The Virtual Institute for Mathematical and Statistical Sciences (VI-MSS)

Description

ICERM's Virtual Institute of Mathematical and Statistical Sciences (VI-MSS) began with a partnership connecting two US mathematical sciences institutes with several mathematics and statistics institutes in India. VI-MSS has sponsored joint workshops, research visits and graduate educational activities with support from the US National Science Foundation, the Indo-US Science and Technology Forum, and the Indian Department of Science and Technology.

VI-MSS at ICERM presently includes jointly funded international collaborations with institutes and institutions in Brazil, China, Israel, Japan, and South Africa. These collaborations create a thriving "virtual" institute in the mathematical and statistical sciences.

Programs

In United States:
  • Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, RI
  • Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC

In Brazil:

  • Instituto Nacional de Matemática Pura e Aplicada (IMPA)

China/Hong Kong:

  • Hong Kong University of Science and Technology (HKUST)

In India:

  • Chennai Mathematical Institute (CMI),
  • Chennai Indian Institute of Science (IISc),
  • Bangalore Indian Institute of Science Education and Research (IISER),
  • Pune Institute of Mathematical Sciences (IMSc),
  • Chennai Indian Statistical Institute (ISI), Kolkata, Delhi, Bangalore
  • Tata Institute of Fundamental Research (TIFR),
  • Mumbai University of Delhi (DU), Delhi
  • ICERM is also collaborating with ICTS.

In Israel:

In Japan:

In South Africa:

  • University of the Witwaterstrand, Johannesburg (WITS) and the African Institute for Mathematical Sciences (AIMS)

The Hong Kong University of Science and Technology-ICERM VI-MSS Program:
Integral Equation Methods, Fast Algorithms and Their Applications to Fluid Dynamics and Materials Science
(HKUST: Dec. 29 2016- Jan. 12, 2017, ICERM: May 30-June 9, 2017)

Description

This program will focus on integral equation methods, fast algorithms and their applications to fluid dynamics and materials science. Integral equation methods have been used for more than a century to establish existence and uniqueness results for a variety of elliptic, parabolic and hyperbolic partial differential equations (PDEs). From a computational perspective, they have been used most extensively in the elliptic (steady state or time harmonic) case, because of their ability to handle complex geometry, unbounded domains and radiation conditions and because of the availability of fast algorithms to reduce the cost of handling the dense matrices that arise from their discretization. These algorithms include fast multipole methods (FMM), methods based on the Fast Fourier Transform (FFT) or the non-uniform FFT (“NUFFT”), and hierarchical compression-based methods (wavelet and SVD-based schemes, H-matrices, HSS-matrices, etc.). The fundamental issue is that discretization of an elliptic boundary integral equation yields a dense N × N matrix, where N denotes the number of degrees of freedom used to describe the unknown. The straightforward application of a dense matrix to a vector requires O(N2) work, while classical Gaussian elimination techniques require O(N3) work to solve the system. The various fast algorithms listed above provide the ability to apply the discretized integral operator to a vector in O(N) or O(N log N) operations. When combined with modern iterative methods (such as GMRES), well-conditioned integral equation formulations such as second kind integral equations (SKIEs) have reduced the total work required to near optimal complexity, bringing large scale simulations within practical reach.

The program will bring a group of researchers from the US and HKUST with common interests and complementary expertise to work intensively on constructing well-conditioned integral equation formulations, developing high-order, fast, and robust algorithms with scalable implementation, and applying them to solve complex, large-scale real physical applications in multiphase flows and dislocation dynamics and to propose positive conditions on the Hadamard conjecture.

Organizing Committee

From HKUST
  • Xiao-Ping Wang
    (Department of Mathematics, HKUST)
  • Yang Xiang
    (Department of Mathematics, HKUST)
From the US
  • Shidong Jiang
    (New Jersey Institute of Technology)
  • Andreas Kloeckner
    (Department of Computer Science,
    University of Illinois at Urbana-Champaign)

Target Applications and Algorithm Developments


Interfacial phenomena play a vital role in technological applications in such diverse areas as fluids, biology and material science. Numerical simulation of interface dynamics is a difficult task. The MBO scheme is an efficient numerical method for simulating interface dynamics driven by surface energy minimization. The scheme is also generalized to wetting dynamics and droplet spreading on rough solid surfaces (X. Xu, D. Wang and X.P. Wang, 2015).

