Grand Challenges in Computational Mathematics: Numerical, Symbolic and Algebraic Computing An NSF View

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Grand Challenges in Computational Mathematics
Numerical, Symbolic and Algebraic ComputingAn
NSF View

• Lenore M. Mullin
• Program Director
• CISE CCF
• Theoretical Foundations Cluster
• National Science Foundation

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Outline

• NSF Overview
• CISE and CCF
• Theoretical Foundations
• Numeric, Symbolic, and Algebraic Computing and
  Optimizations
• Grand Challenges in the Theoretical Foundations
  of Computational Mathematics

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National Science Foundation
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CISE Organization
Office of the Director
Office of the Assistant Director for CISE
CCF Computing and Communications Foundations
CNS Computer and Network Systems
IIS Information and Intelligent Systems
OCI Office of Cyberinfra- structure
(formerly SCI, now an NSF-wide mission,
reporting to Director of NSF since 2006)
Clusters
Clusters
Clusters

• NeTS
• CSR
• CRI

• EMT
• CPA
• TF

• HCC
• III
• RI

Crosscutting CISE Emphasis Areas
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Computing andCommunication Foundations Division
(CCF)

• Emerging Models and Technologies for Computation
  (EMT)
• computational algorithms and simulation
  techniques for nanoscale systems design and
  architecture of systems based on molecular scale
  devices quantum algorithms for computation,
  communication, and coding realization of quantum
  computing algorithms and computational modeling
  of biological processes computing models and
  systems for future technologies.
• Computing Processes and Artifacts (CPA)
• software design methodologies tools for software
  testing, analysis, and verification semantics,
  design, and implementation of programming
  languages micro-architectures memory and I/O
  subsystems application-specific architectures
  performance metrics VLSI electronic design
  analysis, synthesis and simulation algorithms
  system-on-a-chip architecture and design for
  mixed or future media (e.g., nanotechnology).
• Theoretical Foundations (TF)
• models of computation computational complexity
  parallel and distributed computation random and
  approximate algorithms algorithmic algebra,
  geometry, topology, and logic computational
  optimization computational algorithms for
  high-end scientific and engineering applications
  techniques for representing, coding and
  transmitting information mobile communication
  optical communication signal processing systems
  analysis of images, video, and multimedia
  information.

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Computational Discovery
New
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Underlying Themes

• Exploring and modeling natures interactions,
  connections, complex relations, and
  interdependencies, scaling from sub-particles to
  galactic, from cellular to societal, in microns
  to light years, in order to understand them,
  mimic them, synthesize them, and exploit them
  (examples include science of design, theory of
  networked computing, plant genomics, control
  systems, management sciences, prediction, risk
  assessment, decision making, distributed data
  driven application systems, sustainability
  engineering, social, behavioral sciences,
  economics, politics)
• Coupling of the physical world with the cyber
  world, integrating natural sciences with social,
  and computing sciences and engineering (examples
  include logistical systems, supply chains, power
  networks, all sensor related applications, signal
  processing, quantum computing, molecular
  computing, bioinformatics, communications
  systems, cognitive sciences, learning, artificial
  intelligence, biomedical engineering
  applications, human computer interface, virtual
  or smart environments, health systems,
  interactive games)

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Moores Law Data Density Doubles every 18
MonthsEXCEPT Notice flattening of slope due to
Compilers
CMOS ICs
General Architecture
109
106
TX-2
Lattice-Gas Architecture
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QC Roadmap
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MIPS
ENIAC
Quantum Dots
10-3
Conventional Computer Roadmap
10-6
Differential Analyzer
Year
1850
2000
1900
1950
2050
Liquid NMR
Babbage Engine
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Proebstings LawCompiler Advances Double
Computing Power Every 18 YearsThis 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
10-3
Conventional Computer Roadmap
10-6
Differential Analyzer
Year
1850
2000
1900
1950
2050
Liquid NMR
Babbage Engine
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Why do we need Grand Challenges?

