Scalable Parallel-in-Time Integration for Equations of Motion: Particle Production in Analog Cosmologies

The work here, using the functionality of Parareal.jl and PararealGPU.jl, is part of my master's thesis in computational science, finished June 2025.

Abstract

Simulating time-dependent physics has traditionally been constrained to using sequential algorithms, thus not benefiting from advances in parallel computing. Parallel-in-time integration attempts to address this limitation with methods such as the Parareal algorithm. As the performance of the Parareal algorithm scales with the number of processors, it is well-suited to use the massively-parallel nature of graphics processing units. Additional performance gains are seen when the physics is wave-like, as using a spectral method allows for each node in a distributed system to evaluate the Parareal algorithm. Particle production in different cosmologies is used to highlight the performance gains from these methods.

Outline

1. Introduction

2. Background

2.1. Parallel-in-Time Integration (PinT)

2.1.1. Parareal

2.1.2. Multigrid Reduction in Time (MGRIT)

2.1.3. Parallel Full Approximation Scheme in Space and Time (PFASST)

2.2. High-Performance Computing

2.2.1. Multi-threading & GPU Computing

2.2.2. Multi-processing & Distributed Computing

2.3. Equations of Motion

2.3.1. Differential Equations

2.3.2. Traditional Numerical Methods

2.4. Analog Cosmology

2.4.1. BEC Analogs of FLRW Cosmologies

2.4.2. Variable Speed of Sound and Inflation

2.4.3. The Field Equation

2.4.4. Phononic & Free Particle Modes

2.4.5. Initial Conditions

2.4.6. Particle Production

3. Methods

3.1. The Parareal Algorithm

3.2. The Parareal Algorithm at Scale

4. Discussion

4.1. Numerical Analysis

4.1.1. Convergence

4.1.2. Stability

4.1.3. Error

4.2. Algorithm Analysis

4.2.1. Time Complexity

4.2.2. Space Complexity

4.3. Benchmarks

5. Showcases

5.1. Nonlinear ODE: The Pendulum

5.2. PDE: The Wave Equation

5.3. Particle Production in Analog Cosmologies

6. Conclusion

6.1. Future work

Bibliography