Bringing computer simulations closer to reality

This year, 16 mathematicians have been awarded support as part of the Mathematics Program funded by the Knut och Alice Wallenberg Foundation. One of them is Stefano De Marchi.

Scientific computing deals with the development of efficient and safe methods for computer simulations of the reality that surrounds us. The world is too complex to be exactly described, but mathematical models can be used to calculate approximations for various phenomena. The aim of the project is to bring the art of approximation one step closer to reality.

Some things are easy to approximate. For example, the surface of a lake can be approximated using just its average height above sea level and ignoring the small waves and eddies that disturb the surface. Rapid changes in wind speed are more difficult to describe, as they vary from zero at the Earth’s surface to almost constant about 100 metres up. In between, the speed changes in a fairly arbitrary manner. If wind speeds were to vary approximately the same as for height, these phenomena could be approximated using established methods – radial basis functions. However, these are inadequate for describing rapid local variations, such as how the wind shifts across the Earth’s surface or the way sound levels decrease further from their source.

Investigating rational approximations

Instead, the project will investigate rational approximations. These also use radial basis functions, but the numerator and denominator can be changed separately by dividing approximations by each other. This makes the approximations much more flexible. If these rational approximation methods prove successful, the next step will be to create algorithms for computer simulations that are fast enough to be performed in real time, or even faster. There are many potential applications, from better climate models to more efficient machine learning.

Annica Hulth