Experimental Mathematics and High-Performance Computing
David Bailey - Lawrence Berkeley Labs (LBL)
Recent developments in "experimental mathematics" have underscored the value of high-performance computing in modern mathematical research. The most frequent computations that arise here are high-precision (typically several-hundred-digit accuracy) evaluations of integrals and series, together with integer relation detections using the "PSLQ" algorithm. Some recent highlights in this arena include: (2) the discovery of "BBP'-type formulas for various mathematical constants, including pi and log(2); (3) the discovery of analytic evaluations for several classes of multivariate zeta sums; (4) the discovery of Apery-like formulas for the Riemann zeta function at integer arguments; and (5) the discovery of analytic evaluations and linear relations among certain classes of definite integrals that arise in mathematical physics. The talk will include a live demo of the "experimental mathematician's toolkit".
Loosely Dependent Parallel Processes
Robert Bradshaw - University of Washington
Many parallel computational algorithms involve dividing the problem into several smaller tasks and running each task in isolation in parallel. Often these tasks are the same procedure over a set of varying parameters. Inter-process communication might not be needed, but the results of one task may influence what subsequent tasks need to be performed. I will discuss the concept of job generators, or custom-written tasks that generate other tasks and process their feedback. I would discuss this specifically in the context of integer factorization.
Parallel Computation Tools for Research: A Wishlist
Henry Cohn - Microsoft Research
Disk-Based Parallel Computing: A New Paradigm
Gene Cooperman - Northeastern University
One observes that 100 local commodity disks of an array have approximately the same streaming bandwidth as a single RAM subsystem. Hence, it is proposed to treat a cluster as if it were a single computer with tens of terabytes of data, and with RAM serving as cache for disk. This makes feasible the solution of truly large problems that are currently space-limited. We also briefly summarize other recent activities of our working group: lessons from supporting ParGAP and ParGCL; progress toward showing that 20 moves suffice to solve Rubik's cube; lessons about marshalling from support of ParGeant4 (parallelization of a million-line program at CERN); and experiences at the SCIEnce workshop (symbolic-computing.org), part of a 5-year, 3.2 million euro, European Union project. Our new distributed checkpointing package now provides a distributed analog of a SAVE-WORKSPACE command, for use in component-based symbolic software, such as SAGE.
Interactive Parallel Supercomputing: Today: MATLAB(r) and Python coming Cutting Edge: Symbolic Parallelism with Mathematica(r) and MAPLE(r)
Alan Edelman - MIT
Star-P is a unique technology offered by Interactive Supercomputing after
nurturing at MIT. Star-P through its abstractions is solving the ease of use
problem that has plagued supercomputing. Some of the innovative features of
Star-P are the ability to program in MATLAB, hook in task parallel codes
written using a processor free abstraction, hook in existing parallel codes,
and obtain the performance that represents the HPC promise. All this is
through a client/server interface. Other clients such as Python or R could
be possible. The MATLAB, Python, or R becomes the "browser." Parallel
computing remains challenging, compared to serial coding but it is now that
much easier compared to solutions such as MPI. Users of MPI can plug in
their previously written codes and libraries and continue forward in Star-P.
Numerical computing is challenging enough in a parallel environment,
symbolic computing will require even more research and more challenging
problems to be solved. In this talk we will demonstrate the possibilities
and the pitfalls.
Interactive Parallel Computing using Python and IPython
Brian Granger - Tech X Corp.
Interactive computing environments, such as Matlab, IDL and
Mathematica are popular among researchers because their
interactive nature is well matched to the exploratory nature of
research. However, these systems have one critical weakness:
they are not designed to take advantage of parallel computing
hardware such as multi-core CPUs, clusters and supercomputers.
Thus, researchers usually turn to non-interactive compiled
languages, such as C/C++/Fortran when parallelism is needed.
In this talk I will describe recent work on the IPython project
to implement a software architecture that allows parallel
applications to be developed, debugged, tested, executed and
monitored in a fully interactive manner using the Python
programming language. This system is fully functional and allows
many types of parallelism to be expressed, including message
passing (using MPI), task farming, shared memory, and custom user
defined approaches. I will describe the architecture, provide an
overview of its basic usage and then provide more sophisticated
examples of how it can be used in the development of new parallel
algorithms. Because IPython is one of the components of the SAGE
system, I will also discuss how IPython's parallel computing
capabilities can be used in that context.
Science at the petascale: tools in the tool box
Robert Harrison - Oak Ridge National Lab
Petascale computing will require coordinating the actions of 100,000+
processors, and directing the flow of data between up to six levels
of memory hierarchy and along channels that differ by over a factor of
100 in bandwidth. Amdahl's law requires that petascale applications
have less than 0.001% sequential or replicated work in order to
be at least 50% efficient. These are profound challenges for all but
the most regular or embarrassingly parallel applications, yet we also
demand that not just bigger and better, but fundamentally new science.
In this presentation I will discuss how we are attempting to confront
simultaneously the complexities of petascale computation while
increasing our scientific productivity. I hope that I can convince you
that our development of MADNESS (multiresolution adaptive numerical
scientific simulation) is not as crazy as it sounds.
This work is funded by the U.S. Department of Energy, the division of
Basic Energy Science, Office of Science, and was performed in part
using resources of the National Center for Computational Sciences, both
under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory.
Parallel Computation in Number Theory
Bill Hart - Warwick
This talk will have two sections. The first will
introduce a new library for number theory which is
under development, called FLINT. I will discuss the
various algorithms already available in FLINT, compare
them with similar implementations available elsewhere,
and speak about what the future holds for FLINT, with
the focus on parallel processing and integration into
Pari and the SAGE package.
The second part of the talk will focus on low level
implementation details of parallel algorithms in
number theory. In particular I will discuss the design
decisions that we have made so far in the FLINT
library to facilitate multicore and multiprocessor
If time permits, there will be a live demonstration.
Moving Lapack and ScaLapack to Higher Precision without Too Much Work
Yozo Hida - UC Berkeley
I will be discussing recent developments in Lapack and ScaLapack
libraries, along with some recent work on incorporating higher
precision into Lapack and ScaLapack.
Game Theoretical Solutions for Data Replication in Distributed Computing Systems
Samee Khan - University of Texas, Arlington
Data replication is an essential technique employed to reduce the user
perceived access time in distributed computing systems. One can find numerous
algorithms that address the data replication problem (DRP) each contributing in
its own way. These range from the traditional mathematical optimization
techniques, such as, linear programming, dynamic programming, etc. to the
biologically inspired meta-heuristics. We aim to introduce game theory as a new
oracle to tackle the data replication problem. The beauty of the game theory
lies in its flexibility and distributed architecture, which is well-suited to
address the DRP. We will specifically use action theory (a special branch of
game theory) to identify techniques that will effectively and efficiently solve
the DRP. Game theory and its necessary properties are briefly introduced,
followed by a through and detailed mapping of the possible game theoretical
techniques and DRP. As an example, we derive a game theoretical algorithm for
the DRP, and propose several extensions of it. An elaborate experimental setup
is also detailed, where the derived algorithm is comprehensively evaluated
against three conventional techniques, branch and bound, greedy and genetic
Combinatorial Designs: constructions, algorithms and new results
Ilias Kotsireas - Laurier University, Canada
We plan to describe recent progress in the search for combinatorial designs of
high order. This progress has been achieved via some algorithmic concepts, such
as the periodic autocorrelation function, the discrete Fourier transform and
the power spectral density criterion, in conjunction with heuristic
observations on plausible patterns for the locations of zero elements. The
discovery of such patterns is done using meta-programming and automatic code
generation (and perhaps very soon data mining algorithms) and reveals the
remarkable phenomenon of crystalization, which does not yet possess a
satisfactory explanation. The resulting algorithms are amenable to parallelism
and we have implemented them on supercomputers, typically as implicit parallel
Parallel computation of Grobner bases in the Weyl algebra
Anton Leykin - IMA (Minessota)
The usual machinery of Grobner bases can be applied to non-commutative algebras
of the so-called solvable type. One of them, the Weyl algebra, plays the
central role in the computations with $D$-modules. The practical complexity of
the Grobner bases computation in the Weyl algebra is much higher than in the
(commutative) polynomial rings, therefore, calling naturally for parallel
computation. We have developed an algorithm to perform such computation
employing the master-slave paradigm. Our implementation, which has been carried
out in C++ using MPI, draws ideas from both Buchberger algorithm and
Faugere's $F_4$. It exhibits better speedups for the Weyl algebra in
comparison to polynomial problems of the similar size.
