Scalability of High-Performance PDE Solvers

TL;DR

This paper evaluates the performance and scalability of high-performance PDE solvers across different architectures, providing best practices for efficient large-scale simulations and optimization strategies.

Contribution

It introduces a systematic performance analysis framework for PDE solvers, highlighting optimization strategies and trade-offs for various architectures and discretization methods.

Findings

Peak performance varies across architectures and codes.

Optimal code optimization strategies depend on hardware.

Minimum time to solution achieved at 80% parallel efficiency.

Abstract

Performance tests and analyses are critical to effective HPC software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this paper, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for PDEs that govern a wide range of physical applications. We consider a sequence of PDE- motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.