Support Graph Preconditioners for Off-Lattice Cell-Based Models

TL;DR

This paper introduces a graph-based preconditioning method for large, sparse matrices in off-lattice cell-based models, improving the efficiency of solving equations in multicellular system simulations.

Contribution

We extend support graph preconditioners to block-structured matrices and provide theoretical bounds, demonstrating improved performance in agent-based multicellular models.

Findings

Support graph preconditioners reduce iteration counts in conjugate gradient methods.

Theoretical bounds on condition numbers validate the effectiveness of the preconditioners.

Benchmark results show improved computational efficiency over traditional methods.

Abstract

Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies.