Participating Network Members C.M. Elliott [CME] (Sussex), K. Christensen [KC] (Imperial), P. Plechac [PP] (Warwick), A.M. Stuart [AMS] (Warwick), E. Süli [ES] (Oxford), D.D. Vvedensky [DDV] (Imperial).

Modelling in materials science is one of the areas where simulations often need to be performed for the size of a specimen where the full resolution of microscopic scales is prohibitively expensive or impossible. Different approaches have been suggested to overcome this problem. Although they have been successfully implemented for simulations of certain systems, their systematic mathematical analysis is still missing. The activity of the proposed network aims at providing a deeper mathematical insight into coupled macroscopic-microscopic computational paradigms and explore the potentials for improving their efficiency and reliability. Particular objectives and goals of Theme III are:

(1) to develop and analyse new algorithms for numerical coarse-graining;

(2) to improve the mathematical understanding of energy transfer between different scales and its impact on approximation properties of various multi-scale methods currently used;

(3) to explore the application of mesoscopic models with random noise in efficient numerical computations;

(4) to compare deterministic approaches with stochastic simulations in specific examples, such as models for polymeric fluids (Theme II) or modelling of epitaxial growth.

The computational mathematics group in Warwick has a strong background in the analysis of multi-scale problems and the numerical solution of stochastic differential equations.

PP has developed and analysed computational methods for the numerical approximation of microstructures in materials. Recently he has been working, together with Prof. M.A. Katsoulakis, on developing non-equilibrium coarse-grained Monte-Carlo algorithms by numerically solving stochastic mesoscopic equations derived from Ising and Heisenberg-type lattice models. Following the recent work on Ising-type systems ([10]), the mesoscopic kinetic equations are derived in order to guide the design of a coarse-grained stochastic model that allows for efficient MC simulations on larger scales. This novel class of methods is expected to overcome the high computational cost of direct Monte-Carlo simulations on the microscopic lattice models and to enable simulations that include thermal fluctuations for realistic-size specimens. Applications to modelling of processes in finite temperature micromagnetics ([11]) link the methodology with the problems investigated in Theme I.

AMS is working on the development of parameter estimation techniques designed to fit coarse-grained (macroscopic) stochastic models to fine-scale (microscopic) simulation data. This work builds on analytically tractable heat bath models. It involves new theoretical and analytical problems in the estimation process for partially observed hypo-elliptic diffusions. The approach is being applied to problems arising in bio-molecular dynamics. The general framework of parameter estimation appears to be a promising tool for the numerical approximation of microscopic models by stochastic mesoscopic models that describe the behaviour of the system on larger scales while retaining relevant features of the noise.

In the collaboration with the solid-state group at Imperial College PP, AMS, DDV and KC will investigate coarse-graining procedures and mesoscopic computational models for epitaxial growth. The morphological evolution of an epitaxial film is an example of simulation where the kinetics is described by an atomistic process ([12]). The evolution results from adatom motion and step-adatom interactions asserting their influence over macroscopic length- and time-scales of the growth front. Modelling epitaxial phenomena thus necessitates making a compromise between the detailed information provided by first-principles methods, and computational flexibility is afforded by methods such as Monte-Carlo simulations and continuum equations of motion in which atomistic processes are replaced by coarse-grained effective kinetics.

We propose to study multi-scale algorithms that recursively construct a sequence of approximations of the statistical ensemble at increasingly larger (coarser) scales. In the physics terminology this can be seen as numerical approximation of the renormalization group map. After selecting a set of new (coarse-grain) variables on a coarse lattice the new interaction Hamiltonian is effectively approximated from the information gathered on finer lattices. The methods of equilibrium statistical physics and Monte-Carlo algorithms provide a basis for the computational treatment of systems at equilibrium. However, the efficiency of computational algorithms is less understood when dynamic processes on lattices are modelled (e.g. by a kinetic Monte Carlo method).

Modelling of deposition and growth on thin films; polymeric fluids; reversal processes in magnetic materials; crack propagation in elastic solids represent few examples in which correct numerical interface and energy transfer between different scales in numerical approximation is essential. We propose to explore several existing approaches and their application to problems studied in Theme I - III. The framework of ``quasicontiuum method" (QCM) ([13]), ``heterogeneous multi-scale method'' (HMM) and ``coarse-grained molecular dynamics (CGMD) ([14]) combines classical approximations of continuum models (e.g. by using the finite element method (FEM)) with simulation methods developed for atomistic models (e.g. molecular dynamics (MD), kinetic Monte Carlo method (KMC)). The main idea behind these approaches is to sample a small subset of the microscopic degrees of freedom and to extract information that can be passed into the continuum model in the form, for example, of constitutive relations. While the methods provide significant reduction of degrees of freedom needed in computational simulations, their rigorous numerical analysis which is well-developed for their continuum counter-parts is still lacking.