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We introduce a near-linear complexity (geometric and meshless/algebraic)multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficientswith rigorous a-priori accuracy and performance estimates. The method isdiscovered through a decision/game theory formulation of the problems of (1)identifying restriction and interpolation operators (2) recovering a signalfrom incomplete measurements based on norm constraints on its image under alinear operator (3) gambling on the value of the solution of the PDE based on ahierarchy of nested measurements of its solution or source term. The resultingelementary gambles form a hierarchy of (deterministic) basis functions of$H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbandswith respect to the scalar product induced by the energy norm of the PDE (2)enable sparse compression of the solution space in $H^1_0(\Omega)$ (3) inducean orthogonal multiresolution operator decomposition. The operating diagram ofthe multigrid method is that of an inverted pyramid in which gamblets arecomputed locally (by virtue of their exponential decay), hierarchically (fromfine to coarse scales) and the PDE is decomposed into a hierarchy ofindependent linear systems with uniformly bounded condition numbers. Theresulting algorithm is parallelizable both in space (via localization) and inbandwith/subscale (subscales can be computed independently from each other).Although the method is deterministic it has a natural Bayesian interpretationunder the measure of probability emerging (as a mixed strategy) from theinformation game formulation and multiresolution approximations form amartingale with respect to the filtration induced by the hierarchy of nestedmeasurements.

Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games