MPMPLAPACK: The Massively Parallel Multi-Precision Linear Algebra Package
Jason Martin - James Madison University
For several decades, researchers in the applied fields have had access
to powerful linear algebra packages designed to run on massively
parallel systems. Libraries such as ScaLAPACK and PLAPACK provide a
rich set of functions (usually based on BLAS) for performing linear
algebra over single or double precision real or complex data.
However, such libraries are of limited use to researchers in discrete
mathematics who often need to compute with multi-precision data types.
This talk will cover a massively parallel multi-precision linear
algebra package that I am attempting to write. The goal of this C/MPI
library is to provide drop-in parallel functionality to existing
number theory and algebraic geometry programs (such as Pari, Sage, and
Macaulay2) while preserving enough flexibility to eventually become a
full multi-precision version of PLAPACK. I will describe some
architectural assumptions, design descisions, and benchmarks made so
far and actively solicit input from the audience (I'll buy coffee for
the person who suggests the best alternative to the current name).
Component-level Parallelization of Triangular Decompositions
Marc Moreno Maza - Western Ontario
We discuss the parallelization of algorithms for solving polynomial systems symbolically by way of triangular decompositions. We introduce a component-level parallelism for which the number of processors in use depends on the geometry of the solution set of the input system. Our long term goal is to achieve an efficient multi-level parallelism: coarse grained (component) level for tasks computing geometric objects in the solution sets, and medium/fine grained level for polynomial arithmetic such as GCD/resultant computation within each task.
Component-level parallelism belongs to the class of dynamic irregular parallel applications, which leads us to address the following questions: How to discover and use geometrical information, at an early stage of the solving process, that would be favorable to component-level parallel execution and load balancing? How to use this level of parallel execution to effectively eliminate unnecessary computations? What implementation mechanisms are feasible?
We report on the effectiveness of the approaches that we have applied, including "modular methods", "solving by decreasing order of dimension", "task cost estimation for guided scheduling". We have realized a preliminary implementation on a SMP using multiprocessed parallelism in Aldor and shared memory segments for data communication. Our experimentation shows promising speedups for some well-know problems. We expect that this speedup would add a multiple factor to the speedup of medium/fine grained level parallelization as parallel GCD/resultant computations.
Structure and Representations of Real Reductive Lie Groups: A Computational Approach
Alfred Noel - UMass Boston / MIT
I work with David Vogan (MIT) on the Atlas of Lie Groups and Representations. This is a project to make available information about representations of semi-simple Lie groups over real and p-adic fields. Of particular importance is the problem of the unitary dual: classifying all of the irreducible unitary representations of a given Lie group.
I will present some of the main ideas behind the current and very preliminary version of the software. I will provide some examples also. Currently, we are developing sequential algorithms that are implemented in C++. However, because of time and space complexity we are slowly moving in the direction of parallel computation. For example, David Vogan is experimenting with multi-threads in the K-L polynomials computation module.
This talk is in memory of Fokko du Cloux, the French mathematician who, until a few months ago, was the lead developer. He died this past November.
Parallelism perspectives for the LinBox library
Clement Pernet - University of Waterloo
LinBox is a generic library for efficient linear algebra with blackbox
or dense matrices over a finite field or Z. We first present a few
notions of the sequential implementations of selected problems, such
as the system resolution or multiple triangular system resolution, or
the chinese remaindering algorithm. Then we expose perspectives for
incorporating parallelism in LinBox, including multi-prime lifting for
system resolution over Q, or parallel chinese remaindering. This last
problem raises the difficult problem of combining early termination
and work-stealing techniques.
Distributed Computing using SAGE
Yi Qiang - University of Washington
Distributed SAGE (DSAGE) is a distributed computing framework for
SAGE which allows users to easily parallelize computations and
interact with them in a fluid and natural way. This talk will be
focused on the design and implementation of the distributed computing
framework in SAGE. I will describe the application of the
distributed computing framework to several problems, including the
problem of integer factorization and distributed ray tracing.
Demonstrations of using Distributed SAGE to tackle both problems will
be given plus information on how to parallelize your own problems. I
will also talk about design issues and considerations that have been
resolved or are yet unresolved in implementing Distributed SAGE.
