1 / 65

Checking Computation of Numerical Functions by the Use of Functional Equations

Checking Computation of Numerical Functions by the Use of Functional Equations. REC 2006 NSF Workshop on Reliable Engineering Computing F. Vainstein and C. Jones. Presentation Summary. Background Fault tolerance Computing Numerical Functions Theory Finding checking polynomials

zofia
Download Presentation

Checking Computation of Numerical Functions by the Use of Functional Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Checking Computation of Numerical Functions by the Use of Functional Equations REC 2006 NSF Workshop on Reliable Engineering Computing F. Vainstein and C. Jones

  2. Presentation Summary • Background • Fault tolerance • Computing • Numerical Functions • Theory • Finding checking polynomials • The general method • A program developed by this research • Some examples • Considerations for deployment • Future directions

  3. Fault Tolerance Grace in response to the unexpected • Withstands failures • Exhibits desirable behavior • Does not endanger life (military, transportation, medical) • Preserves scientific investment (space, supercomputing) • Meets consumer expectations

  4. Fault Tolerance Can Be Critical Military: Global Hawk Science: Gravity Probe B Exploration: Mars Opportunity Rover Civilian: Airbus A380

  5. Methods for Fault Tolerance • Modular redundancy • Back up systems in the event primary unit fails • Replication with voting • Duplicate function blocks and compare for “majority” wins • Error-correcting codes • Reed-Solomon, parity checks, … • Algorithm-based fault tolerance (ABFT) - Encodes data and augments algorithm to detect errors

  6. A Complex System: The Space Shuttle Total number of parts > 600,000 Total Weight = 4,500,000 pounds Cost to move one pound of cargo = $20,000 Budget = $3.3 billion / year

  7. Modular Redundancy: Space Shuttle Space shuttle avionics from Redundancy Management Techniques for Space Shuttle Computers, Sklaroff, IBM Research Development, 1976.

  8. Replication With Voting: Space Shuttle

  9. Complex System: The Microprocessor Intel Pentium 4 Prescott Core Number of transistors > 125 million Transistor size = 90nm Pipeline = 31 stages Development Budget = $4.2 billion/year “Never in the history of mankind has it been possible to produce so many wrong answers so quickly.” Carl-Erik Froeberg

  10. What Does a Microprocessor Do? • ALU: Arithmetic logic unit performs • math and logic functions. • Math coprocessors were big business • for Intel and others in the 1980s. • Today, most processors incorporate • a math coprocessor or emulator for • numerical calculations. • Move data from one memory location • to another • Make decisions and jump to new • set of instructions

  11. IBM FPU Core Scientific codes typically spend much of their time in common numerical subroutines - about 70% of a phase retrieval application, for example, is spent in the Fast Fourier Transform alone. M. Turmon, Annual Report for FY 2001 Final Report Algorithm-Based Fault Tolerance, Nasa-JPL, Remote Exploration and Experimentation Project. Image Legend:Dark Blue: Interface, Decode and IssuePink: Pipe Management and Data ForwardingYellow: Arithmetic PipeAqua: Load/Store Pipe

  12. Numerical Functions Numbers from numbers • Degrees to radians • Cosine • Hyperbolic Sine • ArcSine • SINC function • Next positive power of 2 • Linear interpolation • Root finding • Gaussian • Mod • Greatest Common Divisor • Absolute value • Minimum • Maximum • Round to next integer • Return the fractional part of a value • Clip in a saturation fashion • Wrapping for integers • Log • Fast Fourier Transform (FFT) • Numerical Differentiation • Kalman Filtering

  13. Numerical Functions in Action: 1 • IMAGE PROCESSING • The FIDO Mars Exploration Rover (MER) • relies on detailed panoramic views in its • operation for near real-time tasks: • Determination of exact location • Navigation • Science target identification • Mapping • WEATHER MODELING • Roe, K., et al., High Resolution Weather Monitoring • for Improved Fire Management, 2001, Maui HPCC • Real-time analysis of environmental information • for prediction of fire behavior

