Projects III: 2024-25 (DW)

Project III (MATH3382) 2024-25

Interval Analysis

Djoko Wirosoetisno

Description

Rigorous proofs and floating-point computation may seem to be diametrically opposite, but recently some difficult mathematical problems have been solved using floating-point computation including Smale's 14th Problem [Tucker, 2002] and the Kepler Conjecture. Here, the trick is to consider an interval instead of a number at every step: by extending arithmetic operations to intervals (carefully taking care of truncation error, roundoffs, etc), one obtains mathematically correct results instead of mere numerical approximations. In this project, we look at the mathematical basis, practical implementations as well as applications of interval arithmetic.

Prerequisites and Corequisites

Numerical Analysis II. Most importantly you should be passionate about numerical analysis and programming. Dynamical Systems III and PDEs III may also be useful.

Resources

Bornemann, F., Laurie, D., Wagon S., Waldvogel, J. (2004) The SIAM 100-Digit Challenge, SIAM (sections 4.3, 4.5, 4.6, 6.5.1, 7.4.2).
Hansen, E. (1980) Global optimization using interval analysis - the multi-dimensional case, Numerische Mathematik, 34, 247-270.
Kearfott, R.B. (1996) Rigorous Global Search: Continuous Problems, Springer.
Moore, R.E., Kearfott, R.B., Cloud, M.J. (2009) Introduction to Interval Analysis, SIAM.
Neumaier, A. (1990) Interval Methods for Systems of Equations, Cambridge.
Tucker, W. (2002) A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2, 53-117.
Other books and journal articles as appropriate.

email: Djoko Wirosoetisno