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Simulating the dynamics of discretized interacting structures whose relationship is dictated by a kernel function gives rise to a large dense matrix. We propose a multigrid solver for such a matrix that exploits not only its data-sparsity resulting from the decay of the kernel function but also the regularity of the geometry of the structures and the quantities of interest distributed on them. Like the well-known multigrid method for large sparse matrices arising from boundary-value problems, our method requires a smoother for removing high-frequency terms in solution errors, a strategy for coarsening a grid, and a pair of transfer operators for exchanging information between two grids. We develop new techniques for these processes that are tailored to a kernel function acting on discretized interacting structures. They are matrix-free in the sense that there is no need to construct the large dense matrix. Numerical experiments on a variety of bio-inspired microswimmers immersed in a Stokes flow demonstrate the effectiveness and efficiency of the proposed multigrid solver. In the case of free swimmers that must maintain force and torque balance, additional sparse rows and columns need to be appended to the dense matrix above. We develop a matrix-free fast solver for this bordered matrix as well, in which the multigrid method is a key component. 

Cells rely on their cytoskeleton for key processes including division and directed motility. Actin filaments are a primary constituent of the cytoskeleton. Although actin filaments can create a variety of network architectures linked to distinct cell functions, the microscale molecular interactions that give rise to these macroscale structures are not well understood. In this work, we investigate the microscale mechanisms that produce different branched actin network structures using an iterative classification approach. First, we employ a simple yet comprehensive agent-based model that produces synthetic actin networks with precise control over the microscale dynamics. Then we apply machine learning techniques to classify actin networks based on measurable network density and geometry, identifying key mechanistic processes that lead to particular branched actin network architectures. Extensive computational experiments reveal that the most accurate method uses a combination of supervised learning based on network density and unsupervised learning based on network symmetry. This framework can potentially serve as a powerful tool to discover the molecular interactions that produce the wide variety of actin network configurations associated with normal development as well as pathological conditions such as cancer.