Abstract. We propose a model for describing the parallel performance of multigrid software on distributed memory architectures. The goal of the model is to allow reliable predictions to be made as to the execution time of a given code on a large number of processors, of a given parallel system, by only benchmarking the code on small numbers of processors. This has potential applications for the scheduling of jobs in a Grid computing environment where reliable predictions as to execution times on different systems will be valuable. The model is tested for two different multigrid codes running on two different parallel architectures and the results obtained are discussed.

We propose a model for describing and predicting the performance of parallel numerical software on distributed memory architectures within a multi-cluster environment. The goal of the model is to allow reliable predictions to be made as to the execution time of a given code on a large number of processors of a given parallel system, and on a combination of systems, by only benchmarking the code on small numbers of processors. This has potential applications for the scheduling of jobs in a Grid computing environment where informed decisions about which resources to use in order to maximize the performance and/or minimize the cost of a job will be valuable. The methodology is built and tested for a particular class of numerical code, based upon the multilevel solution of discretized partial differential equations, and despite its simplicity it is demonstrated to be extremely accurate and robust with respect to both the processor and communications architectures considered. Furthermore, results are also presented which demonstrate that excellent predictions may also be obtained for numerical algorithms that are more general than the pure multigrid solver used to motivate the methodology. These are based upon the use of a practical parallel engineering code that is briefly described. The potential significance of this work is illustrated via two scenarios which consider a Grid user who wishes to use the available resources either (i) to obtain a particular result as quickly as possible, or (ii) to obtain results to different levels of accuracy.

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It is generally accepted that colorectal cancer is initiated in the small pits, called crypts, that line the colon. Normal crypts exhibit a regular pit pattern, similar in twodimensions to a U-shape, but aberrant crypts display different patterns, and in some cases show bifurcation. According to several medical articles, there is an interest in correlating pit patterns and the cellular kinetics, namely of proliferative and apoptotic cells, in colonic crypts. This paper proposes and implements a hybrid convection-diffusion-shape model for simulating and predicting what has been validated medically, with respect to some aberrant colonic crypt morphogenesis. The model demonstrates crypt fission, in which a single crypt starts dividing into two crypts, when there is an increase of proliferative cells. The overall model couples the cell movement and proliferation equations with the crypt geometry. It relies on classical continuum transport/mass conservation laws and the changes in the crypt shape are driven by the pressure exerted by the cells on the crypt wall. This pressure is related to the cell velocity by a Darcy-type law. Numerical Communicated by Gabriel Wittum. simulations are conducted and comparisons with the medical results are shown.

We propose a model for describing and predicting the performance of practical parallel engineering numerical software in a multi-cluster environment with different distributed memory architectures. The goal of the model is to allow reliable predictions to be made as to the execution time of a given code on a large number of processors of a given parallel system, by only benchmarking the code on small numbers of processors. The model is tested using a a practical engineering multilevel code. Despite its simplicity the model is shown to be accurate and robust with respect the cluster architectures considered.