Title: Regularized dynamical systems associated with structured monotone inclusions

Speaker: Pham Ky Anh (Vietnam National University)

Date and Time: Wed Mar 30 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

In this report, we consider two dynamical systems associated with additively structured monotone inclusions involving a multi-valued maximally monotone operator A and a single-valued operator B in real Hilbert spaces. We established strong convergence of the regularized forward-backward and regularized forward – backward–forward dynamics to an “optimal” solution of the original inclusion under a weak assumption on the single-valued operator B. Convergence estimates are obtained if the composite operator A + B is maximally monotone and strongly (pseudo)monotone. Time-discretization of the corresponding continuous dynamics provides an iterative regularization forward-backward method or an iterative regularization forward-backward-forward method with relaxation parameters. Some simple numerical examples were given to illustrate the agreement between analytical and numerical results as well as the performance of the proposed algorithms.

Title: Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors

Speaker: Shawn Wang (The University of British Columbia)

Date and Time: Wed Mar 23 2022, 11:00 AEST (Register here for remote connection via Zoom)

Abstract:

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. We extend Simons’s lemma to closed convex functions, show its connections to Attouch-Théra duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings. Joint work with H. Bauschke

Title: Generalized Gearhart-Koshy acceleration for the Kaczmarz method

Speaker: Janosch Rieger (Monash University)

Date and Time: Wed Mar 16 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart and Koshy have developed an acceleration technique for the Kaczmarz method for homogeneous linear systems that minimises the distance to the desired solution in the direction of a full Kaczmarz step. Matthew Tam has recently generalised this acceleration technique to inhomogeneous linear systems.

In this talk, I will develop this technique into an acceleration scheme that minimises the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently.