Finite difference methods fengshui: alignment through a mathematics of arrays

Abstract

Numerous scientific-computational domains make use of array data. The core computing of the numerical methods and the algorithms involved is related to multi-dimensional array manipulation. Memory layout and the access patterns of that data are crucial to the optimal performance of the array-based computations. As we move towards exascale computing, writing portable code for efficient data parallel computations is increasingly requiring an abstract productive working environment. To that end, we present the design of a framework for optimizing scientific array-based computations, building a case study for a Partial Differential Equations solver. By embedding the Mathematics of Arrays formalism in the Magnolia programming language, we assemble a software stack capable of abstracting the continuous high-level application layer from the discrete formulation of the collective array-based numerical methods and algorithms and the final detailed low-level code. The case study lays the groundwork for achieving optimized memory layout and efficient computations while preserving a stable abstraction layer independent of underlying algorithms and changes in the architecture.CCS Concepts • Software and its engineering → Software design engineering;

P3 problem and Magnolia language: Specializing array computations for emerging architectures

“…Burrows et al (2018) identified an array API for FDM solvers. Chetioui et al (2019) followed up with a formalization of the identified array API using MoA. We will first give an overview of PDE solvers as described by Burrows et al (2018), and an introduction to the corresponding MoA theory.…”

“…of Chetioui et al (2019), and implement hardware-agnostic and hardware-dependent rewriting rules. We show how they can be applied to our Magnolia program, and measure the resulting performance improvements.…”

“…Reasoning at the DNF level is thus completely hardware agnostic. By repeatedly applying a set of terminating rewrite rules, any MoA expression can be reduced to its DNF (Mullin and Thibault, 1994;Chetioui et al, 2019)where the resulting array is described at each index by indexing into the input arrays and scalar-level operations.…”

“…They conjectured that ψ-reduction is canonical (i.e., it is terminating and confluent)-and thus takes any MoA expression to its unique DNF. In their work on embedding Burrows et al's (2018) array API for FDM solvers in MoA, Chetioui et al (2019) outline a rewriting system sufficient to transform a program based on this API to its DNF and show that this rewriting system is indeed canonical. This draws appeal to MoA as a framework for the optimization of PDE solvers based on FDM.…”

“…The topic of optimizing the performance of array computations is therefore extremely relevant to the P 3 problem. We have previously explored the Mathematics of Arrays (MoA) formalism (Mullin, 1988) as a tool to optimize array computations for different hardware architectures, first through their Denotational Normal Form (DNF; Chetioui et al, 2019) and then through their Operational Forms (OFs; Chetioui et al, 2021). A thorough mathematical understanding of a given domain is key to enabling domain-specific semantic-preserving rewrites-and therefore optimizations.…”