Research

My research program concerns the analysis of nonlinear solvers and their applications. The topic of my dissertation was the analysis and implementation of Anderson acceleration to Newton’s method. This work lead to two publications ([1] and [2] below) which are summarized here.

Lately, my energy has been devoted to generalizing theoretical results concerning Anderson acceleration applied to a broader set of Newton-like methods, and further developing gamma-safeguarding. This was a safeguarding scheme introduced in [1], and generalized in [2], to obtain a local convergence result for Anderson accelerated Newton’s method. Rather than simply a theoretical tool, numerical results in [2] indicate the potential for adaptive gamma-safeguarded Anderson to be a viable method in its own right.

I am also working on some collaborative projects that apply Anderson and gamma-safeguarding to more complex problems than the standard benchmarks I have used so far.

Publications

[2] M. Dallas, S. Pollock, and L.G. Rebholz, Analysis of an adaptive safeguarded Newton-Anderson algorithm of depth one with applications to fluid problems, Adv. Comput. Sci. Eng., 2 (2024), pp. 246-270. pdf

[1] M. Dallas and S. Pollock, Newton-Anderson at Singular Points, Int. J. Numer. Anal. Mod., 20 (2023), pp. 667-692. pdf