Seven of the 78 scheduled minisymposia were interval oriented, giving a well-balanced overview of recent progress in interval computations, representing major strides forward in every direction, since the early beginnings. The talks were well-attended and well-received.
Martin Berz described a method used by his COSY INFINITY software for the accurate computation of particle trajectories in accelerator rings. His method handles the wrapping effect in modest systems of ODEs by expanding the solution in a multi-dimensional Taylor series in time and in the initial conditions. He does not propagate an enclosing box from one step to the next. Instead, the mapping of an initial box, induced by the differential flow, is followed by multivariate Taylor polynomials with point coefficients plus small interval remainders, effectively eliminating "the wrapping effect".
Pascal Van Hentenryck, Frederic Benhamou, and Jean-Francois Puget described the commercial Numerica package, which uses constraint processing techniques to solve nonlinear systems and global optimization problems, perhaps with constraints. Numerica solves many problems at a time proportional to problem dimension and has solved problems in several hundred dimensions. Jean-Francois described applications solved for Chrysler's Neon and Ford's transmission design. Pascal intends to extend some constraint processing techniques to validated solution of ordinary differential equations.
Rudolf Lohner, Ole Stauning, and Ned Nedialkov reported on software for validated solution of ODEs and strategies for step size control and control of the wrapping effect. The old restriction by constant a priori bounds to Euler steps has been replaced by several variants of a priori polynomial enclosures and validation by Taylor series plus remainder. Ned Nedialkov and Robert Rihm described validated methods suitable for stiff ODEs based on implicit Taylor series methods. Ned showed several examples for which his implicit Hermite-Obreshkov methods are more efficient than forward interval Taylor series methods.
John Pryce and Weldon Lodwick showed how validated methods can also be applied to differential algebraic equations (DAEs). Pryce extracts a certain linear programming problem from the DAE. The solution to the LP tells us the sequence in which to compute the Taylor coefficients for differential and algebraic components. Then, high order Taylor polynomials can be generated and passed to the ODE stepping components of software by Lohner, Stauning, or Nedialkov.
Baker Kearfott presented the SunSoft Globsol package for validated solution of nonlinear systems and global optimization problems, perhaps with constraints. GlobSol has solved simple problems in up to 256 dimensions and more realistic problems in as many as 100 dimensions. Paul Thalacker, Paulina Chin, and George Corliss presented prototypical industrial applications of importance in finance, mechanical and electrical engineering, medicine, and symbolic computations.
Zhe Liu, in a minisymposium on numerical analysis, discussed the C++ implementation of interval arithmetic operations in the IEEE floating-point standard, following the interval exception-handling model proposed by Chiraev and Walster at Sun.
I. Burkhan Turksen discussed uses of interval computation in the theory of fuzzy sets, logic, and measures, and Ulrike Storck discussed adaptive validated quadrature algorithms.
In many ways, it was the applications of validated techniques we found most exciting. In addition to the particle accelerators of Berz and the GlobSol applications already mentioned, Vladik Kreinovich talked about aerospace applications, John Funge about robots and intelligent agents, Jon Rokne about geometric primitives in computer graphics, Luis Seco about quantum mechanics, and Ramon Moore about practical applications from range safety at the Pacific Missile Range in the 1960's to quantum mechanics, atomic physics, vortex sheet motions, and beam physics in the 1990s. He also proposed an application to the calculation of closest approaches of asteroids and comets.
We are already organizing minisymposia and talks for next year's SIAM Annual Meeting in Atlanta, May 12-15, and interval sessions for several applications areas. Please feel welcome to participate and spread the word, or contact George Corliss for suggestions.
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