We study the numerical integration problem for functions with infinitely many variables which stem from a weighted reproducing kernel Hilbert space. We study the worst case ε \varepsilon -complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most ε \varepsilon . Here we assume that the cost of evaluating a function depends polynomially on the number of active variables. The infinite-dimensional integration problem is tractable if the ε \varepsilon -complexity is bounded by a constant times a power of 1 / ε 1/\varepsilon . The smallest such power is called the exponent of tractability. We provide improved lower bounds for the exponent of tractability for general finite-order weights and, with the help of multilevel algorithms, improved upper bounds for three newly defined classes of finite-order weights. The newly defined finite-intersection weights model the situation where each group of variables interacts with at most ρ \rho other groups of variables, ρ \rho some fixed number. For these weights we obtain sharp upper bounds for any decay of the weights and any polynomial degree of the cost function. For the other two classes of finite-order weights our upper bounds are sharp if, e.g., the decay of the weights is fast or slow enough. Furthermore, we deduce a lower bound for the exponent of tractability for arbitrary weights and a constructive upper bound for product weights.
We study the numerical integration problem for functions with infinitely many variables which stem from a weighted reproducing kernel Hilbert space. We study the worst case ε \varepsilon -complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most ε \varepsilon . Here we assume that the cost of evaluating a function depends polynomially on the number of active variables. The infinite-dimensional integration problem is tractable if the ε \varepsilon -complexity is bounded by a constant times a power of 1 / ε 1/\varepsilon . The smallest such power is called the exponent of tractability. We provide improved lower bounds for the exponent of tractability for general finite-order weights and, with the help of multilevel algorithms, improved upper bounds for three newly defined classes of finite-order weights. The newly defined finite-intersection weights model the situation where each group of variables interacts with at most ρ \rho other groups of variables, ρ \rho some fixed number. For these weights we obtain sharp upper bounds for any decay of the weights and any polynomial degree of the cost function. For the other two classes of finite-order weights our upper bounds are sharp if, e.g., the decay of the weights is fast or slow enough. Furthermore, we deduce a lower bound for the exponent of tractability for arbitrary weights and a constructive upper bound for product weights.
Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation. Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor ε \varepsilon is bounded by C ε − p C \varepsilon ^{-p} for some exponent p p and some positive constant C C . The ε \varepsilon -exponent of tractability is defined as the smallest power of ε − 1 \varepsilon ^{-1} in these bounds. It is shown by using Monte Carlo quadrature that the ε \varepsilon -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the ε \varepsilon -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.
We study lower bounds on the worst-case error of numerical integration in tensor product spaces. As reference we use the N-th minimal error of linear rules that use N function values. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error. We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality. Under the assumption of the existence of a worst-case function for the uni-variate problem we present two methods for providing good lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function. This method can be seen as a generalization of the method of decomposable reproducing kernels, that is often successfully applied when integration in Hilbert spaces with a reproducing kernel is studied. The second method, although only applicable for positive quadrature rules, has the advantage, that it does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. The methods presented can be applied to problems beyond the Hilbert space setting. For demonstration purposes we apply them to several examples, notably to uniform integration over the unit-cube, weighted integration over the whole space, and integration of infinitely smooth functions over the cube. Some of these results have interesting consequences in discrepancy theory.
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first nontrivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or nonlinear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter, J. Complexity, 26 (2010), pp. 229--254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi--Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first nontrivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or nonlinear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter, J. Complexity, 26 (2010), pp. 229--254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi--Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.
Multi- and Infinite-variate Integration and L2-Approximation on Hilbert Spaces with Gaussian Kernels
We study integration and L2-approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor product form. In the infinite-variate case, for both computational problems, we establish matching upper and lower bounds for the polynomial convergence rate of the n-th minimal error. In the multivariate case, we improve several tractability results for the integration problem. For the proofs, we establish the following transference result together with an explicit construction: Each of the computational problems on a space with a Gaussian kernel is equivalent on the level of algorithms to the same problem on a Hermite space with suitable parameters.
We study integration and L 2 L^2 -approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor product form. In the infinite-variate case, for both computational problems, we establish matching upper and lower bounds for the polynomial convergence rate of the n n -th minimal error. In the multivariate case, we improve several tractability results for the integration problem. For the proofs, we establish the following transference result together with an explicit construction: Each of the computational problems on a space with a Gaussian kernel is equivalent on the level of algorithms to the same problem on a Hermite space with suitable parameters.