Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multidimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed.

Cells rely on an interplay of subcellular elements for motility and migration. Certain regions of motile cells, such as the lamellipodium, are made of a complex mixture of actin monomers and filaments, which polymerize at the front of the cell, close to the cell membrane, and depolymerize at the rear. The dynamic actin turnover induces the so-called intracellular retrograde flow, and it is a fundamental process for cell motility. Apart from some comprehensive mathematical models, the computational modelling of actin treadmilling has been based on simpler biophysical models. Here, we adopt a highly detailed theoretical model of the actin treadmilling process and develop a coupled unsteady finite element formulation. We clearly describe the structure and implementation of the coupled problem within the finite element method. Our numerical results show an excellent correlation with experimental results from literature and with previous models. We include time dependent effects and convective transport terms, which expose puzzling dynamics in the retrograde flow. We propose several biological scenarios to analyze the behavior of the actin treadmilling along space and time. We observed response times of the main density variables in the order of seconds. Compared with previous analytical solutions, which make assumptions related to convective transport, transient dynamics, and actin fluxes, the generic solution can have significant influence on the retrograde flow. All together, our results unveil a promising applicability of classical finite element methods to derive an in silico testing platform for the actin treadmilling processes in motile cells, which could allow for an extension to other biophysical effects.

Typical strategies for reducing the computational cost of contact mechanics models use low‐rank approximations. The underlying hypothesis is the existence of a low‐dimensional subspace for the displacement field and a non‐negative low‐dimensional subcone for the contact pressure. However, given the local nature of contact, it seems natural to wonder whether low‐rank approximations are a good fit for contact mechanics or not. In this article, we investigate some of their limitations and provide numerical evidence showing that contact pressure is linearly inseparable in many practical cases. To this end, we consider various mechanical problems involving nonadhesive frictionless contacts and analyze the performance of the low‐rank models in terms of three different criteria, namely, compactness, generalization, and specificity.