathematics, physics, information, materials and biological sciences. He will focus on new algorithmic development and analysis which have the potential to significantly improve the usual practice on the modeling and simulation of rare events. He will consider some specific and important applications, including systems of interacting particles and interfaces in geometrically confined and frustrated configurations or deformable geometry which arise in many areas of physics, chemistry and biology (such as formation of nano-clusters, bimolecular conformation, vesicle mediated interactions, and critical nucleation in solid state transformations). He will attempt to draw strong connections with some of the algorithms developed by practitioners implemented in existing software codes such as those for first principle calculation and computational chemistry. Most of the problems involved in the research project are associated with either infinite dimensional spaces such as deterministic or stochastic partial differential equations or finite dimensional spaces with high dimensions (discretization of differential equations or particle systems involving a large number of particles), which lead to many computational challenges. Various mathematical and numerical issues will be studied, ranging from efficient local saddle point search and its robust numerical implementation to rigorous analysis and effective multiscale simulations of relevant dynamics and rare events. It is expected that the progress made during the project will have a broad impact on the community interested in the study of rare events. The project will also contribute to education and training as it will provide students valuable training ground and research experience in an interdisciplinary environment. Much effort will be devoted to promoting active engagement of student participation at all levels and integrating research findings into teaching and training.

DMREF: Deblurring our View of Atomic Arrangements in Complex Materials for Advanced Technologies

Non-technical Description: As we try and find new technologies to solve some of mankind's toughest challenges such as abundant sustainable energy, environmental remediation, and health, we are increasingly seeking more and more complex materials. We already have devices that turn sunlight into electricity and use sunlight to split water into precious hydrogen fuel, but issues such as device efficiency and cost mean that the current technologies cannot be taken to the vast scale needed for our modern needs. This puzzle may be solved by the use of advanced materials that perform their tasks - energy conversion, cancer cell killer, or whatever it may be - with greater efficiency. This is inevitably leading us towards more complicated materials that consist of many different chemical elements and have engineered structures on multiple different length-scales from the atomic to the nano- and meso-scales all the way to macroscopic scales. The problem is that, because of their complexity, it becomes very difficult to even characterize these materials when we have made them, let alone design and engineer them at the nanoscale. Our usual tools based on the scattering of x-rays by crystals stop working for such nanoscale structures. The problem is not that we lack powerful enough x-ray beams. The problem is that the x-ray scattering signal from these complicated materials doesn't contain enough information to allow us to find a unique structure solution. It is as if we are looking at complex patterns of atomic arrangements through blurry, steamed up glasses. This project will bring greater clarity to this situation by marrying together advances in applied mathematics from diverse areas such as image recognition, information theory and machine learning, which are having transformative impacts in commerce, law enforcement and so on, and applying them to the problem of recognizing atomic arrangements in materials of the highest complexity.

Technical Description: The approach will to solve multi-scale structures of materials by marrying together the latest advances in the processing of x-ray scattering data from nanomaterials, such as atomic pair distribution function (PDF) analysis, with other sources of input information such as small angle scattering, EXAFS and other spectroscopies, as well as inputs from first principle theory such as DFT, but place them in a rigorous mathematical framework and a robust computational framework such that the information content in the data may be utilized to the greatest extent possible whilst taking into account uncertainties from statistical and systematic uncertainties. The mathematical framework will utilize the latest developments in stochastic optimization, uncertainty quantification including function-space Bayesian methods, machine learning and image recognition.

Computational fluid dynamics is an important research field that plays a crucial role in the understanding of fluid flows appearing in many mechanical, hydrodynamic and biophysical processes. It occupies a central place in the development of computational science. The proposed research intends to help designing effective numerical algorithms, particularly those related to the so-called smoothed particle hydrodynamics (SPH), for modeling complex fluids and interfacial phenomena. The overall research objective is consistent with the long term vision of predictive and reliable computational science, and in the near term, it serves to complement ongoing research on SPH related methods and their applications currently being carried out by various academic institutions and national laboratories. The PI will not only work to facilitate the research effort but also to strengthen the training and education of young students and junior researchers. He will team up with collaborators to ensure the timely translation and integration of new theoretical findings into enhanced simulation capability for a variety of applications such as those involving heterogeneous transport in underground, atmospheric and biophysical systems, energy and high-strength materials, which are highly relevant to important national and societal interests.

