Applications of Interval Analysis to Real Analysis and the Analysis of PDEs

From CS Colloquium

Monday, 16 June, 1PM, COMP 221 Roumen Anguelov [1], University of Pretoria

Applications of Interval Analysis to Real Analysis and the Analysis of PDEs

Host: Vladik Kreinovich [2]

The talk gives an introduction to the spaces of interval valued functions, which have emerged in connection with applications to real analysis, approximation theory and partial diferential equations. These spaces are all based on extending the concept of continuity of real functions to interval functions. Central to this development is the concept of Hausdorf continuity. The Hausdorf continuous functions are not unlike the usual real valued continuous functions. For instance, they assume real values on a dense subset of the domain and are completely determined by the values on this subset. However, these functions may also assume interval values on a certain subset of the domain. Hence the concept of Hausdorf continuity is formulated within the realm of interval functions. It turns out that the order, the topology and the algebraic operations on the set C(X) of continuous functions can be extended in a natural way to the set of Hausdorf continuous functions. The application of Hausdorf continuous functions to Analysis and to Nonlinear PDEs, are based on the quite extraor- dinary fact that the the order, topological and algebraic structures on the set of Hausdorf continuous functions are all complete. We discuss the following applications:

² Dedekind order completion of C(X).
² Rational extensions of C(X) and their metric completion
² Hausdorf Continuous Solutions of Nonlinear PDEs through the Order Completion Method
² Hausdorf continuous viscosity solutions of Hamilton-Jacobi equations