Robust and scalable nonlinear solvers for finite element discretizations of biological transportation networks

2025-04-06

AI Summary

This work addresses the numerical simulation challenge of gradient flows driven by nonconvex energy functionals in modeling biological transport networks. Methodologically, it introduces the first robust, fully implicit finite element solver supporting large-scale 3D parallel computation: backward Euler time discretization is combined with a discontinuous Galerkin spatial formulation; a novel positive semidefiniteness-preserving strategy for discontinuous conductivity tensors ensures numerical stability throughout the gradient flow evolution; and a customized linear preconditioner together with high-concurrency HPC implementation is developed. Contributions include: the first 3D numerical simulation of biological network formation models; rigorous verification of strong scalability and robustness on both 2D and 3D meshes; discovery of pronounced sensitivity of emergent network structures to mesh resolution and topology; and achievement of near-optimal parallel performance.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Medical and Biological Imaging
• Machine Learning: Bio-inspired Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

We develop robust and scalable fully implicit nonlinear finite element solvers for the simulations of biological transportation networks driven by the gradient flow minimization of a non-convex energy cost functional. Our approach employs a discontinuous space for the conductivity tensor that allows us to guarantee the preservation of its positive semi-definiteness throughout the entire minimization procedure arising from the time integration of the gradient flow dynamics using a backward Euler scheme. Extensive tests in two and three dimensions demonstrate the robustness and performance of the solver, highlight the sensitivity of the emergent network structures to mesh resolution and topology, and validate the resilience of the linear preconditioner to the ill-conditioning of the model. The implementation achieves near-optimal parallel scaling on large-scale, high-performance computing platforms. To the best of our knowledge, the network formation system has never been simulated in three dimensions before. Consequently, our three-dimensional results are the first of their kind.

Problem

• Develop robust nonlinear solvers for biological transport networks
• Ensure conductivity tensor positivity in gradient flow minimization
• Achieve scalable 3D simulation of network formation

Innovation

• Discontinuous space for conductivity tensor preservation
• Robust fully implicit nonlinear finite element solvers
• Near-optimal parallel scaling on HPC platforms