Applied Analysis Seminar

Tuesday July 9 in K9509, SFU

Speaker and Title: 2:00 - 2:45: Philip Loewen (UBC, Mathematics) "Smooth Bumps and Their Gradients"

Abstract: A bump is a real-valued function whose support is nonempty and bounded; continuously differentiable bumps are called smooth.

In this talk I will outline the results of recent investigations (joint with J. Borwein, I. Kortezov, and M. Fabian) into the kinds of sets obtainable as the range of the gradient for a smooth bump. I will show how to build a smooth bump on the plane whose gradient range is not simply connected. Passing to infinite dimensional Banach spaces, I will explain how to use a given smooth bump to build another one whose gradient range exactly reproduces the closure of a preassigned convex neighbourhood of the origin; the same procedure covers other reasonable shapes. Some tantalizing open problems that motivated this research remain unsolved: I'll mention one of these and its current status.

Speaker and Title: 2:45 - 3:30: Jon Borwein (CECM) "Bregman Monotone Optimization Algorithms"

Abstract: The notions of Bregman distance and ofBregman monotonicity provide a unified way of analysing many non-linear iterative methods for reconstruction and feasibility problems.

I shall survey my recent work on this subject with Heinz Bauschke and Patrick Combettes. I'll also describe some related work on barrier functions with Jon Vanderwerff.