Millican Colloquium

Low-Rank Anderson Acceleration

In this talk, we present the low rank Anderson Acceleration (lrAA), a numerical method that directly computes low rank solutions to nonlinear equations. In many applications (e.g. nonlinear diffusion), the approximate solution, when represented as a matrix, is approximately low rank. It is challenging to design a numerical scheme for nonlinear equations that work directly with low rank. A principal challenge is that if nonlinearities are evaluated element-wise for all matrix elements then the computational savings, from quadratic to linear in the grid points per dimension, is lost.

We propose lrAA, which is based on Anderson Acceleration (AA), a well known technique for accelerating Picard iteration for fixed point problems. We couple AA with low rank truncation and cross approximations. We develop a new method for matrix cross approximation, Cross-DEIM, that uses the discrete empirical interpolation method (DEIM) based index selection to achieve effective cross approximations throughout the iterations. We show that lrAA works well for benchmark problems, such as Bratu problem and Allen-Cahn equations. This is joint work with Daniel Appelö (VT).

Increasing Subsequences, Matrix Loci, and Shadow play

An increasing subsequence in a permutation $w = [w(1), \dots, w(n)]$ is a sequence $1 \leq i_1 < \cdots < i_k \leq n$ such that $w(i_1) < \cdots < w(i_k)$. Increasing subsequences in permutations play important roles in representation theory and probability. We describe a graded ring whose algebraic structure is governed by the combinatorics of increasing subsequences through the shadow line construction of X. Viennot. This ring will come from the set of $n \times n$ permutation matrices, thought of as points in matrix space. 

Maurice Rojas, Texas A&M University

Prime Powers and Locating Roots of Equations