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.

This chapter showshow derive operator-adapted wavelets for nonsymmetric operators by (possibly nonstandard) symmetrization.

The minmax solution of the mixed extension of an optimal recovery game is evaluated in terms of the optimal recovery splines. Furthermore, the latter are given the interpretation of optimal bets in a mixed extension of the game corresponding to elementary components of the values of the measurement functions, giving rise to the interpretation of these optimal recovery splines as elementary bets, or gamblets. The screening effect is described and a rigorous proof of it established using exponential decay properties of the gamblets generated by Dirac delta measurement functions.

This chapter introduces the Sobolev spaces and self-adjointellipticoperators on those spaces that will be used through the book. It also introduces basic concepts and tools such as Gelfand triples, the Sobolev embedding theorem, the equivalence between energy norm and the Sobolev space norm, the dual norm, the Green's function, and eigenfunctions.

This chapter establishes the exponential decay of gamblets under an appropriate notion of distance derived from subspace decompositionin a way that generalizesdomain decomposition in the computation of PDEs.The first stepspresent sufficient conditions forlocalizationbased on a generalization of the Schwarz subspace decomposition and iterative correction methodintroduced by Kornhuber and Yserentantand the LOD method of Malqvist and Peterseim. However,when equipped withnonconforming measurement functions, one cannot directly work in the primal space, but instead one has to find ways to work in the dual space. Therefore, the next steps presentnecessary and sufficient conditions expressed as frame inequalities in dual spaces that, in applications to linear operators on Sobolev spaces,are expressed as Poincaré, inverse Poincaré, and frame inequalities.

This chapter reviews the application of gamblets to the opening of the complexity bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients. In this setting, the spatial component of the PDE is an elliptic operator on a Sobolev space, and since the natural truncationof the gamblet decomposition of the operator is rank-revealing, in that it is adapted to the eigensubspaces of the operator, these gamblet decompositionscan be naturally applied to the simulation of time-dependent operators. These gamblets are not only adapted to the coefficients of the underlying PDE but also to the numerical scheme used forits time discretization. For higher order PDEs these gamblets may be complex-valued.