Large scale computing

The increase of speed of mathematical software, based on so-called linear solvers has kept pace with the ever faster hardware, well-known as Moore’s Law. The numerical methods used today can typically be a factor 16 million faster than those used 36 years ago. A mathematical model normally consists of relations between a large number of variables or unknowns. A typical example is to solve heat problems: how is the heat distributed in a rod if we know the temperature at both ends? For a “standard” numerical method, it would typically require one ten thousandth of a second, based on present technology. For a similar three dimensional object, say a cube, the computing time would be about 4 months. This example demonstrates that straightforward approaches, even for one of the simplest problems (like this) do not work in practice.
Fortunately, mathematicians have invented much faster methods to solve a variety of problems. The increase of speed of such methods is comparable with the hardware improvement, as predicted by Moore’s Law, indeed. Using fast methods like Multi-grid the three dimensional heat problem can be solved in less than a second.

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