Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Phenology for tropoclimatological surveys and large-scale mapping

Biotic and abiotic phenological observations can be collected from continental to local spatial scale. Plant phenological observations may only be recorded wherever there is vegetation. Fog, snow and ice are available as phenological para-meters wherever they appear. The singularity of phenological observations is the possibility of spatial intensification to a microclimatic scale where the equipment of meteorological measurements is too expensive for intensive campaigning. The omnipresence of region-specific phenological parameters allows monitoring for a spatially much more detailed assessment of climate change than with weather data. We demonstrate this concept with phenological observations with the use of a special network in the Canton of Berne, Switzerland, with up to 600 observations sites (more than 1 to 10 km² of the inhabited area). Classic cartography, gridding, the integration into a Geographic Information System GIS and large-scale analysis are the steps to a detailed knowledge of topoclimatic conditions of a mountainous area. Examples of urban phenology provide other types of spatially detailed applications. Large potential in phenological mapping in future analyses lies in combining traditionally observed species-specific phenology with remotely sensed and modelled phenology that provide strong spatial information. This is a long history from cartographic intuition to algorithm-based representations of phenology.