Support Graph Preconditioners for Off-Lattice Cell-Based Models

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.

💡 Research Summary

The paper addresses a critical computational bottleneck in off‑lattice agent‑based cell models: solving the large, sparse, symmetric positive‑definite friction matrix Γ that arises from the overdamped equation of motion Γ v = F. In these models each cell is represented by a 3‑D object (sphere, capsule, ellipsoid, etc.) and the interaction between any contacting pair of cells i and j contributes a 3 × 3 friction block Γ_ccij that depends on the contact area A_ij and parallel/perpendicular friction coefficients γ∥, γ⊥. The global matrix Γ therefore has a block‑Laplacian structure: its off‑diagonal blocks are negative‑definite (the pairwise friction tensors) and its diagonal blocks collect cell‑substrate friction contributions. Because the contact pattern changes over time, both the sparsity pattern and the condition number κ(Γ) vary dynamically, often degrading CG convergence.

The authors propose a graph‑based preconditioning strategy that is both matrix‑free and well‑suited to the block‑structured nature of Γ. They reinterpret the non‑zero pattern of Γ as a weighted undirected graph C (the “collision graph”), where vertices correspond to cells and edges carry the 3 × 3 friction tensors as matrix‑valued weights. The friction matrix Γ is exactly the block Laplacian of C. Building on Vaidya’s support‑graph theory, they select a subgraph H of C that is a maximum spanning tree (MST) with respect to the Frobenius norm of the edge weights. The Laplacian of this tree, L_T, is used as a preconditioner: they factor L_T = E Eᵀ, solve the transformed system Eᵀ Γ E ẑ = Eᵀ F, and recover the solution v = E ẑ.

The theoretical contribution consists of extending support‑graph analysis from scalar‑weighted graphs to matrix‑weighted (block) graphs. The authors prove that if the block Laplacian satisfies a generalized block‑diagonal dominance condition (which holds for the friction matrices because each block is symmetric positive‑definite), then the eigenvalues of the preconditioned system are bounded by constants that depend only on the ratios γ_max/γ_min and on the extremal edge conductances of the collision graph. In particular, they obtain \