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Architecture Aware TensorBased Computing Challenges for the Computer Science and Mathematics Communi

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Title: Architecture Aware TensorBased Computing Challenges for the Computer Science and Mathematics Communi


1
Architecture Aware Tensor-Based
Computing Challenges for the Computer Science and
Mathematics Communities CISE CCF Algorithmic
Foundations Moores Law and Verifiable,
Scalable, Portable, and Reproducible Matrix and
Tensor Software
  • Lenore Mullin
  • Program Director
  • CISE CCF Algorithmic Foundations
  • National Science Foundation
  • lmullin_at_nsf.gov

2
Outline
  • NSF and CISE
  • CCF Algorithmic Foundations and Beyond
  • Challenges and Open Questions
  • Conclusions

3
National Science Foundation
National Science
Office of Inspector General
Board
Administrative Offices
Office of the Director
Directorate for Mathematical Physical Sciences
Directorate for Biological
Sciences
Directorate for Social, Behavioral Economic
Sciences
Directorate for Computer
Information Science Engineering
Office Cyberinfrastructure
Directorate for Education
Human Resources
Office of International Science and Engineering
Directorate for Engineering
Directorate for Geosciences
Office of Polar Programs
4
CISE Goals
  • Enable the United States to remain competitive in
    computing, communications, and information
    science and engineering
  • Promote understanding of the principles and uses
    of advanced computing, communications, and
    information systems in service to society
  • Contribute to universal, transparent, and
    affordable participation in an information-based
    society

5
Achieving CISE Goals
  • CISE supports investigator initiated research in
    all areas of computer and information science and
    engineering
  • CISE helps develop and maintain cutting-edge
    national computing and information infrastructure
    for research and education
  • CISE contributes to the education and training of
    the next generation of computer scientists and
    engineers.

6
CISE Organization http//www.nsf.gov/cise/about/or
g_chart.jsp
Assistant Director Jeannette Wing Deputy Assist
Dir Deborah Crawford
Div Dir Taieb Znati
Div Dir Haym Hirsh
Div Dir Sampath Kannan
7
CCF Computing and Communication Foundations
Division http//www.nsf.gov/div/index.jsp?divCCF
  • Emerging Models and Technologies for Computation
  • Computational biology quantum computing
    nano-scale computing biologically-inspired
    computing
  • Foundations of Computing Processes and Artifacts
  • Advanced computation research compilers
    computer architecture design automation
    (micro/nano) graphics visualization software
    engineering languages
  • Theoretical/Algorithmic Foundations
  • Computer science and communication theory
    numeric symbolic/graphic computation theory of
    computing computational algebra and geometry
    signal processing

8
Theoretical/Algorithmic Foundations Numeric,
Symbolic and Algebraic Computing
  • Investigations into new data structures and
    algorithms that yield optimizations for
    particular applications are encouraged.
  • This includes the design and construction of high
    quality scientific software ideally adept across
    numerous scientific domains. Tensors are
    pervasive throughout NSF disciplines.
  • Specific research topics of interest include, but
    are not limited to, the following numerical
    linear and multi-linear algebras, tensor algebras
    and decompositions used in memory hierarchy
    mappings linear and non-linear optimization
    modeling and simulation of complex processes and
    numerical solutions of differential equations and
    PDEs. Research in numerical computing and
    optimization has natural interdisciplinary
    applications. In fact, this program seeks
    applications in science and engineering whose
    basic problems actually require the development
    of new numerical and optimization methods.

9
Theoretical/ Algorithmic Foundations Numeric,
Symbolic, and Algebraic Computing
  • Research focused on finding powerful methods for
    symbolically solving algebraic - numeric systems
    that combine differential, integral and
    polynomial equations is required. Interests
    include foundational research in algorithms and
    their efficient execution.
  • Basic research topics include computational
    algebra and analysis, computational number theory
    and algebraic geometry, integration of numeric
    and symbolic techniques, symbolic scientific
    applications and software. Fruitful application
    areas for symbolic computation include the
    solution of complex equation sets.
  • Symbolic/Numeric manipulation and Tensors
  • composition of tensor operations(symbolic) and
    numeric instantiation e.g. SAGE, Matlab,
    Mathematica, Maple, Expression Templates, XML,
    compilers, interpreters, …
  • Tensors are n-d arrays

