Using Interval Analysis to Bound Numerical Errors in Scientific Computing

In scientific computing, the approximation of continuous physical phenomena by floating-point numbers gives rise to rounding error. The behaviour of rounding errors is difficult to predict, and most scientific applications ignore it. For applications in which the accuracy of the result is critical, this is not an acceptable choice. Interval analysis is an alternative to conventional floating-point computation that offers guaranteed error bounds. Despite this advantage, interval methods have rarely been applied in high performance scientific computing. In part, this is because of the additional cost associatedwith performing interval operations over the corresponding floating-point operations. Another issue is the lack of example applications of interval analysis; many scientific users simply do not know that an alternative to floating-point computation exists. This thesis develops and demonstrates techniques by which interval analysis can be feasibly applied to scientific computing. Methods are shown by which the performance of interval codes may be significantly improved on the UltraSPARC platform: hand-optimised codes for vector functions are developed that are around 60% faster than the Sun interval implementation, and an alternative method of interval multiplication is developed that is around 30% faster than the method used by other implementations. The benefits of interval analysis are demonstrated through application to an example scientific problem: the evaluation of electrostatic potentials. Alternative methods of evaluation are analysed: two new accurate methods of summation are proposed and proven to reduce rounding error in this application. This demonstration includes what is believed to be the first interval implementation of the fast multipole method. The implementation is used to explore the balance between rounding and truncation errors in the method, which has previously been ignored. The results suggest that rounding error may affect the accuracy that can be practically achieved using the method. The results also provide guidance in choosing parameters to minimise error.

[1]  R. Beatson,et al.  A short course on fast multipole methods , 1997 .

[2]  Siegfried M. Rump,et al.  Accurate Sum and Dot Product , 2005, SIAM J. Sci. Comput..

[3]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .

[4]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[5]  Michael J. Schulte,et al.  The Interval-Enhanced GNU Fortran Compiler , 1999, Reliab. Comput..

[6]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[7]  Yozo Hida,et al.  Accurate Floating Point Summation , 2002 .

[8]  Rajat P. Garg,et al.  Techniques for Optimizing Applications: High Performance Computing , 2001 .

[9]  Intel Corportation,et al.  IA-32 Intel Architecture Software Developers Manual , 2004 .

[10]  George T. Gillies,et al.  The Newtonian gravitational constant: recent measurements and related studies , 1997 .

[11]  S. Gill,et al.  A process for the step-by-step integration of differential equations in an automatic digital computing machine , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  Tiraz R. Birdie,et al.  The Use of Interval Analysis in Hydrologic Systems , 1998, Reliab. Comput..

[13]  Jack M. Wolfe.  Reducing truncation errors by programming , 1964, CACM.

[14]  Walter Krämer,et al.  Automatic Forward Error Analysis for Floating Point Algorithms , 2001, Reliab. Comput..

[15]  Charles Fefferman,et al.  Interval Arithmetic in Quantum Mechanics , 1996 .

[16]  Nobito Yamamoto,et al.  A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem , 2004 .

[17]  Martin Head-Gordon,et al.  Derivation and efficient implementation of the fast multipole method , 1994 .

[18]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[19]  John Michael McNamee,et al.  A comparison of methods for accurate summation , 2004, SIGS.

[20]  Svetoslav Markov,et al.  On the algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic , 2001 .

[21]  U. Kulisch,et al.  Scientific Computing with Automatic Result Verification. , 1994 .

[22]  Richard Goodman,et al.  Convergence Estimates for the Distribution of Trailing Digits , 1976, JACM.

[23]  J. Vignes,et al.  An Efficient Stochastic Method for Round-Off Error Analysis , 1985, Accurate Scientific Computations.

[24]  Martin Berz,et al.  Efficient Control of the Dependency Problem Based on Taylor Model Methods , 1999, Reliab. Comput..

[25]  J. Hass,et al.  The double bubble conjecture , 1995 .

[26]  Bruno Lang,et al.  Newton's constant of gravitation and verified numerical quadrature , 1996, Reliab. Comput..

[27]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[28]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[29]  Mark A. Stadtherr,et al.  ROBUST PHASE STABILITY ANALYSIS USING INTERVAL METHODS , 1998 .

[30]  Charles L. Lawson,et al.  Basic Linear Algebra Subprograms for Fortran Usage , 1979, TOMS.

[31]  Arnold Neumaier,et al.  Taylor Forms—Use and Limits , 2003, Reliab. Comput..

[32]  David L. Weaver,et al.  The SPARC Architecture Manual , 2003 .

[33]  Guido D. Salvucci,et al.  Ieee standard for binary floating-point arithmetic , 1985 .

[34]  David L Weaver,et al.  The SPARC architecture manual : version 9 , 1994 .

[35]  C. Fefferman,et al.  Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential. , 1993 .

[36]  Y. Lin,et al.  Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering , 2004, J. Glob. Optim..

[37]  Jack J. Dongarra,et al.  An extended set of FORTRAN basic linear algebra subprograms , 1988, TOMS.

[38]  Jon G. Rokne,et al.  Complex interval arithmetic , 1971, CACM.

[39]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[40]  Martin Berz,et al.  Controlling the Wrapping Effect in the Solution of ODEs for Asteroids , 2003, Reliab. Comput..

[41]  Michael J. Schulte,et al.  A combined interval and floating point multiplier , 1998, Proceedings of the 8th Great Lakes Symposium on VLSI (Cat. No.98TB100222).

[42]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[43]  Ulrich W. Kulisch,et al.  Proposal for Accurate Floating-Point Vector Arithmetic , 1993 .

[44]  Eldon R. Hansen,et al.  Sharpness in Interval Computations , 1997, Reliab. Comput..

[45]  Arnold Neumaier,et al.  Grand Challenges and Scientific Standards in Interval Analysis , 2002, Reliab. Comput..

[46]  Ole Møller.  Quasi double-precision in floating point addition , 1965 .

[47]  William Kahan,et al.  Pracniques: further remarks on reducing truncation errors , 1965, CACM.

[48]  Michael A. Malcolm,et al.  On accurate floating-point summation , 1971, CACM.