In this program, we will develop integral equation formulations and design NUFFT-based fast algorithms for these problems. As compared with existing numerical schemes, the proposed method restricts the computation to a small neighborhood of the material interfaces, provides high-order discretization for smooth or piecewise smooth cases, and achieves near optimal complexity. We will also pay particular attention to scalable implementation to take advantage of modern heterogeneous parallel computers so that real physical 3D problems can be simulated and studied within minutes and hours instead of days and weeks.

Possible Projects:

  • Develop integral equation formulations and design NUFFT-based fast algorithms for problems with complex and rough boundaries.
  • Generalize the MBO-type schemes to multiphase fluid flow (i.e. couple with the Navier-Stokes equations).
  • Develop parallel algorithms for 3D interface dynamic problems.

Dislocations are line defects in crystals. The climb motion of dislocations is assisted by diffusion and emission and/or absorption of vacancies or interstitials, and plays crucial roles in the plastic deformation of crystalline materials at high temperature. Numerical simulation of the dynamics of dislocation lines, or discrete dislocation dynamics (DDD), is an important tool for the study of plasticity. However, in early DDD simulations, the formulations of dislocation climb were based only on several special cases (single, straight dislocations) for which analytical solutions are available.

Recently, Prof. Yang Xiang and collaborators derived a Green's function formulation for the climb of curved, multiple dislocations in three-dimensions (Three-dimensional formulation of dislocation climb, Y.J. Gu, Y. Xiang, S.S. Quek, and D.J. Srolovitz, J. Mech. Phys. Solids, 83, 319-337, 2015). This new formulation is able to capture the long-range contribution to the dislocation climb velocity associated with vacancy diffusion, which was missing in the previous DDD simulation models.

In this proposed collaborative research workshop, robust numerical methods will be discussed and implemented to calculate the dislocation climb velocity accurately and efficiently, including boundary integral equation methods, NUFFT method, accurate dislocation discretization method, etc. Parallelization and other high performing methods will also be discussed for dislocation climb simulations in large systems.

Possible Projects:

  • Construct second kind integral equation formulation for the dislocation climb in the equilibrium state of vacancy diffusion.
  • Develop and implementing FMM/NUFFT based fast algorithms for solving the integral equation formulation for the vacancy diffusion equation in dislocation climb.
  • Apply the above algorithm to study large-scale dislocation climb problems.

In 1908, Hadamard conjectured that the Green's function for the clamped plate problem, or mathematically, the first Dirichlet problem of the biharmonic equation on a convex domain is nonnegative. However, after 1949 numerous counterexamples disproved the positivity conjecture of Hadamard. The first result in this direction came by Duffin (1949), who showed that the Green function changes sign on a long rectangle. Garabedian then showed change of sign of the Green function in ellipses with ratio of half axes ≈ 1.6 (1951). Hedenmalm, Jakobsson and Shimorin (2002) mention that sign change occurs already in ellipses with ratio of half axes ≈ 1.2. Sign change is also proven by Coffman-Duffin (1980) in any bounded domain containing a corner, the angle of which is not too large. Their arguments are based on previous results by Osher and Seif (1973) and cover, in particular, squares. This means that neither in arbitrarily smooth uniformly convex nor in rather symmetric domains the Green function needs to be positive. In fact, convexity is neither sufficient nor necessary for a positive Green function. And the question of under what geometry the Green function will be nonnegative remains open.

