Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators).
In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems. I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).
Main preprint: https://arxiv.org/abs/2409.14438.
WoodburyLS preprint: https://arxiv.org/abs/2406.15120