Tractability of multivariate analytic problems

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions

Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces

Some Results on the Complexity of Numerical Integration

This is a survey (21 pages, 124 references) written for the MCQMC 2014 conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimension and on the complexity of oscillatory integrals. Some small errors of earlier versions are corrected

Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the authors derived an upper estimate for the $n$th minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-$1$ lattice rule that obtains a rate of convergence arbitrarily close to $\mathcal{O}(n^{-\alpha})$, where $\alpha>1/2$ denotes the smoothness of our function space and $n$ is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension $d$ is significantly improved

Tractability of the Fredholm problem of the second kind

We study the tractability of computing ε-approximations of the Fredholm problem of the second kind: given f ∈ Fd and q ∈ Q2d, find u ∈ L2(Id) satisfying u(x)− q(x,y)u(y)dy=f(x) ∀x∈Id =[0,1]d. Id Here, Fd and Q2d are spaces of d-variate right hand functions and 2d-variate kernels that are continuously embedded in L2(Id) and L2(I2d), respectively. We consider the worst case setting, measuring the approximation error for the solution u in the L2 (I d )-sense. We say that a problem is tractable if the minimal number of information operations of f and q needed to obtain an ε- approximation is sub-exponential in ε−1 and d. One information operation corresponds to the evaluation of one linear functional or one function value. The lack of sub-exponential behavior may be defined in various ways, and so we have various kinds of tractability. In particular, the problem is strongly polynomially tractable if the minimal number of information operations is bounded by a polynomial in ε−1 for all d. We show that tractability (of any kind whatsoever) for the Fredholm problem is equivalent to tractability of the L2-approximation problems over the spaces of right-hand sides and kernel functions. So (for example) if both these approximation problems are strongly polynomially tractable, so is the Fredholm problem. In general, the upper bound provided by this proof is essentially non-constructive, since it involves an interpolator algorithm that exactly solves the Fredholm problem (albeit for finite-rank approximations of f and q). However, if linear functionals are permissible and that Fd and Q2d are tensor product spaces, we are able to surmount this obstacle; that is, we provide a fully-constructive algorithm that provides an approximation with nearly-optimal cost, i.e., one whose cost is within a factor ln ε−1 of being optimal