  14. Numerical Functions in Action: 2 NON-LINEAR CONTROL SYSTEMS Brennan, S., Integrated Chassis Control for Vehicles, 2000 SCIENTIFIC SUPERCOMPUTING U. Landman, et al., Large-scale classical molecular dynamics, 2001, Georgia Tech

  15. Background Summary: • Computing is at the heart of most modern systems • Fault tolerance is a concern – especially for mission and safety critical systems • The computation of numerical functions is a critical area of computing

  16. Notable Work in Numerical Result Checking • M. Blum, R. Rubinfeld • - Self-Testing/Correcting with Application to Numerical • Problems, 1990 • M. Blum, H. Wasserman • - Reflections on the Pentium Division Bug, 1995, • - Software Reliability Via Runtime Result Checking, 1997 • Promoted numerical checking • A motivation for result checking Used functional equations but no general method existed.

  17. An Algebraic Method for Fault Tolerance 1991 – Feodor Vainstein, Georgia Tech Error Detection and Correction in Numerical Computations by Algebraic Methods Developed a general theory for generating functional equations. Showed that many numerical functions have functional equations and that computations of such numerical functions could be verified by checking polynomials – a novel technique based upon algebraic concepts such as the transcendental degree of field extensions.

  18. Contribution of This Work:A Method for Practical Numerical Checking • Developed software method for finding checking • polynomials. • Treated the case of functions that are not polynomially • checkable. • User-friendly program for hardware/software engineering • Design considerations

  19. Polynomial Numerical Checking Example: 1

  20. Polynomial Numerical Checking Example: 2

  21. Polynomial Numerical Checking Example: 3

  22. Polynomial Numerical Checking Example: 4

  23. Algebra*: Fields *S. Lang, Algebra, Addison-Wesley, 1965

  24. Algebra: Algebraically Dependent

  25. Algebra: Transcendental Degree of Field Extension

  26. Algebra: Algebraically Closed and Algebraic Closure

  27. Algebra: Linear Independence

  28. Theory: Polynomially Checkable

  29. Theorem:

  30. Theory: Example and Generality

  31. Theory: Linearly Checkable

  32. Theory: Other Cases We also considered • Functions over various fields • PC and LC functions of several variables • Partially polynomially checkable functions The focus of the present work is on finding a practical method for determining approximate checking polynomials for PC and non-PC functions for real-valued functions of a single variable.

  33. Least Squares Estimation The least squares estimation technique is used to compute estimations of parameters and to fit data. Since some functions are not PC we can generalize to approximate for non-PC functions. • There are other methods but this was chosen to • Add robustness • Develop a practical process • Treat all polynomially checkable functions

  34. Application of Least Squares Estimation: 1 The problem of finding a checking polynomial can be reduced to the following optimization problem. Let

  35. Application of Least Squares Estimation: 2

  36. Application of Least Squares Estimation: 3

  37. Application of Least Squares Estimation: 4

  38. Software Implementation of Least Squares Estimation: 1 Solve the matrix equation:

  39. Software Implementation of Least Squares Estimation: 2 The coefficients of the checking polynomial are then in vector X: Those values can be used to find the value of the delta function: Deviation shows how good is our approximation

  40. The Matlab Function: • Solves the least squares estimation problem • Finds the delta function value for a range of k • Returns the checking polynomial coefficients • for the best (smallest error) delta function • Plots the error over the function domain for • the best delta • Plots deviation for a range of k • Simulink, DSP Builder generates VHDL and • deploys to Altera FPGA (Xilinx similar)

  41. Function Input

  42. Function Output

  43. Example: SINE Function Output

  44. Example: SINE Function Plots The sine function is linearly checkable (LC)

  45. The Logarithm Function: Output

  46. The Logarithm Function: Plots

  47. The Logarithm Function: k = [1…40]

  48. Checking Polynomials: Simple Functions

  49. Checking Polynomials: Compound Functions

  50. Why Matlab Matlab (MATrix LABoratory) • Matrix-oriented programming environment • Code can compile to C/C++ • Built-in routines for data analysis and visualization • GUI/Web publishing support • A popular environment for technical computing http://www.gtrep.gatech.edu/undergradlabs/labman/CheckingPolynomial

More Related