Adaptive Multiscale Methods for Approximation and Preconditioning

Multiscale modelling is ubiquitous in science, technology and social science. Models with multiple time or length scales arise, for example, in modelling pollutant transport in groundwater, turbulent fluid flow in reactor cooling systems, high-frequency waves in radar and sonar and atomistic-continuum models in material science. Because of the highly varying scales involved in multiscale problems, the accurate modelling of all scales is often outside the reach of even the largest supercomputers. A suitable goal of computation is then to compute a solution on the finest computationally affordable grid, in such a way that the accuracy is not polluted by the fine scales which remain unresolved. More precisely, the aim is to solve the multiscale problem on a coarse mesh in such a way that the error is of the same order (with respect to the number of degrees of freedom) as if the problem were smooth, and moreover, the accuracy does not degrade if the finest scale present in the model decreases. Such computational methods are often called ``robust''. There are two paradigms for the construction of robust numerical methods for multiscale problems. The first is to replace the multiscale problem with a (nearby) smooth problem and then solve the latter numerically. Examples of this approach include upscaling in porous medium flow and transport or the use of geometric theory of diffraction and ray-tracing in high-frequency wave propagation problems. The basic difficulty with this approach is that these approximations tend to be valid only when the fine scale small parameter is sufficiently small (in order to make some sort of averaging valid), and, moreover, their rigorous analysis requires simplifying assumptions (such as periodicity and scale separation in homogenization theory).An alternative approach is to devise problem-adapted numerical methods which are targeted to the type of multiscale behaviour arising in the particular application, and are capable of resolving it robustly on a coarse mesh. This usually involves replacing the (piecewise) polynomial approximations at the heart of classical numerical methods with problem-adapted bases which are better able to reflect the solution behaviour on coarse meshes. Examples of this type of approach in application areas include sub-grid scale modelling in large eddy simulation, and the modelling of localised convective storms in large-scale weather prediction software. This is a new collaboration between the PI and the proposed VF which will not take place without the requested EPSRC support. We will produce new results on methods of the second type. Our methods will be adaptive (i.e. the non-polynomial bases will be computed automatically, rather than designed in detail by the practitioner) and they will work well both in the presence of small lengthscale (small wavelength of data) as well as large contrast (large amplitude of data). We will test our methods on systems arising from problems with random data with small lengthscale and large variance (leading to small wavelength and large amplitude). We will also investigate the application of the same ideas in the design of robust preconditioners for conventional discretisations of multiscale problems including those which approximate equations describing high frequency wave phenomena.

The robust solution of problems where the coefficients vary on scales finer than any feasible mesh is an urgent (in general) unsolved problem in numerical analysis, the solution of which has far-reaching consequences in applications. Particular areas where this issue is of great importance include the hydrocarbon and water resources industries and in oceanography/numerical weather forecasting. Exactly this problem is a central theme on regular conferences of people whose main field lies outside mathematics, such as the SIAM Geosciences Series. The results which we shall obtain will be published in leading numerical analysis journals and will be presented in international meetings at which the PI and VF are regularly invited. For example the PI will give talks in areas related to this research proposal at the International Conference on Sparse Representation of Multiscale Data and Images: Theory and Applications, at NTU, Singapore, 14-17 December 2009 and at the SIAM UK and Republic of Ireland Section Annual Meeting, Edinburgh January 2010. The results of this research will be discussed at the LMS Durham Symposium on Numerical analysis of Multiscale Problems at which there will be an ``Industry Day'' with representatives of companies like Chevron, Exxon Mobil, Schlumberger, and the Met Office, all of whom are interested in these issues. The PI has extensive experience of promoting his research at high-level international meetings and collaborating with industry (see ``Track Record''). For example, he was co-organiser of a Workshop in Bath in 2006 in which representatives from Schlumberger, Institut Francais du Petrole and SINTEF Oslo took part along with academics from around the world.