Universal Envelopes and Applications

Abstract: Computable analysis formalizes computation with continuous objects such as real numbers and functions and hence offers itself as a rigorous foundation for numerical analysis and scientific computing. It turns out, however, that a great many computational problems of practical interest fail to admit computable solutions in the sense of computable analysis because their solutions do not depend continuously on the problem instance, and computable functions must be continuous. This generally does not deter practitioners from "computing" solutions to such problems in some (typically not rigorously specified) sense.

If we simultaneously believe that computable analysis offers the "correct" definition of computable function, that the treatment of problems of interest in scientific computing might require discontinuous operations, and that computable analysis can indeed serve as a rigorous foundation of scientific computing, then we are led to the problem of defining what it might mean to "compute" discontinuous functions from the perspective of computable analysis.

I will present "universal envelopes" as an attempt at providing such a definition. To any function f between admissible represented spaces, one can assign a continuous "universal envelope", which intuitively speaking captures the largest amount of "continuously available" information on f. I will discuss how universal envelopes can be used for the stability analysis of numerical programs and present some applications of universal envelopes to systems verification.

logicnumerical analysis

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic