# Advances in Implementation, Theoretical Motivation, and Numerical Results for the Nested Iteration with Range Decomposition Algorithm

## TL;DR

This paper enhances the nested iteration with range decomposition algorithm (NIRD) for elliptic PDEs, improving its accuracy, scalability, and performance on complex problems through algorithmic modifications, convergence analysis, and extensive numerical testing.

## Contribution

It introduces improvements to NIRD, including adaptivity and wider partitioning options, along with a new convergence proof and performance model, demonstrating enhanced scalability and effectiveness.

## Key findings

- NIRD converges rapidly with high accuracy on complex elliptic problems.
- Algorithmic improvements lead to better scalability and robustness.
- Numerical results confirm NIRD's competitive performance on large-scale problems.

## Abstract

This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD converges to a high level of accuracy within a small, fixed number of iterations (usually one or two) when applied to simple elliptic problems. This paper makes some improvements to the NIRD algorithm (including the addition of adaptivity during preprocessing, wider choice of partitioning functions, and modified error measurement) that enhance the method's accuracy and scalability, especially on more difficult problems. In addition, an updated convergence proof is presented based on heuristic assumptions that are supported by numerical evidence. Furthermore, a new performance model is developed that shows increased performance benefits for NIRD when problems are more expensive to solve using traditional methods. Finally, extensive testing on a variety of elliptic problems provides additional insight into the behavior of NIRD and additional evidence that NIRD achieves excellent convergence on a wide class of elliptic PDE's and, as such, should be a very competitive method for solving PDE's on large parallel computers.