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.

This chapter generalizes and extends the treatment of optimal recovery splines from Sobolev spaces to Banach spaces equipped with quadratic norm and nonstandard dual pairing.

Hierarchical optimalrecovery 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 interms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioningwith respect to linear measurements producesoptimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioningthe Gaussian field on the filtration formed by those measurements.

This chapter presents the theory of optimal recovery in the setting of Sobolev spaces andthe context of information-based complexity. It also describes optimal recovery splines, their variational properties, and their minmax optimality characterization.

This chapter bounds the condition numbers of thestiffness matrix of operator-adapted wavelets within each subband (scale). These resulting bounds are characterized through weak alignment conditions between measurement functions and eigensubspaces of the underlying operator. In Sobolev spaces, these alignment conditions translate into approximate error estimates associated with variational splines andscattered data approximation. These estimates are established for the three primary examples, subsampled Diracs, Haar prewavelets, and local polynomials,of hierarchies of measurement functions in Sobolev spaces.

The game theoretic interpretation of gamblets is extended from Sobolev spacesto the Banach space setting. Identities for conditional covariances are presented in this generalized setting.

This chapter reviews and summarizesapplications of gamblets to the compression, inversion, and approximate PCA of dense kernel matrices.

This chapter introduces two-person zero-sum games, optimal recovery games, and their lifts tomixed extended games and defines saddle points and minmax solutions.The optimal mixed strategy for the mixed extension of the optimal recovery games is generated by conditioning the canonical Gaussian field associated with the energy norm. Since the dependence ofthese optimal solutions on the measurement functions is through the conditioning process only, the canonical Gaussian field is referred to as a universal field. This fact demonstrates that the optimal solutions generated from a nested hierarchy of measurement functions form a martingale.

This chapter transitions the presentation to Banach spaces equipped with a quadraticnorm defined by a symmetric positive linear operator. Basic terminology and results are established (using a representation of the dual product that is distinct from the one obtained from the Riesz representation associated with such Banach spaces).

Wavelets adapted to a given self-adjoint elliptic operator are characterized by the requirement that they block-diagonalize the operator intouniformly well-conditioned and sparse blocks. These operator-adapted wavelets (gamblets) are constructed as orthogonalized hierarchies of nested optimal recovery splines obtained fromclassical/simple prewavelets(e.g., ~Haar) used as hierarchies of measurement functions. The resulting gamblet decomposition of an element in a Sobolev space is described andanalyzed.