Particle based computational methods such as the Smoothed Particle Hydrodynamics (SPH) and related methods offer great flexibility in numerical simulations and are becoming widely used in various scientific and engineering applications. As these techniques get populated into major simulation codes to be used by a large computational science and engineering community, it is imperative to carry out a more quantitative assessment and mathematical analysis as part of the rigorous validation and verification process. Assessing SPH based simulations is challenging since these methods have been historically applied to solve complex problems where either traditional methods do not work well or the formal accuracy is of secondary concern. Studies based on conventional numerical analysis techniques may not always produce mathematical findings that are strongly relevant in practice. Our proposed research is to improve the theoretical understanding of SPH and related methods. A novelty of our approach draws on the recently developed nonlocal models and their numerical approximations by the PI's group. It leads to new avenues to analyze SPH type methods by both distinguishing and relating the different roles of integral kernel representations/approximations of the locally defined spatial derivatives and numerical discretization of the resulting nonlocal operators. Indeed, inappropriate nonlocal relaxations of differential operators on the continuum level may be the root cause of some problematic issues inherent to particle based simulations. Incompatible discretization can also contribute to the loss of fidelity and stability. By taking nonlocal integral operators and nonlocal continuum formulations as bridges connecting continuum PDE models and particle like discrete approximations, our approach represents a significant departure from conventional numerical analysis that compares the discrete schemes with the underlying continuum PDEs directly. The focus on algorithm robustness is particularly relevant to SPH like methods given their intended application to complex systems involving multiple scales and extreme operating conditions. Specific objectives for the next few years include: developing continuum reformulations of SPH like methods with nonlocal/integral operators; and studying SPH type methods using discretization of nonlocal models as a bridge. In carrying out the proposed work, we use an integrated analytical and computational approach to provide both mathematical infrastructure needed for theoretical analyses and practical insight for code development. We pay close attention to techniques that work for solutions lacking regularity or exhibiting strong variations and for particle distributions and boundary conditions that are frequently encountered in practical implementations.

In recent decades, scientific and technological fields have experienced "data moments" as researchers recognized the potential of drawing new types of inferences by applying techniques from computational statistics and machine learning to ever-growing datasets. At the same time, everyday life is increasingly saturated with products of data analysis: search engines, recommendation systems, autonomous vehicles, etc. These developments raise fundamental methodological questions, including how to collect and pre- pare data for analysis, and how to transform statistical inferences into effective action and new statistical inquiries. To address these questions, it is necessary to develop theoretical foundations for the practice of data science, and to provide practitioners with sound and practically relevant methodological training. The Columbia TRIPODS Institute pursues these goals through an integrated program of research in data science foundations, curriculum development, and center-building activities. The research program seeks to provide theoretical understanding of practical heuristics, develop modular and well-structured toolkits of computational primitives for data science, and to support the entirety of the data science cycle, from data collection and annotation, to the assessment of the analysis product.

The Institute pursues programs of research, education and center-building aimed at articulating theoretical foundations for data science. Its activities aim to have a major impact in shaping this emerging field. The research directions include understanding tractable classes of optimization problems, developing primitives that support efficient computation on data, and developing methodological foundations for interactive protocols in data science. These directions address challenging problems at the interface between theory and practice, the solutions of which require ideas spanning mathematics, statistics, and computing. The educational activities articulate model curricula in data science at the MS/professional and PhD levels, including interdisciplinary courses aimed at building a common language for a new generation of scientists and engineers. Center building activities are organized around cross-disciplinary themes and structured to encourage interaction across disciplines and to develop a common methodological community in Foundations of Data Science. These research and educational activities-including workshops, summer schools, distinguished lecture series, long-term visits, and outreach- help to further define and disseminate a common language for foundational research and education, and to increase diverse participation in data science. Located within the Center for Foundations of Data Science in the Data Science Institute at Columbia University, the Institute is at the center of the Northeast Big Data Hub. This position supports expansion of activities within Columbia, and also with other Big Data Hub members, TRIPODS Institutes, and research/industry organizations. Funds for the project come from CISE Computing and Communications Foundations, CISE Information Technology Research, MPS Division of Mathematical Sciences, and MPS Office of Multidisciplinary Activities.

The study of nonlocal models has attracted much attention in many science and engineering disciplines such as materials science, mechanics, biology, and social science, and they are therefore of interest to applied and computational mathematics. Nonlocal models differ from the more common local models because they account for the factors active on a range rather than only at a point at which they are considered. The project is aimed at advancing the mathematical and numerical analysis of robust and effective numerical methods for those nonlocal models with a finite range of interactions. The research will complement the ongoing development of effective simulation platforms for nonlocal modeling in various application domains. It will also contribute to the integrated interdisciplinary education and research training of students.

An important class of robust numerical schemes for nonlocal models is provided by asymptotically compatible (AC) discretization schemes. The latter are designed to assure the convergence of approximate solutions, as numerical resolution gets refined, to correct physical solutions for problems with changing or even diminishing ranges of nonlocal interactions. The project will include a comprehensive study of AC schemes for nonlocal problems with heterogeneously distributed ranges of nonlocal interactions and/or having boundary/interfaces. Further investigations of AC schemes will be carried out for problems involving coupled local/nonlocal models and nonlinear problems motivated by important applications. The focus on robust discretization methods like the AC schemes is particularly relevant to reliable and efficient simulations of nonlocal models with application to complex physical systems involving multiscale, singular, and anomalous behaviors. An integrated analytical and computational approach will be used to develop both fundamental ideas and practical insight so that the research findings will not only enrich the mathematical theory of AC schemes but also offer guidance to their practical applications.

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.