Distributed model order reduction of a model for microtubule-based cell polarization using HAPOD

In this contribution we investigate in mathematical modeling and efficient simulation of biological cells with a particular emphasis on effective modeling of structural properties that originate from active forces generated from polymerization and depolymerization of cytoskeletal components. In detail, we propose a nonlinear continuum approach to model microtubule-based forces which have recently been established as central components of cell mechanics during early fruit fly wing development. The model is discretized in space using the finite-element method. Although the individual equations are decoupled by a semi-implicit time discretization, the discrete model is still computationally demanding. In addition, the parameters needed for the effective model equations are not easily available and have to be estimated or determined by repeatedly solving the model and fitting the results to measurements. This drastically increases the computational cost. Reduced basis methods have been used successfully to speed up such repeated solves, often by several orders of magnitude. However, for the complex nonlinear models regarded here, the application of these model order reduction methods is not always straight-forward and comes with its own set of challenges. In particular, subspace construction using the Proper Orthogonal Decomposition (POD) becomes prohibitively expensive for reasonably fine grids. We thus propose to combine the Hierarchical Approximate POD, which is a general, easy-to-implement approach to compute an approximate POD, with an Empirical Interpolation Method to efficiently generate a fast to evaluate reduced order model. Numerical experiments are given to demonstrate the applicability and efficiency of the proposed modeling and simulation approach.