Accurate macroscale modelling of spatial dynamics in multiple dimensions

Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction-diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the modelling of systems by a type of finite elements, and the `equation free' macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/PDE to as high an order as desired. Computer algebra handles the considerable algebraic details as seen in the specific application to the Ginzburg--Landau PDE. However, higher order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction-diffusion PDEs and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv admin note: substantial text overlap with arXiv:0904.085

An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

Comparing the full time-dependent Bogoliubov--de-Gennes equations to their linear approximation: A numerical investigation

In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen--Cooper--Schrieffer (BCS) equations, often also denoted as Bogoliubov--de--Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg--Landau-type of equation, in accordance with a recent theoretical results. We include a full description on the numerical implementation of the partial differential BCS\ BdG equations.Comment: 10 pages, to appear in EP

Global topological control for synchronized dynamics on networks

A general scheme is proposed and tested to control the symmetry breaking instability of a homogeneous solution of a spatially extended multispecies model, defined on a network. The inherent discreteness of the space makes it possible to act on the topology of the inter-nodes contacts to achieve the desired degree of stabilization, without altering the dynamical parameters of the model. Both symmetric and asymmetric couplings are considered. In this latter setting the web of contacts is assumed to be balanced, for the homogeneous equilibrium to exist. The performance of the proposed method are assessed, assuming the Complex Ginzburg-Landau equation as a reference model. In this case, the implemented control allows one to stabilize the synchronous limit cycle, hence time-dependent, uniform solution. A system of coupled real Ginzburg-Landau equations is also investigated to obtain the topological stabilization of a homogeneous and constant fixed point

Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.Comment: 14 page