10
CCF Theoretical/Algorithmic Foundations (AF)
  • Cluster supports research in the following areas
  • Models of computation
  • Computational complexity
  • Parallel and distributed computation
  • Random and approximate algorithms
  • Algorithmic algebra, geometry, topology, and
    logic
  • Computational optimization
  • Techniques for representing, coding and
    transmitting information
  • 30M/Year
  • New TF Program Solicitation NSF 08-518
  • Due Date March 12, 2008 - March 19, 2008
  • http//www.nsf.gov/pubs/2008/nsf08518/nsf08518.htm
  • TR Program Officers John Cozzens, Lenore Mullin,
    Richard Biegel,
  • Sirin Tekinay, Robert Grafton, EK Park

11
Theoretical/Algorithmic Foundations and BEYOND!!!
  • How can we create transformational science when
    we cant verify scientific software?
  • How can domain scientists doing computational
    experiments achieve reproducibility
  • Same answer and is that answer correct?
  • Are the resources used the same?
  • Can the software scale to todays and tomorrows
    hardware?
  • Can we produce software that is optimal?

12
Theoretical/Algorithmic Foundations and BEYOND!!!
  • Optimality and Large Data Sets
  • Optimality and Data Locality across
    processor/memory hierarchy
  • Peta-Scale Computing and Beyond scalability and
    portability
  • Algebra of Arrays to build ANY Tensor based
    application
  • Must be a closed algebra without anomalies for
    verification
  • No language today has such an algebra

13
Moores Law Data Density Doubles every 18
Months EXCEPT Notice flattening of slope due to
Compilers
CMOS ICs
General Architecture
109
106
TX-2
Lattice-Gas Architecture
103
QC Roadmap
1
MIPS
ENIAC
Quantum Dots
Conventional Computer Roadmap
10-3
10-6
Differential Analyzer
Year
1850
2000
1900
1950
2050
Liquid NMR
Babbage Engine
14
Proebstings Law Compiler Advances Double
Computing Power Every 18 Years This means that
while hardware computing horsepower increases at
roughly 60/year, compiler optimizations
contribute only 4.
General Architecture
109
CMOS ICs
106
Lattice-Gas Architecture
TX-2
103
QC Roadmap
1
MIPS
ENIAC
Quantum Dots
Conventional Computer Roadmap
10-3
10-6
Differential Analyzer
Year
1850
2000
1900
1950
2050
Liquid NMR
Babbage Engine
15
What is Computational Science and
Engineering?
Computer Science and Engineering
Physical Sciences and Biological
Sciences
X
Mathematics
X The Intersection of Domain Sciences,
Mathematics and Computer Science and Engineering
16
What can we do?
  • Recent Award for mini-symposia at the 2009 SIAM
    Annual meeting (Lenore Cowen, Tufts Uniting
    Discrete Methods, optimizations and CISE
    Community with the Community studying Matrix
    Operations, Tensors, Verifiable Computational
    Experiments and Scalability) in which Computer
    Scientists and students will be funded to attend
    and interact. This was initiated due to numerous
    tensor sessions at the 2008 SIAM Annual meeting.
  • Tensor Decompositions Solving Fundamental
    Problems in Chemistry
  • Tensor Decompositions for Large-Scale Date
    Applications
  • A Novel Higher-order Generalized Singular Value
    Decomposition for Comparative Analysis of DNA
    Microarray Data from Different Organisms
  • Tensor Algebraic Methods and Their Application to
    High-Dimensional Multi-Modal Data
  • TensorFaces Multilinear (Tensor) Decomposition
    of Image Ensembles
  • Multilinear (Tensor) Independent Component
    Analysis
  • Modeling of Epileptic Seizures using Tensor
    Analysis
  • On a Generalization of Sylvester Methods for
    Symmetric Tensor Decomposition

17
What can we do?
  • A mini-symposium at the 2008 SIAM Annual meeting
    (MS3) entitled Architecture-Aware Scientific
    Computing. Organizers and Presenters L. Mullin
    (NSF) and Padma Raghavan (NSF PI).
  • Plans to have an invitation only workshop with
    Frank Olken (IIS) are planned for spring 2009 to
    bring together experts in Knowledge
    Representation, Tensors, Algorithms and other
    related areas in Computer Science. Charles Van
    Loan, Cornell
  • Recent Award for a workshop at the Courant
    Institute to bring together Mathematicians and
    Computer Scientists to discuss scalable
    algorithms for PDEs on parallel, distributed, and
    multi-core algorithms. ODEs and PDEs can be
    represented as matrix and tensor operations.
    Numeric and symbolic environments are growing in
    popularity to combine verification and optimal
    implementations.
  • Career Application 2008 BIO and CCF AF (Orly
    Alter Integrative and Comparative Tensor Algebra
    Models of DNA Microarray Data from Different
    Studies of the Cell Cycle)

18
What can we do?
  • Milestones in Computer Algebra 2008 (Invitation
    only workshop)
  • Systematic Tensor Simplification a Diagramatic
    Approach by A. D. Kennedy and T. Reiter.
  • This workshop illustrated the need for combined
    numeric and symbolic environments to compute and
    symbolically prove correctness of designs.
    Numerous articles from this workshop discussed
    the need to combine environments, which was
    validated by an NSF supported workshop report
    written by E. Kaltofen (one of the organizers)
    November 2007 at NSF in Arlington.
    Symbolic/Numeric proposals entered in the 2008
    NSG solicitation showed a 100 growth over 2007.

19
What can we do?
  • SCAN 2008 expected outcomes
  • Hardware and software support for verification
    tools
  • Theory, algorithms and arithmetic for verified
    numerical computations
  • Supercomputing and reliability
  • Dynamical systems and verified numerical
    computation
  • Global optimization and verified numerical
    computation
  • Programming tools for verified numerical
    computation
  • Computer aided proofs
  • Industrial and scientific applications of
    verified numerical computations
  • CoProD 08 expected outcomes
  • Definition of new directions for combining
    numeric and symbolic approaches in solving
    constraints and optimization problems in
    particular and in decision making in general.

20
What can we do?
  • Matrix and Tensor operations are pervasive in
    science and engineering
  • Tensors are n-d arrays, but n-d arrays are more
    general
  • Generalized multi-dimensional Inner and Outer
    products
  • Summations of multi-dimensional arrays
  • Projection operators
  • AXB like problems
  • Coupled differential and integral equations,
    eigenvalue problems generally translate to
    matrix problems
  • Even non-linear operations iterative solutions
  • Linear and Multilinear Algebra is not enough!
  • Scalars, anomalies
  • Existing languages are not enough!

21
Possible Solutions
  • Identify a closed algebra that subsumes important
    matrix operations
  • Augment existing languages with this algebra
    optional use
  • Solve a few important problems completely
  • Use the same algebra to map to processor memory
    hierarchies
  • Use the same algebra to abstract machines
  • These concepts proposed at Sandia Workshop on
    Memory Hierarchy Optimizations for Scientific
    Software, January 2008

22
Possible Solutions
  • Synergize Mathematicians, Computer Scientists
    and Domain Scientists to collaborate
  • Create a new community that solves these open
    questions
  • Revisit and reinvent
  • Community then creates a research base for
    funding agencies
  • Workshops and Colloquia
  • Supplements and/or new grants

23
Numeric, Symbolic and Algebraic Computing Program
in TF
  • These issues appear in last years and this
    years solicitations
  • MPS and CISE cooperating programs
  • Hope to develop new solicitations
  • Attend SIAM, SC, APS, MRS, etc.
  • Raise the consciousness of computational
    scientists in these communities
  • After solving small number of algorithms within
    this algebra, identify what to do next.
  • Can be used in existing programs.

24
Thank You Questions? Lenore Mullin Program
Director National Science Foundation Computer
Information Science Engineering
Directorate Division of Computer and
Communications Foundations Algorithmic
Foundations Cluster lmullin_at_nsf.gov
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