Bill Walster

1
Computing with Intervals Recent
Developments

• Bill Walster
• June 15, 2006

2
Overview

• Floating-point arithmetic flaws
• NNSA's Predictive Science
• Intervals solve critical computing problems
• Intervals are here and are here to stay
• Sun leads commercial interval development
• There is still lots to be done
• Computing with intervals is big science

3
Floating-point Problem

• Computational science is stuck
• Floating-point numbers contain no accuracy
  information
• Impossible to represent and track input data
  inaccuracies
• Impossible to perform rigorous error and
  sensitivity analyses
• Floating-point numbers are a finite set of real
  numbers
• Impossible to numerically safely solve
  important problems
• Single and multi-objective (linear and nonlinear)
  function optimization
• Nonlinear systems of equations
• Over- and under-determined systems of linear and
  nonlinear interval equations
• Ordinary and partial differential equations
• Speed kills
• The faster machines get, the more chance for
  silent catastrophic errors

4
National Nuclear Security Administration (NNSA)
Predictive Science

• Definition
• ... the application of verified and validated
  computational simulations to predict reactions
  within complex systems where routine experimental
  tests are not feasible.

5
Some Problems Solved with Intervals

• Proved stability of Tevatron The world's
  largest and most powerful particle
  accelerator html
• Avionics control system design html
  ,html, html
• Solid modeling html and graphics rendering html,
  html
• Numerical proofs html , html
  , html
• Robot control html, html , motion
  planning pdf
• Measuring physical constants html, html

6
Large, High-Value Opportunities

• Enhance software productivity and reliability
• Algorithm and code error detection
• High abstraction level for application developers
• Code transparency
• Safely reduce product liability in
  mission-critical applications
• Input data uncertainty propagation
• Sensitivity analysis
• Increase speed
• Interval bounds minimize unnecessary computing
• Important generic interval algorithms scale
  horizontally
• Enhance IC design optimization

7
Interval Computing Suppliers

• Sun Microsystems html, html
• Intel pdf
• Maple html
• Mathematica html
• MuPad html
• MatLab html

8
Sun Leads Interval Computing

• World class consultants
• Ray Moore, Eldon Hansen
• Foundation for exception-free hardware
• World class interval compiler support
• World class interval math library

9
Sun Labs Interval Aplications

• Second order partial differential equations
  (PDEs)
• Elliptic Laplace's equation
• Example Robot motion planning pdf
• Parabolic Heat diffusion
• Hyperbolic Nonlinear equations leading to shocks
• Second kind integral equations
• Electrostatics and Electrodynamics
• Antenna design
• Radar cross section
• Integrated circuit design optimization

10
Plans

• Develop solutions to new problems in key areas
• Partial differential and integral equations (PDEs
  and IEs)
• Design optimization examples using PDEs and IEs
• Airfoil shape
• Radar cross section
• Antenna
• Demonstrate interval benefits relative to
  alternatives
• Integrate hardware and software support into
  existing processors and compilers

11
Interval Innovation Opportunities

• Data-type compiler support alone is insufficient
• Additional integrated compiler support
• Using dependence to remove unnecessary width
• Integrating symbolic and numerical mathematics
• Interval computing
• Speed and ease-of-use
• Interval hardware, tools, solvers, and commercial
  applications
• Successes with demonstrated benefits
• Instructional textbooks and software

12
Big Science Interval Opportunities

• Applications, Applications, Applications
• Given applications, interval speed
• Solver library algorithms
• Compiler support for speed and narrow width
• Hardware for basic interval arithmetic operations

13
Computing with Intervals Recent
Developments

• bill.walster_at_sun.com