# Tractability of Multivariate Problems: Volume II: Standard Information for Functionals

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*Erich Novak; Henryk Woźniakowski*

A publication of the European Mathematical Society

This is the second volume of a three-volume set comprising a
comprehensive study of the tractability of multivariate problems. The
second volume deals with algorithms using standard information
consisting of function values for the approximation of linear and
selected nonlinear functionals. An important example is numerical
multivariate integration.

The proof techniques used in volumes I and II are quite different. It is
especially hard to establish meaningful lower error bounds for the
approximation of functionals by using finitely many function values.
Here, the concept of decomposable reproducing kernels is helpful, allowing it
to find matching lower and upper error bounds for some linear functionals. It
is then possible to conclude tractability results from such error bounds.

Tractability results, even for linear functionals, are very rich in variety.
There are infinite-dimensional Hilbert spaces for which the approximation with
an arbitrarily small error of all linear functionals requires only one function
value. There are Hilbert spaces for which all nontrivial linear functionals
suffer from the curse of dimensionality. This holds for unweighted spaces,
where the role of all variables and groups of variables is the same. For
weighted spaces one can monitor the role of all variables and groups of
variables. Necessary and sufficient conditions on the decay of the weights are
given to obtain various notions of tractability.

The text contains extensive chapters on discrepancy and integration,
decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms,
lattice rules and the CBC (component-by-component) algorithms. This is done in
various settings. Path integration and quantum computation are also
discussed.

This volume is of interest to researchers working in computational
mathematics, especially in approximation of high-dimensional problems. It is
also well suited for graduate courses and seminars. There are 61 open problems
listed to stimulate future research in tractability.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in computational mathematics.