in \[sec:contcase\] to the numerical homogenization of and the discovery of interpolation interpolators. In particular, it will be shown that the resulting elementary gambles form a set of deterministic basis functions (gamblets) characterized by (1) optimal recovery and accuracy properties (2) exponential decay (enabling their localized computation) (3) robustness to high contrast.

To compute fast, the game presented above must not be limited to filling the information gap between $u_m\in \R^m$ and $u\in H^1_0(\Omega)$. This game must be played (and repeated) over hierarchies of levels of complexity (e.g. one must fill information gaps between $\R^{4}$ and $\R^{16}$, then $\R^{16}$ and $\R^{64}$, etc...). We will therefore, in Section \[sechnh\], consider the (hierarchical) game where Player I chooses the r.h.s of and Player II must (iteratively) gamble on the value of its solution based on a hierarchy of nested measurements of $u$ (from coarse to fine measurements). Under Player II’s mixed strategy (identified in Section \[sec:problem\] and used in Section \[sec:contcase\]), the resulting sequence of multi-resolution approximations forms a martingale. Conditioning and the independence of martingale increments lead to the hierarchy of nested interpolation operators and to the multiresolution orthogonal decomposition of into independent linear systems of uniformly bounded condition numbers. The 