Grand Challenges in Computational Mathematics:

Numerical, Symbolic and Algebraic Computing

An NSF ViewLenore M. MullinProgram Director

CISE CCFTheoretical 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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CCFComputing and

CommunicationsFoundations

CNSComputer and

NetworkSystems

IISInformation and

IntelligentSystems

Office of theAssistant Director

for CISE

OCIOffice of

Cyberinfra-structure

(formerly SCI, now an NSF-wide mission, reporting to Director of NSF

since 2006)

Office of the Director

Clusters ClustersClusters

Crosscutting CISE Emphasis Areas

• EMT• CPA• TF

• NeTS• CSR• CRI

• HCC• III• RI

CISE Organization

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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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NewPhysics, Biology, Chemistry, Economics, Geosciences, Statistics…

Computational Discovery

Data

Core Concept ExperimentTheory

Visualization, simulation, Computational Science

Interpretation

InsightsDomains of inquiry

ManufacturingProcesses

Statistical learning

DNA Transcription

Current

Computational Computational DiscoveryDiscovery

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Underlying Themes

• Exploring and modeling nature’s 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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Moore’s Law: Data Density Doubles every 18 Months

EXCEPT Notice flattening of slope due to Compilers

1850 1950 20001900 2050

10-6

103

1

10-3

106

109

Babbage Engine

CMOS ICs

TX-2

ENIAC

Differential Analyzer

GeneralArchitecture

Lattice-GasArchitecture

Quantum Dots

Liquid NMR

Conve

ntio

nal C

ompu

ter R

oadm

ap

QC

Roa

dmap

MIPS

Year

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Proebsting’s 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%.

1850 1950 20001900 2050

10-6

103

1

10-3

106

109

Babbage Engine

CMOS ICs

TX-2

ENIAC

Differential Analyzer

GeneralArchitecture

Lattice-GasArchitecture

Quantum Dots

Liquid NMR

Conve

ntio

nal C

ompu

ter R

oadm

ap

QC

Roa

dmap

MIPS

Year

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Why do we need Grand Challenges?

• Moore’s Law slope flattens out• Moore’s 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 Moore’s 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?

Mathematics

Physical Sciences and Biological Sciences

Computer Science and Engineering

X

X = The Intersection of Domain Sciences, Mathematics andComputer 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/networks 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:   10:30 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

(703) 292-8910lmullin@nsf.gov