Processor oblivious parallel algorithms with provable performances: applications
Jean-Louis Roch - ID-IMAG (France)
Based on a work-stealing schedule, the on-line coupling of two algorithms
(one sequential; the other one recursive parallel and fine grain) enables
the design of programs that scale with provable performances on various
parallel architectures, from multi-core machines to heterogeneous grids,
including processors with changing speeds. After presenting a generic scheme
and framework, on top of the middleware KAAPI/Athapascan that efficiently
supports work-stealing, we present practical applications such as: prefix
computation, real time 3D-reconstruction, Chinese remainder modular lifting
with early termination, data compression.
Combinatorial designs and code synchronization
Vladimir Tonchev - Michigan Tech
Difference systems of sets are combinatorial designs that arise in connection
with code synchronization. Algebraic constructions based on cyclic difference
sets and finite geometry and algorithms for finding optimal difference systems
of sets are discussed.
Parallel Homotopy Algorithms to Solve Polynomial Systems
Jan Verschelde - UIC
A homotopy is a family of polynomial systems which defines a deformation
from a system with known solutions to a system whose solutions are needed.
Via dynamic load balancing we may distribute the solution paths so that a
close to optimal speed up is achieved. Polynomial systems -- such as the
9-point problem in mechanical design leading to 286,720 paths -- whose
solving required real supercomputers twenty years ago can now be handled
by modest personal cluster computers, and soon by multicore multiprocessor
workstations. Larger polynomial systems however may lead to more
numerical difficulties which may skew the timing results, so that
attention must be given to "quality up" as well. Modern homotopy methods
consist of sequences of different families of polynomial systems so that
not only the solution paths but also parametric polynomial systems must be
Parallel sparsening and simplification of systems of equations
Thomas Wolf and Winfried Neun
In a Groebner Basis computation the guiding principle for pairing and
`reducing' equations is a total ordering of monomials or of derivatives for
differential Groebner Bases. If reduction based on an ordering is replaced by
reduction to minimize the number of terms of an equation through another
equation then on the downside the resulting (shorter) system does depend on the
order of pairing of equations for shortening but on the upside there are number
of advantages that makes this procedure a perfect addition/companion to the
Groebner Basis computation. Such features are:
- In contrast to Groebner Basis computations, this algorithm is safe in the sense that it does not need any significant amount of memory, even not temporarily.
- It is self-enforcing, i.e. the shorter equations become, the more useful for shortening other equations they potentially get.
- Equations in a sparse system are less coupled and a cost effective elimination strategy (ordering) is much easier to spot (for humans and computers) than for a dense system.
- Statistical tests show that the probability of random polynomials to factorize increases drastically the fewer terms a polynomial has.
- By experience the shortening of partial differential equations increases their chance to become ordinary differential equations which are usually easier to solve explicitly.
- The likelihood of shortenings to be possible is especially high for large overdetermined systems. This is because the number of pairings goes quadratically with the number of equations but for overdetermined systems, more equations does not automatically mean more unknowns to occur which potentially obstruct shortening by introducing terms that can not cancel.
- The algorithm offers a fine grain parallelization in the computation to shorten one equation with another one and a coarse grain parallelization in that any pair of two equations of a larger system can be processed in parallel. In the talk we will present the algorithm, show examples supporting the above statements and give a short demo.
Programming Models for Parallel Computing
Kathy Yelick - UC Berkeley
The introduction of multicore processors into mainstream computing is
creating a revolution in software development. While Moore's
Law continues to hold, most of the increases in transistor density will be
used for explicit, software-visible parallelism, rather than increasing
clock rate. The major open question is how these machines will be
In this talk I will give an overview of some of the hardware trends, and
describe programming techniques using Partitioned Global Address Space
languages. PGAS languages have emerged as a viable alternative to message
passing programming models for large-scale parallel machines and clusters.
They also offer an alternative to shared memory programming models (such as
threads and OpenMP) and the possibility of a single programming model that
will work well across a wide range of shared and distributed memory
PGAS languages provide a shared memory abstraction with support for locality
through the user of distributed data types. The three most mature PGAS
languages (UPC, CAF and Titanium) offer a statically partitioned global
address space with a static SPMD control model, while languages emerging
from the DARPA HPCS program are more dynamic. I will describe these
languages as well as our experience using them in both numeric and