In this program, we will study this problem numerically, and we will aim for our numerical results to provide new insights and lead to some positive conjectures on the geometry of the domain for which the Green function for the clamped plate problem is nonnegative. Our numerical study is based on a second kind integral equation formulation for the clamped plate problem. As the negative value of the Green function often first appears on the points close to the boundary, one needs high-order quadrature scheme in order to be able to compute the involved nearly singular integrals accurately. Finally, since one needs to scan many different geometries, fast algorithms are also in urgent need. We plan to develop and apply the FMM-accelerated QBX (quadrature by expansion) scheme for the discretization and evaluation of associated integral operators.

Possible Projects:

  • Implement parallel FMM-accelerated QBX scheme with arbitrary precision.
  • Study the Hadamard conjecture in the case of ellipses and identifying the ellipse with smallest ratio of half axes where the Green function becomes negative.
  • Study the Hadamard conjecture in the case of an arbitrary convex smooth curve to provide some conjecture and insight about this case.

Metamaterials are materials whose properties at the scale of a propagating wave with which they are interacting (that is, above the atomic scale, but well below the macroscopic bulk scale) determine key properties of the interaction. Metamaterials often consist of periodic arrays of structures, with purposefully placed ‘defects’ in the periodic lattice to achieve desired properties. For example, photonic crystals typically consist of a unit cell designed such that the bulk materials has one or multiple band gaps, that is, a range of wavelengths at which EM waves will not propagate in the bulk material. Defects then allow the creation of carefully designed propagation pathways within the non-propagating bulk materials. Because of the mismatch in scale between the wavelength of light and the scale of the bulk structure, simulation of metamaterials is a formidable task that could stand to benefit tremendously from better computational tools. Material interfaces in metamaterials are typically sharp, leading to PDE BVPs with piecewise constant coefficients, which makes these problems very amenable to being studied with the help of boundary integral equation methods. High-order accuracy, as provided by the QBX quadrature scheme, is a key ingredient, since the propagation properties of EM waves are sensitive to small perturbations, making simulation at low orders of accuracy unaffordable. The quasi-periodization method of Barnett and Greengard (2010) is another recent advance that has the (thus far unexploited) potential to aid in the study of metamaterials. In this program, we propose to create a computational toolkit for the study of metamaterials. Envisioned contributions extending beyond the creation of the toolkit include enabling the study of materials with bulk-scale defects as well as efficient and accurate numerical methods for the truncation of such calculations in the integral equation setting.

Possible Projects:

  • Design and implement a scheme to couple the QBX quadrature scheme embedded in a high-frequency FMM in three dimensions with a variant of the Barnett/Greengard periodization method.
  • Investigate truncation and acceleration schemes for bulk calculations.
  • Develop distributed-memory parallel algorithms for approaching large-scale metamaterial simulations.

Mathematical and Statistical Aspects of Cryptography
(January 12-14, 2012) (in Kolkata, India)

Description

This workshop focuses on mathematical and statistical aspects of public key cryptography. The main ingredients from mathematics so far include discrete logarithms and factoring over the integers, generalizations of the discrete logarithm to elliptic curves, hyperelliptic curves and further generalizations, aspects of infinite non-abelian groups, and closest vector problems (CVP) in integer lattices. Cryptanalysis in all of these areas can involve analyses of patterns in vast amounts of data, hence the need for statistical methods. One goal of this workshop, though not the only one, is to focus attention on the problem of quantifying the complexity of lattice-based problems, for example extrapolating the difficulty of solving a CVP in an integer lattice as a function of its dimension and other parameters.

A copy of the presentations given at this workshop is available as a PDF document.

Organizing Committee

Winter School and Conference on Computational Aspects of Neural Engineering
(December 10-21, 2012) (in Bangalore, India)

Description

We are pleased to announce the first joint IMI-ICERM Winter School on Computational Aspects of Neural Engineering. The course is directed at graduate students, postdoctoral fellows, and other researchers from the physical sciences (e.g. physics, mathematics, computer science, engineering) and the life sciences (e.g. neuroscience, biology, physiology). The course will offer participants the opportunity to learn about the foundations of neural engineering and brain-computer interfacing, and develop their skills in computational analysis of neural data for the control of external devices. The topics will range from primers on neuroscience, signal processing, and machine learning to brain-computer interfacing based on multi neuronal activity, electrocorticography (ECoG), and electroencephalography (EEG).

