This book is the second volume of a three volume set comprising study of the tractability of multivariate problems. Recall that in the first volume the authors primarily studied problems specified by linear operators and and algorithms that use arbitrary linear information given by arbitrary linear functionals.
In the second volume the authors study multivariate problem specified by linear functionals and a few selected nonlinear functionals, and algorithms that use standard information given by function values. More precisely, Volume II consists of twelve chapters (numbered from 9 till 20). The book starts with a notion of discrepancy and integration in Chapter 9. In Chapter 10 general linear functionals are studied, whereas in Chapter 11 the idea of decomposable kernels is explained and lower error bounds presented. This also serves as a motivation for switching to weighted spaces in Chapter 12. In Chapter 13 is analyzed average case setting, while in Chapter 13 probabilistic setting is studied. The relations between the average case, probabilistic case and worst case are covered in Chapter 15 and Chapter 16, concentrating especially on the worst case. Chapter 16 deals also with the integration, emphasis being on constructive lattice rules. Chapter 17 turns back to the probabilistic rules and the influence of the dimension is discussed.Tractability conditions on weights for the standard Monte carlo algorithms are given and compared with other situations. In Chapter 18 tractability of a few selected nonlinear functionals in the worst case and randomized setting are studied. Finally, in Chapter 19 the authors briefly mention two generalizations of the material of previous chapters, i.e. switching from finite \(d\) to \(d=\infty\). The point is illustrated on path integration and integration over a Sobolev space of functions depending on infinitely many variables. In Chapter 20 a summary of tractability results is presented for multivariate integration over three standard weighted Sobolev spaces.