What is an "Optimally Transported Scheme" ? It is a numerical technique used to approximate (compute) a physical model with a computer. There exists numerous numerical techniques to do that job. An "Optimal Schemes" is the ideal scheme, proved to be optimal from an algorithmic complexity point of view. The "Optimally Transported Schemes" project tries to be as close as possible to such schemes: i.e. we are looking to the fastest, or equivalently the most accurate, numerical technique possible.

Well. Interesting. But what it is for ? From a business point of view, this project targets potentially every industry to which numerical analysis is an IT cost problem, or an impeachment toward a diversification of their product offering, due to performance issues of standard numerical schemes technology. To these industry, we propose either to optimize their computational resources, or to innovate considering opened numerical problems.

Examples of straightforward applications are located on the left panel: Investment Banking, Signal Processing,  Metrology Industry or Front Tracking.

So what is the idea behind Optimal Schemes ? Physical phenomena's may usually be described with functions or measures: mass density, momentum, etc.. A function is not understandable to a computer, because they are finite machines: a computer knows how to handle only a finite number of points. The Optimal Scheme project tries to answer and to use a fundamental construction: the best representation of a measure by a finite number of points. This is our optimality criteria, and why we believe that this approach could lead to "optimal schemes".

I am a mathematician, can you tell me more ? More precisely we are working into the field of numerical analysis of PDE (Partial Differential Equations) or SDE (Stochastic Differential Equations). These fields are worked out with tools coming from Optimal Transport Theory and Optimal Quantization.

You may wish to have a look to a first  white paper related to Optimal Schemes.

These numerical analysis techniques has been conceived primarily to tackle high-dimensional problems coming from stochastic considerations arising in mathematical finance, see Investment Banking Applications, but applies to a wider class of problems.

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ast modified: 03/14/09