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This chapter generalizes and extends the development of operator-adapted wavelets (gamblets) and their resulting multiresolution decompositions from Sobolev spaces to Banach spaces equipped with a quadratic norm and a nonstandard dual pairing. The fundamental importance of the Schur complement is elucidated and the geometric nature of gamblets is presented from two views: one regarding basis transformations derived from the nesting, and the other the linear transformations associated with these basis transformations. A table of gamblet identities is presented.
This chapter extends the presentation of Gaussian measures, cylinder measures, and fields from Sobolev spaces to Banach spaces with quadratic energy norm. The relationship between weak distributions and cylinder measures is elucidated as is the relationship between Gaussian cylinder measures and Gaussian fields.
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.
At the cost of some redundancy, to facilitate accessibility, the multiresolution decomposition and inversion of symmetric positive definite (SPD) matrices on finite-dimensional Euclidean space are developed in the Gamblet Transform and Decomposition framework.
By representing the operator as independent, sparse, well-conditioned linear systems, the operator-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 is illustrated through a numerical application to a second-order divergence form elliptic operator with rough coefficients.
Hierarchical optimal recovery games are defined using a hierarchy of measurement functions. The sequence of optimal mixed minmax solutions is shown to be a martingale. Sparse rank-revealing representations of Gaussian fields are established.
This chapter introduces optimal recovery games on Banach spaces, presents their natural lift to mixed strategies, and then characterizes their saddle points in terms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioning with respect to linear measurements produces optimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioning the Gaussian field on the filtration formed by those measurements.