Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher Order Convergence

Abstract

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.

References in corpus (4)

• Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order
• Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules
• The decay of the Walsh coefficients of smooth functions
• Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition

Cited by in corpus (11)

• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
• Embeddings of Weighted Hilbert Spaces and Applications to Multivariate and Infinite-Dimensional Integration
• Embeddings for Infinite-Dimensional Integration and $L_2$-Approximation with Increasing Smoothness
• MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM
• Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration
• Infinite-dimensional integration and $L^2$-approximation on Hermite spaces
• MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM
• Hyperbolic cross approximation in infinite dimensions
• Some Results on the Complexity of Numerical Integration
• Infinite-Variate $L^2$-Approximation with Nested Subspace Sampling