The course will consist of 3 hours of lectures each morning, followed by a 3-hour MATLAB-based computer laboratory in the afternoon. Participants will pair up for these laboratories, and an effort will be made to pair someone from the life sciences with someone from the physical sciences. All classes and laboratories will be held on the campus of the Indian Institute of Science (IISc).

This program is part of the IISc Mathematics Initiative (IMI) at the Indian Institute of Science and the VIMSS program at ICERM.

Organizing Committee

Workshop and Conference on Limit Theorems in Probability
(January 2-11, 2013) (in Bangalore, India)

Description

Ever since Jakob Bernoulli proved the law of large numbers for Bernoulli random variables in 1713, the subject of limit theorems has been a driving force for the development of probability theory as a whole. The elucidation of different flavours of laws of large number, central limit theorems and laws of iterated logarithm, their extensions to Markov chains or sums of weakly dependent or stationary processes, limit theorems for Banach space valued random variables, etc., have given rise to a rich theory as well as the basic tools for tackling any problem involving randomness.

Today, 300 years after the landmark result of Bernoulli, it is fruitful to look back at the way in which search for limit theorems has shaped the subject. It is also fruitful to consider how the emphasis has evolved over time from simple limit theorems to getting bounds on the rates of convergence or obtaining inequalities, which are of more immediate relevance in applications to nite samples. The current workshop and conference will focus on some of these topics, and also more broadly on issues of current interest in probability theory.

The workshop (January 2-8, 2013) will consist of five short courses on a variety of topics, aimed at the level of graduate students but also of potential interest to researchers in probability and related fields. After the workshop the conference (January 9-11, 2013) will have lectures on recent developments in various relevant fields of probability.

Organizing Committee

Computational Topology and Data Analysis Workshop
(November 17-21 2014)

Description

The review of Mathematical Sciences research at South African universities commissioned by the National Research Foundation highlighted the isolation of South African mathematics from its applications and related disciplines and not being fully distributed across different areas of mathematics. In particular it noted that there are contemporary mainstream subfields that are not represented and some research is disconnected from areas of contemporary interest. The newly established Centre for Mathematical and Computational Sciences and the African Institute for Mathematical Sciences are collaborating to address some of these gaps by co-organising workshops that will introduce new areas of study to the South African Mathematical Sciences Research landscape.

There is heightened awareness and renewed interest in (Big) Data Analysis since the announcement that South Africa together with Australia would be hosting the Square Kilometre Array project. One of the programmes to be pursued by the Centre for Mathematical and Computational Sciences is the Mathematical and Statistical underpinnings of Big Data.

Computational Topology or Applied Algebraic Topology is a fairly new line of study that combines topological results with efficient computational tools to analyse data and solve problems in many fields, including sensor networks, clustering, robotics, protein biochemistry, computer graphics and image analysis etc. The main objectives of the workshop are to (a) is to introduce the relatively new area of Computational Topology to the attendees and to ‘seed’ this area in the mathematical research landscape in South Africa; (b) give an overview of some of the most important developments and results; (c) discuss some of the contemporary issues, promising directions and open problems and questions. It is hoped that at the end of the workshop researchers in the mathematical sciences and related disciplines will have acquired the basic knowledge prerequisite to undertake research in Topological Data Analysis. The target audience will be researchers from the mathematical, statistical and computational sciences who may want to incorporate into their research aspects or computational topology; postgraduate students who might want to undertake a doctoral project in this area and practitioners from public or private sector.

A typical day will consist of two lectures in the morning and one lecture in the afternoon, each of one hour duration followed by thirty minutes of discussion, brainstorming or hands-one activities. There will be a 90-minute session in the afternoon which will vary from short presentations by young mathematicians; case study presentations by practitioners; panel discussion by experts from academia, private and public sectors.

View the slides from one of the lectures from this event: Nonlinear Dynamics in a Time of High Dimensional Nonlinear Data.

