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LANS Informal Seminar: Fred J. Hickernell
July 25, 2012 @ 15:00 CDT
Seminar Title: Automatic Numerical Algorithms with Performance Guarantees
Speaker: Fred J. Hickernell , Professor and Chair Department of Applied Mathematics, Illinois Institute of Technology
Date/Time: 2012-07-25 15:00
Location: Building 240, 1404-1405
Description:
Extreme scale computing provides very cheap computations, which allow us to simulate very complex systems. These simulations depend on building blocks that compute approximate answers to a specified tolerance. Given an error tolerance, eps, the algorithm should automatically determine the number and type of numerical data needed to provide the approximation in a reasonable amount of time. While automatic numerical algorithms exist, e.g., quad in MATLAB, we do not understand well under what conditions the can be fooled. This talk addresses this dilemma, in particular for integration and function recovery.
The error analysis justifying a numerical algorithm typically assumes input functions to lie a ball in a Banach space of some radius, r. To determine the number of function values needed to guarantee that the error is small enough, one must know r, which means that one needs to know the semi-norm of the input function, e.g., || f’ ||. An automatic algorithm approximates this semi-norm, but typically there is no rigorous justification for the estimate. The approach here assumes that f lies in a cone, not a ball. This allows a rigorous justification of the semi-norm of the function and a guarantee that the automatic algorithm returns the correct answer.