• Moores Law slope flattens out
• Moores Law slope eventually declines
• Software can not keep up with hardware advances
• How can we put a stop to these declines?
• How can we verify correctness of
• Semantics
• Performance
• Time, Space, Power, Heat, etc.

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Grand Challenge Motivating Questions

• What have we learned (to date) about
  Computational Mathematics?
• Are programming languages closed under an
  algebra?
• For numerical computing
• For symbolic computing
• For algebraic computing
• For optimizations in all the above
• Can we verify programs?
• Semantically?
• Operationally?

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Grand Challenge Motivating Questions

• Are there data structures with deterministic
  characteristics?
• For Layout and storage
• That are pervasive across scientific disciplines
• DSP
• Computational Quantum Mechanics
• That are Closed under one algebra
• Can we describe decomposition and mappings of
  such data structures to processor/memory
  hierarchies using the same algebra?
• For Block, cyclic, block-cyclic, etc
  decompositions
• Over Cache, Main, Shared, Distributed, Grid, etc.
  memories

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Grand Challenge Motivating Questions

• Can we abstract computing architectures using the
  same algebra?
• For RASCs?
• Quantum Computers?
• Combined RASC/Quantum/ Computers
• For FPGA and ASICS?
• Can we create tools that can theoretically
  predict performance attributes prior to
  execution?
• That Interface to compilers or translators?
• That are Domain specific?
• Experimental Methods?
• Can we create Reproducible computational
  experiments?
• In time, space, power, etc.
• Provide Numerical stability when there are
  enormous numbers of processors and communications
  networks working on one problem?

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Grand Challenge Motivating Questions

• Can we build software to keep up with Moores
  Law?

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Where is the Research Needed?

• What disciplines?
• How do they work together?
• What theories? New?
• What curriculums?
• BS, MS, PhD
• Within existing university department structures?
• K-12?

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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
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Theoretical Grand Challengesfor Computational
Mathematics Numerical, Symbolic, and Algebraic
Computing

• The Theory of Computing
• Mathematical Models of Computation
• Is the Turing Model sufficient for complex
  parallel and distributed multilevel-memory
  architectures and grids?
• Is the Turing Model sufficient for Quantum
  Computers?
• What are the data structures, algorithms, and
  algebras pervasive in science worthy of domain
  specific languages, tools, and architectures/netwo
  rks such that a deterministic analysis is
  possible?
• Could we then theorize about performance?
  Predictable reproducible performance? On any
  machine/network? Verify semantics as well as
  operational costs?

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NSF and the Research Community

• Need the Research community to address questions
  posed
• Need the Research community to cross disciplinary
  lines
• Need the Academic community to cross disciplinary
  lines
• Develop Academic and Research Programs to address
  initiatives

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NSF and the International Community

• OISE
• Small research initiation with funding
  organizations in other countries
• Promote collaborations, teams
• Example This week at NSF
• Title  How to Cooperate with European
  Commission Research Programs
• What are the European Union research
  programs? 
• What is Framework Programme VII
  (FP7)? 
• What is the new European Research
  Council (ERC)?
• Come and find out at panel discussion featuring
          Lou Brown, GEO         Carmen Huber,
  DMR/ MPS         Jeanne Hudson, OISE/O/D
          Suzi Iacono, CNS/CISE
• Where  Room 375
• When  Monday, April 23
• Time   1030 a.m.

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NSF and the International Community

• Add-ons to individual reseach grants
• Student/faculty exchanges
• Conferences and Workshops
• Jointly with EC, e.g. initial workshop in Europe.
• Fund researchers from US to Europe
• Foster connections with researchers in European
  Research Agencies
• EC.

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Contact Information

• Lenore M. Mullin
• CISE/CCF
• Theoretical Foundations
• [phone]
• lmullin_at_nsf.gov