Complexity order of multiple resource algorithms

TL;DR

This paper introduces a new way to measure the efficiency of algorithms that use multiple resources, predicting the optimal complexity order and validating it with numerical examples in stochastic differential equations.

Contribution

It defines the complexity order for multi-resource algorithms and derives a formula for the optimal order, validated through numerical experiments.

Findings

Optimal complexity order equals the inverse sum of individual resource orders.

The theory accurately predicts the efficiency of various algorithms.

Numerical results align well with the analytic predictions.

Abstract

Algorithmic efficiency is essential to reducing energy and time usage for computational problems. Optimizing efficiency is important for tasks involving multiple resources, for example in stochastic calculations where the size of the random ensemble competes with the time-step. We define the complexity order of an algorithm needing multiple resources as the exponent of inverse total error with respect to the total resources used. The optimum order is predicted for independent, factorable resources. We show that it equals the inverse sum of the inverse resource orders. This is applied to computing averages in a stochastic differential equation. We treat numerical examples for multiple different algorithms and for stochastic partial differential equations, all giving quantitative results in excellent agreement with our more general analytic theory.