# MiniSymposium-6: Emerging Topics in Numerical Linear Algebra

The incorporation of GPU’s in peta- and exascale computers has already rapidly changed the traditional landscape of high-performance computing (HPC) environments used for large-scale scientific simulations. However, crucial components of simulation software, such as current-generation state-of-the-art numerical linear algebra tools, are still not able to fully utilize the new heterogeneous architectures. They fail to meet the expected efficiency and storage requirements, ensure resilience to common system failures, or other, often conflicting, requirements. We see a growing demand for reduced or mixed precision computations, new algorithms for decentralized processing and communication abilities to work in truly autonomous device networks or techniques allowing discipline scientists and engineers to significantly speed-up their simulations. The purpose of this minisymposium is to present some of the research addressing recent challenges emerging in front of the numerical linear algebra community.

## Organizers:

**Paul Cazeaux**, University of Kansas**Agnieszka Miedlar**, University of Kansas

## Saturday, October 2, 2021 at 4:40 – 6:00 pm (CST)

**Sara Pollock**

University of Florida

4:40 pm

*Mode-damping Extrapolation Methods for Eigenvalue Problems*

We will discuss a novel depth-one extrapolation method that can transform the standard power iteration into a robust and efficient method for converging to a dominant eigenpair. The method uses a linear combination of current and previous update steps to form a better approximation of the dominant eigenvector, which leads to the improved performance. We will overview the theory, demonstrate the method numerically, and see some extensions to an extrapolated Arnoldi method for recovering multiple eigenpairs.

**Ning Wan**

University of Kansas

5:00 pm

*Numerical Study of Nonlinear Acceleration Methods for Solving Large Scale Eigenvalue Problems*

Many applications in science and engineering lead to models that require solving large scale eigenproblems, especially nonlinear eigenvalue/eigenvector problems. A fixed-point iteration, known as the self-consistent field (SCF) iteration, is used to solve a class of nonlinear eigenproblems. However, the SCF iteration is usually slow, and not always work. In this talk, we will discuss different methods and acceleration techniques to handle such problems.

**Alyson L. Fox**

Lawrence Livermore National Laboratory

5:20 pm

*Current State of the Error Analysis Thrust of ZFP Lossy Compression*

The data-movement bottleneck problem is a growing concern for many HPC applications. Without changing the underlying data format, applications can only reduce the latency costs by reducing the number of messages or changing the communication topology. Mixed-precision methods can reduce both latency and bandwidth concerns but are restricted to a finite set of precisions. Instead, we proposed to use ZFP, a compressed format for floating-point data, as a new data-type that offers additional flexibility than the traditional IEEE data-types. As ZFP is a lossy compression method is extremely important to understand how the error impacts the accuracy of the solution within an application.We will discuss the current error analysis work for ZFP and present a new flexible precision option for ZFP.

**Aaron Barrett**

University of Kansas

5:40 pm

*Redundant Computations in a Fixed Point Linear Solver for Collaborative Autonomy*

Unreliable computing environments impose several additional constraints on algorithmic approaches, including the presence of communication delays among network elements, heterogeneous computing architectures, and the need for fully decentralized algorithms. Our goal is to adapt the asynchronous parallel Jacobi method to fully utilize these environments by incorporating the idea of redundant computations. The redundant asynchronous parallel Jacobi algorithm partitions a linear system among computing elements, e.g., processors, while allowing overlap such that each component of the solution vector is updated by at least one other processor. Redundant computations are introduced primarily to address the issue of delayed communication; in order to improve the quality of the iterative scheme, we loosen the constraint that the performance of the algorithm relies on the performance of the slowest processor. We present a new framework for analyzing redundant asynchronous linear system solvers and present numerical results for the case where the Jacobi iteration matrix is nonnegative.