Support Graph Preconditioners for Off-Lattice Cell-Based Models

TL;DR

This work addresses the computational bottleneck of solving large, sparse, symmetric positive definite linear systems arising from off-lattice cell-based models by introducing graph-based, block-structured preconditioners derived from the collision graph. By extending Vaidya’s support graph preconditioners to block Laplacians and proving eigenvalue bounds, the authors show that maximum spanning tree (MST) preconditioners markedly reduce CG iterations and wall-clock time compared to standard strategies, across diverse cell geometries. The approach enables matrix-free preconditioning that aligns with collision-detection data, with near-linear-time MST construction and factorization, and yields substantial practical gains for large-scale in-silico tissue simulations. The results advocate MST-based preconditioning as a robust, scalable solution for efficient simulation of thousands to millions of cells, and point to future hierarchical and parallel extensions to further improve performance.

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.

Figures (4)

• Figure 1: Graphical overview of our preconditioner construction. The figure illustrates the step-by-step construction of our proposed preconditioner, proceeding from left to right. A) Agent configuration. The initial setup showing individual agents (cells) in a spatial arrangement. B) The collision graph. Using collision detection algorithms, we construct a graph where nodes represent agents and edges represent collisions or friction interactions between them. The friction matrix is the graph Laplacian of the collision graph. C) The maximum spanning tree. Using Prim's algorithm, we construct a maximum spanning tree from the collision graph. The graph Laplacian of this tree, is used as the preconditioner.
• Figure 2: An example of the elimination game being played on the same graph with different orderings. Graph edges and matrix entries are in solid black and denoted by crosses respectively. Fill edges and entries are in dotted blue and blue circles respectively.
• Figure 3: Preconditioner convergence behavior for different cell arrangements: hexagonal lattices with varying disorder (top row), random packing (bottom left), and bridged spheroids (bottom right). The insets show cross-sections through the 3D cell configurations. The red dashed line in each plot indicates the solver relative tolerance $10^{-5}$. Legend: Identity/no preconditioner (orange dotted line); block Jacobi (green long dash-dot); diagonally-shifted block IC(0) (dashed red); block Gauss-Seidel (cyan dash-dot); MST preconditioner (solid blue); augmented MST (gray circular markers). Top Row: Cells arranged on a three-dimensional hexagonal lattice with positional noise of mean zero and standard deviation $0.3r$ applied to each cell. Each configuration contains $n \approx 50\,000$ cells. The edge-to-vertex ratio is reported for each configuration. Left: $\gamma_{\mathrm{med}} = 3 \times 10^4$. Right: $\gamma_{\mathrm{med}} = 3 \times 10^3$. $\kappa$ in the title is the full friction matrix condition number. Bottom Left: $n=50\,000$ cells randomly packed in a spheroidal domain with $\gamma_{\mathrm{med}} = 3 \times 10^4$. Bottom Right: Two densely packed spheroids connected by a bridge of randomly placed cells, $n \approx 50\,000$ total, $\gamma_{\mathrm{med}} = 3 \times 10^4$.
• Figure 4: Performance comparison of preconditioners for block-structured linear systems arising from $n$ cells arranged in a hexagonal lattice with positional noise (30% of cell radius). Each data point represents the average of 5 independent experiments. The friction parameters are $\gamma_{\mathrm{med}} = 3 \times 10^4$ with $\gamma_{\parallel} = 2 \times 10^6$ and $\gamma_{\perp} = 8 \times 10^7$. (Left) Total wall-clock time as a function of problem size $n$ for solving the linear system to relative tolerance $10^{-5}$. Times include all computational costs: matrix assembly (where required), preconditioner construction and factorization, and conjugate gradient iterations. The MST preconditioner achieves a speedup of $3.5\times$ compared to unpreconditioned CG at $n = 10^5$. The direct solver (Eigen LDLT) excels for $n < 10^3$ but becomes catastrophically slow for larger problems. Preconditioners tested: (1) identity (no preconditioning), (2) block Jacobi with $LDL^T$ factorization of diagonal blocks, (3) MST with tree construction and factorization, (4) augmented-MST with tree construction, augmentation and nested (flexible) conjugate gradient (inner relative tolerance $\delta = 0.1$), (5) IC(0) with diagonal shift and incomplete factorization, (6) block Gauss-Seidel with setup, and (7) Eigen's sparse $LDL^T$ direct solver including assembly and factorization. The MST preconditioner achieves the fastest total solution times. (Right) Iteration counts for the same preconditioners as a function of $n$.

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Definition 2.2: Matrix-weighted graph
• Definition 2.3: Block Laplacian
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 3.3