Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
  We show how the discovery of robust scalable numerical solvers for arbitrary
bounded linear operators can be automated as a Game 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 is a Banach space $B$ endowed with a quadratic
norm $\|\cdot\|$, the optimal measure (mixed strategy) for such games (e.g. the
adversarial recovery of $u\in B$, given partial measurements $[\phi_i, u]$ with
$\phi_i\in B^*$, using relative error in $\|\cdot\|$-norm as a loss) is a
centered Gaussian field $\xi$ solely determined by the norm $\|\cdot\|$, whose
conditioning (on measurements) produces optimal bets. When measurements are
hierarchical, the process of conditioning this Gaussian field produces a
hierarchy of elementary bets (gamblets). These gamblets generalize the notion
of Wavelets and Wannier functions in the sense that they are adapted to the
norm $\|\cdot\|$ and induce a multi-resolution decomposition of $B$ that is
adapted to the eigensubspaces of the operator defining the norm $\|\cdot\|$.
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 continuous linear operators
(including arbitrary continuous linear bijections from $H^s_0$ to $H^{-s}$ or
to $L^2$) into a sequence of independent linear systems with uniformly bounded
condition numbers and leads to $\mathcal{O}(N \operatorname{polylog} N)$
solvers and eigenspace adapted Multiresolution Analysis (resulting in near
linear complexity approximation of all eigensubspaces).
