HPC Technologies New architectures
and models for programming Numerical models
and mathematical methods Laboratoire en Informatique
Haute Performance
pour Le Calcul et la Simulation

A number of observations arise out of the experience gained at the CEA’s HPC complex over the past 20 years.
Increasing user requirements have resulted in a rapid increase in demand for processing power and data management capabilities. Changing physical models are generating significantly larger flows of data from the computer to the storage and visualisation systems. Moore’s law on the rate of technological advancement in microprocessors is no longer yielding increases in processor frequency, but increases in the number of cores per processor.
The unit calculation capacity remains more or less constant, only parallelism makes it possible to increase the processing capacity of a machine.

Calculation software (codes) are the translation, through numerical algorithms, of the mathematical formulations of the physical models studied. Upstream and downstream of the calculation, the environment software manages several complex calculation preparation and analysis operations.

The numerical simulation of complex multi-physical phenomena, while respecting the scales in space and time, requires numerous calculations, which use and generate large volumes of data, on power computers: this is high performance computing.
The power of individual processors remains essentially constant; currently, the only way to increase a computer’s processing power is to scale the number of processors.
Clearly, developing future generations of supercomputers calls for disruptive technologies, in particular in the area of electric power consumption management. To this end, the CEA decided to adopt a pro-active co-design methodology for future computers, alongside Atos/Bull, with which the CEA has built up a partnership over more than 15 years.

The agreement between CEA and Atos/Bull to develop an exascale supercomputer by 2020 sets out the R&D objectives that must be achieved in order to address major challenges in areas such as energy performance, operating reliability and mass interconnection.

When working with high-performance numerical simulation, developing a new program code is not simply a case of writing a set of programming instructions. The program designer must also take account of the architecture of the massively parallel supercomputers, so that the calculations are efficiently distributed over the hundreds, or even thousands, of processors and the exchange of information is optimised.
Mastering the complexity of the modelling, the mathematical methods and algorithms, the relevant computer science and the validation techniques as well as the software production are all key factors in the success of a new numerical simulation project.

Numerical models and mathematical methods

Together with the increase in computing power (Moore's Law), the mathematical and numerical techniques used to resolve the equations from physical models have progressed greatly, despite the many difficulties related to the complexity and non-linearity of these models.

Examples include the methods for hyperbolic systems and gas dynamics, for transport and diffusion equations and for coupled systems. New techniques have been able to improve precision whilst preserving robustness. Dynamic adaptation of meshes, by refining or unrefining, is a good example. Front-capture techniques are another example, without forgetting the various finite element methods. For transport equations, improvements relate to both the statistical (Monte Carlo) and deterministic approaches. We could also consider the algorithms for efficient resolution of large linear systems, or the techniques for evaluating the uncertainties and sensitivity of the simulation parameters.

This progress in numerical models and mathematical methods comes from a continuous research effort aimed at applied mathematics, numerical analysis and computer science. This research is performed in cooperation with the various research communities concerned: university laboratories, other higher education institutes and research organisations. This can lead to more formal collaborations. The research is illustrated by internationally reviewed publications and conference presentations. It is also manifest in the supervision of trainees (Masters and third year engineering students), graduate students and postdoctoral researchers.