The introduction reviews, summarizes, and illustrates fundamental connections among Bayesian inference, numerical quadrature, Gausssian process regression, polyharmonic splines, information-based complexity, optimal recovery, and game theory that form the basis for the book. This is followed by describing a sample of the results derived from these interplays; including those in numerical homogenization, operator-adapted wavelets, fast solvers, and Gaussian process regression. It finishes with an outline of the structure of the book.

By representing the operator as independent, sparse, well-conditioned linear systems, theoperator-adapted wavelet (gamblet) transform of the previous chapter naturally leads to a scalable linear solver with some degree of universality. The near-linear complexity of the solver and the Fast Gamblet Transform are based on the nesting, exponential localization, and Riesz stability of the underlying wavelets. The representation of the Green's function in the basis formed by these wavelets is sparse and rank-revealing.The algorithm isillustrated through a numerical application toa second-order divergence form elliptic operator with rough coefficients.