Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis Houman Owhadi California Institute of Technology ACM We show how, the discovery/design of robust scalable numerical solvers for arbitrary bounded linear operators, can, to some degree, be addressed/automated as a Game/Decision Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space, BB, is a Banach space endowed with a quadratic norm ∥⋅∥‖⋅‖, optimal mixed and pure strategies coincide and are obtained by conditioning Gaussian fields on BB with respect to computable information and elementary gambles/bets (gamblets) form a multi-resolution basis of BB. These gamblets generalize the notion of Wavelets and Wannier basis functions in the sense that they are adapted to the norm ∥⋅∥‖⋅‖ and induce a multi-resolution decomposition of BB that is adapted to the eigensubspaces of the operator Q:B∗→BQ:B∗→B defining the norm ∥⋅∥‖⋅‖. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of bounded linear operator (which include arbitrary bounded invertible differential operators mapping Hs0(Ω)H0s(Ω) to H−s(Ω)H−s(Ω) or Hs0(Ω)H0s(Ω) to L2(Ω)L2(Ω)) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to O(Npolylog(N))O(Npolylog⁡(N)) solvers and eigenspace adapted MRA (resulting in near linear complexity PCA, SVD, active subspace decomposition, etc...).