Graduate Student Team-Based Research: Computational Symplectic Topology
(May 17 – May 26, 2015 in Tel-Aviv, and July 27 – August, 5, 2015 at ICERM)

Description

Symplectic and contact geometry and topology, which provide a natural setting for Hamiltonian dynamics, comprise a broad spectrum of interrelated disciplines in the mainstream of modern mathematics. The past two decades gave rise to several exciting developments in these fields: on one hand, powerful new mathematical tools and concepts were introduced, solving long-standing problems that were previously unattainable; and on the other hand, challenging and exciting new questions arose for future research. Presently, symplectic and contact geometry have connections with an amazingly wide range of areas in mathematics and physics: differential and algebraic geometry, complex analysis, dynamical systems, low-dimensional topology, quantum mechanics, and string theory.

The research program will address a number of cutting-edge research topics within symplectic and Hamiltonian dynamics, with a special focus on computational and experimental aspects.

Program Structure

Several projects will be developed by the faculty organizers. Graduate students will be collaborating in teams formed around each project. All graduate students will participate in both sessions of the program: 05/17/2015 - 05/26/2015 in Tel-Aviv, and 07/27/2015 - 08/05/2015 at ICERM. Between the site visits, the teams will continue collaborating remotely via email and video-conferencing.

Selection and financial arrangements

We plan to have 14 graduate students participate in the program (7 from US universities and 7 Israeli students). Review of applications will begin on January 2, 2015 and close when the positions are filled. Accepted US-based graduate students will be reimbursed for travel (to ICERM and to Tel-Aviv) and for local accommodations (shared housing). A meal allowance is included.

Organizing Committee

  • Daniel Alvarez-Gavela
    (Stanford University)
  • Gautam Banhatti
    (University of Muenster)
  • Matthew Strom Borman
    (IAS, Princeton and Stanford University)
  • Robert Castellano
    (Columbia University)
  • Yaniv Ganor
    (Tel-Aviv University)
  • Richard Hind
    (Notre Dame)
  • Victoria Kaminker
    (Tel-Aviv University)
  • Michael Khanevsky
    (University of Chicago)
  • Asaf Kislev
    (Tel-Aviv University)
  • Konstantin Kliakhandler
    (Tel-Aviv University)
  • Jeremy Lane
    (University of Toronto)
  • Andrei Pavlichenko
    (University of Missouri - Columbia)
  • Christopher Policastro
    (University of California, Berkeley)
  • Leonid Polterovich
    (Tel-Aviv University)
  • Itamar Rauch
    (Tel-Aviv University)
  • Lorenzo Rigolli
    (Ruhr-Universitat Bochum)
  • Daniel Rosen
    (Tel Aviv University)
  • Karina Samvelyan
    (Tel-Aviv University)
  • Ood Shabtai
    (Tel-Aviv University)
  • Kyler Bryce Siegel
    (Stanford University)
  • Richard (Bret) Stevenson
    (University of Georgia)
  • Emmanuel Tsukerman
    (University of California, Berkeley)
  • Shira Tanny
    (Tel-Aviv University)
  • Jun Zhang
    (University of Georgia)
Monday July 27, 2015
Time Description Speaker Location Abstracts Slides
9:00 - 10:00Welcome and Breakfast11th Floor Lecture Hall
10:00 - 11:00Progress report: Project 1 (capacities)11th floor lecture hall
11:30 - 12:30Progress Report: Project 2 (egg-beaters)
12:30 - 2:30Lunch
2:30 - 3:30Khanevsky - 111th floor lecture hall
4:00 - 5:00Borman - 111th floor lecture hall

Tuesday July 28, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 11:00Khanevsky - 210th floor classroom
11:30 - 12:30Borman - 210th floor classroom
12:30 - 2:30Lunch
2:30 - 5:00Work in GroupsGroup #1: 11th floor conference room Group #2: 10th floor collaborative space
5:00 - 6:30Welcome Reception11th Floor Collaborative Space

Wednesday July 29, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 5:00Work in GroupsGroup #1: 11th floor conference room Group #2: 10th floor collaborative space

