energy scalar product and over which has uniformly bounded condition numbers. It is, to some degree, surprising that this decomposition can be achieved in near linear complexity and not in the complexity of an eigenspace decomposition. Naturally [@OwZh:2016], this decomposition can be applied to the fast simulation of the wave and parabolic equations associated to or to its fast diagonalization.

The essential step behind the automation of the discovery/design of scalable numerical solvers is the observation that fast computation requires repeated computation with partial information (and limited resources) over hierarchies of levels of complexity and the reformulation of this process as that of playing underlying hierarchies of adversarial information games [@VNeumann28; @VonNeumann:1944].

Although the problem of finding a fast solver for may appear disconnected from that of finding statistical estimators or making decisions from data sampled from an underlying unknown probability distribution, the proposed game theoretic reformulation is, to some degree, analogous to the one developed in Wald’s Decision Theory [@Wald:1945], evidently influenced by Von Neumann’s Game Theory [@VNeumann28; @VonNeumann:1944] (the generalization of worst case Uncertainty Quantification analysis [@ouq2010] to sample data/model uncertainty requires an analogous game theoretic formulation [@OwhadiScovel:2015w], see also [@OwhadiScovel:2013] for how the underlying calculus could be 