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.

The purpose of this chapter is to introduce the canonical Gaussian field defined bythe energy norm of the operator,which will play a central role in the interplay among the results of the previous chapters, Gaussian process regression, and game theory. The chapter begins witha presentation of basic definitions and results related to Gaussian random variables, Gaussian vectors, Gaussian spaces, Gaussian conditioning, Gaussian processes, Gaussian measures, and Gaussian fields.

The computation ofgamblets is accelerated by localizing their computation in a hierarchical manner (using a hierarchy of distances), and the approximation errors caused by these localization steps are bounded based on three properties: nesting, the well-conditioned nature of the linear systems solved in the Gamblet Transform, and theexponential decay of the gamblets. These efficiently computed, accurate, andlocalized gamblets are shown to producea Fast Gamblet Transform of near-linear complexity. Application to the three primary classes ofmeasurement functions in Sobolev spaces are developed.

This chapter reviews classical homogenizationconcepts such as the cell problem; correctors; compactness by compensation; oscillating test functions; H, G, and Gamma convergence; and periodic and stochastic homogenization. Numerical homogenization is presented as the problem of identifying basis functions that are both as accurate and as localized as possible. Optimal recovery splines constructed from simple measurement functions (Diracs, indicator functions, and local polynomials) provide a simple to solution to this problem: they achieve the Kolmogorov n-width optimal accuracy (up to a constant) and they are exponentially localized. Current numerical homogenization methods are reviewed. Gamblets, the LOD method, the variational multiscale method, andpolyharmonic splines are shown to have a common characterization as optimal recovery splines.

After a brief discussion of invariant measures and entropy, we introduce semidynamical and dynamical systems. We use Koopmanism to show how to obtain semigroups/groups of linear operators on function spaces, when we have a quasi-invariant measure. Applications are given to solution flows of differential equations. In the last part we discuss “dilations” as a mathematical approach to the origins of irreversibility.

We survey compact, trace class, and Hilbert–Schmidt operators. Mercer’s theorem is discussed and applied to convolution semigroups on the circle. Density operators and the quantum Liouville equation for mixed states in quantum theory are introduced.