Stanford 50: State of the Art and Future Directions of Computational Mathematics and Numerical Computing


  • March 30, 2007
  • 10:15 am - 10:40 am

A best approximation problem with application to parallel computing

Martin Gander (Université de Genève)

The classical best approximation problem is the following: given a real-valued continuous function on a compact interval and a class of functions defined on the same interval, find an element in the class that realizes the distance of the function to the class. If the class is the linear space of polynomials of degree less than or equal to n, and the distance is measured in the L-∞ norm, then the approximation problem is called a Chebyshev best approximation problem.

We are interested in a best approximation problem in a more general setting: we search for a given function f: C → C the polynomial sn* of degree less than or equal to n that minimizes over all s of degree less than or equal to n the quantity

sup z ∈ K |(s(z) - f(z)) / (s(z) + f(z)) exp(-l f(z))|

where K is a compact set in C, and l is a non-negative real parameter. The solution of this best approximation problem is important in parallel computing: it leads to the fastest iterative domain decomposition methods.

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