Student Applied Math Talk: Adaptive compression for eigenvalue problems

Seminar | March 23 | 5-6 p.m. | 736 Evans Hall

 Michael Lindsey, UC Berkeley

 Department of Mathematics

Large, dense Hermitian eigenvalue problems play a central role in computational quantum chemistry. Often the matrices for these problems are of the form $A+B$, where multiplication by $A$ is cheap, multiplication by $B$ (and hence also by $A+B$) is costly, and $\Vert A \Vert_2 \gg \Vert B \Vert_2$. Although $B$ is less significant than $A$, it is still too significant to be neglected, and the evaluation of matrix-vector products $Bv$ typically constitutes the vast majority of the computational cost of standard iterative approaches. While $B$ itself cannot necessarily be approximated well by a low-rank matrix, it can be compressed adaptively with respect to a low-dimensional subspace that is iteratively updated until convergence to the desired eigenspace is achieved. We will discuss the properties of the aforementioned ‘adaptive compression’ operation, as well as the convergence of the associated adaptive compression method for solving eigenvalue problems, which has been adopted in community electronic structure software packages such as Quantum ESPRESSO. In particular, we will explain how to prove local convergence with an asymptotic rate, as well as global convergence that holds generically in a strong sense. The proof proceeds by studying the adaptive compression method as a dynamical system and ultimately takes some surprising turns through rather diverse fields of math.