Digital inversive vectors can achieve strong polynomial tractability for the weighted star discrepancy and for multivariate integration
For researchers in high-dimensional numerical integration, this work demonstrates that pseudorandom vectors can provably achieve tractability, addressing the practical need for deterministic methods.
The paper shows that digital inversive vectors can achieve strong polynomial tractability for the weighted star discrepancy and for multivariate integration of certain function classes, providing a deterministic alternative to random nodes.
We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not explode exponentially with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier- and cosine series, where the proposed method is successful.