Sparse Polynomial Approximation of High-Dimensional Functions

The front matter includes the title page, series page, copyright page, dedication, TOC, preface, notation, abbreviations

In this chapter, we introduce the main ideas and themes of this book.

In this chapter, we introduce the key tools for multivariate polynomial approximation that will be used throughout the remainder of the book.

In Chapter 2 we introduced the main concepts behind sparse high-dimensional approximation via orthogonal polynomials. In this chapter, we present a series of convergence rates for the best s-term approximation error, introduced previously in §2.4. We also discuss rates of convergence for the two modifications introduced therein: namely, best s-term approximation in lower sets and weighted best (k;w)-term approximation. Throughout, our main assumption is the existence of a holomorphic extension of the target function f, in the sense made precise in Assumption 2.3.

In Chapter 3, we presented different types of best s-term decay rates for the approximation of high- or infinite-dimensional functions via Chebyshev and Legendre polynomials. These results rely on the assumption that the target function admits a holomorphic extension to an open neighborhood of a filled-in Bernstein polyellipse. The aim of this chapter is to illustrate how this assumption naturally arises in applications where the function of interest is the solution map (or a quantity of interest) of a parametric model involving DEs.

In Chapter 3 we identified conditions under which a high-dimensional function can be approximated with a certain rate by its best s-term expansion in an orthonormal polynomial basis. We now turn our attention to the task of computing approximations from a finite number of sample values. Following the dichotomy introduced in §1.7, in this chapter we deal with the case of known anisotropy, in which a target set of multi-indices S is known. Our focus is on least-squares approximation.

In this chapter, we present a short introduction to compressed sensing theory, focusing in particular on those aspects most relevant to polynomial approximation.

In this chapter, we use compressed sensing to construct polynomial approximations of smooth functions. In §7.1 we first demonstrate how to recast the polynomial approximation problem as a compressed sensing problem. Although this was already done in §6.1, we now no longer assume f is a sparse polynomial. Next, we discuss error analysis. A key component of this is the truncation error, which arises from restricting the infinite expansion of f to a finite expansion in the large, finite index set Λ.

High-dimensional approximation using concepts such as sparsity and best s-term approximation is a thriving area of research. In this final chapter, we briefly describe several further lines of investigation in this area, as well as a number of open problems.

We have presented various numerical examples throughout this book. As noted in §1.9.2, we elected to discuss the results of these experiments at a high level and therefore deliberately avoided explaining too many of the technical details in the individual chapters. The role of this appendix is to give the reader more technical information about the numerical experiments and the rationale behind them. We recall that all the figures can be reproduced using the MATLAB code available on the book's companion website:

www.sparse-hd-book.com

In this appendix, we provide a number of bounds and asymptotic estimates for cardinality of the total degree and hyperbolic cross index sets introduced in §2.3. We present the proofs for several of these results.

The back matter includes bibliography, index, and back cover.