Thursday July 30, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 12:30Work in GroupsGroup #1: 11th floor conference room Group #2: 10th floor collaborative space
12:30 - 2:30Lunch
2:30 - 3:30Progress Report: Project 1 (capacities)10th Floor Classroom
4:00 - 5:00Progress Report: Project 2 (egg-beaters)10th Floor Classroom

Friday July 31, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 5:00Work in GroupsGroup #1: 11th floor conference room Group #2: 10th floor collaborative space

Monday August 3, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 5:00Work in GroupsGroup #1: 11th floor conference room Group #2: 10th floor collaborative space

Tuesday August 4, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 12:00Concluding Report: Project 1 (capacities)11th floor lecture hall
12:00 - 12:05Computational Symplectic Topology Group Picture11th Floor Lecture Hall
12:00 - 2:00Lunch
2:00 - 4:00Concluding Report: Project 2 (egg-beaters)11th floor lecture hall

Wednesday August 5, 2015
Time Description Speaker Location Abstracts Slides
10:00 - 12:00Work in Groups
12:00 - 2:00Lunch
2:00 - 4:30Conclusion

D. Alvarez-Gavela, V. Kaminker, A. Kislev, K. Kliakhandler, A. Pavlichenko, L. Rigolli, D. Rosen, O. Shabtai, B. Stevenson, J. Zhang. Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms. Posted on arXiv.

Brown-ICERM-Kobe Simulation Summer School
(August 17 - 31, 2015 split between Providence, RI, USA and Kobe, Japan)

Description Goals

This program has three objectives. First, it provides graduate students with opportunities to acquire fundamental knowledge and skills in high performance computing, including parallel computing and visualization in 3D caves, and to expose them to the research carried out in these areas at Brown and Kobe Universities. Second, graduate students will learn how to work collaboratively in teams, thus preparing them for the changing nature of research. Finally, the program will provide students with opportunities to develop a global perspective and mindset through participation in a culturally rich and diverse program.

Format

The summer school will take place during 17-31 August 2015 (not counting travel before and after the program). During the first week in Providence, students will attend mini-courses that provide an introduction to numerical algorithms, parallel computing, training on the FX-10 supercomputer in Kobe, and application areas. Simultaneously, student teams, led by advanced graduate students, postdocs, and faculty, will begin to work on their projects. During the second week in Kobe, the student teams will continue to work on their projects, run simulation on Kobe’s FX-10 (which has the same architecture as RIKEN’s K computer), and visualize results and data on Kobe’s 3D visualization system. Teams present their results on the last day to an audience of administrators and research faculty at Kobe University.

Distinctive Features

The program is distinguished by (i) the small number of participants and their teams, which allows for individual instruction, mentoring, and support, (ii) a two-week intensive research summer school which enhances multi-cultural competencies among students, and (iii) the participation of distinctive researchers from Brown, Kobe, and the RIKEN Advanced Institute for Computational Science as guest lecturers.

Prerequisites

This program is open to MSc and first- to second-year graduate students. Prior exposure to scientific computing and programming is useful but not required. Online resources and lectures will be offered during July 2015 prior to the program.

Organizing Committee

Group Projects
Group Project 1: Fracture mechanics of brittle materials using peridynamics
Group Project 2: Self-assembly of micelles using dissipative particle dynamics
Group Project 3: Vortex dynamics of turbulent channel flow using pseudospectral methods
History

The first two Brown-ICERM-Kobe Simulation Schools ran in August 2013 and 2014, each time with 3 research teams, consisting of one team leader and 4-5 team members. In 2014, the projects were

  • Peridynamic Theory of Solid Mechanics
  • Dissipative Particle Dynamics Simulation
  • Direct Numerical Simulation of Turbulent Channel Flow

Schedule
LocationDate
Providence, RI, USA 17-21 August 2015
Travel 22-24 August 2015
Excursion 25 August 2015
Kobe, Japan 26-30 August 2015
Kobe, Japan 31 August 2015 (Final Presentations)