Exponential Decay
Houman Owhadi, Clint Scovel · Cambridge University Press eBooks · 2019
This chapter establishes the exponential decay of gamblets under an appropriate notion of distance derived from subspace decomposition in a way that generalizes domain decomposition in the computation of PDEs. The first steps present sufficient conditions for localization based on a generalization of the Schwarz subspace decomposition and iterative correction method introduced by Kornhuber and Yserentant and the LOD method of Malqvist and Peterseim. However, when equipped with nonconforming measurement functions, one cannot directly work in the primal space, but instead one has to find ways to work in the dual space. Therefore, the next steps present necessary and sufficient conditions expressed as frame inequalities in dual spaces that, in applications to linear operators on Sobolev spaces, are expressed as Poincaré, inverse Poincaré, and frame inequalities.