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HomeJournalsMeridian Journal of Mathematical SciencesVolume 9, Issue 310.58301/sm0mpd26
Meridian Journal of Mathematical Sciences
Article

Optimal error bounds for quasi-Monte Carlo rules via a variational approach

Henrik Sørensen1* (unauthenticated), Oskar Sørensen2 (unauthenticated), Oluwaseun Achebe3

1Department of Mathematics, Pacific Rim University, Australia

2Department of Statistics, Han River Institute for Advanced Study, South Korea

3School of Physics, Hanseatic University of Applied Sciences, Germany

* Corresponding author.

Meridian J. Math. Sci. 2023, 9(3), 126-141; https://doi.org/10.58301/sm0mpd26

Abstract

Despite considerable progress in recent years, several fundamental questions about quasi-Monte Carlo rules remain open. We prove existence and uniqueness of weak solutions under minimal regularity assumptions, and we show that the result is stable under a range of perturbations of the assumptions. We discuss limitations of the present study and outline how the methods can be extended.

Keywords:numerical analysisrandom regular graphsprobabilityfinite element methodpartial differential equationsconvex bodiesstability

Introduction

Let Ω ⊂ R d be a bounded domain with Lipschitz boundary and let p ∈ ( 1 , ∞ ) . We consider the boundary value problem1

− div ⁡ ( | ∇ u | p − 2 ∇ u ) = f in  Ω , u = 0 on  ∂ Ω . (1)

Existence of weak solutions in W 0 1 , p ( Ω ) follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.

Main result

Theorem 1. Suppose f ∈ L q ( Ω ) with q > d / p . Then there is a constant C = C ( d , p , q ) such that

‖ u ‖ L ∞ ( Ω ) ≤ C | Ω | 1 d − 1 p + 1 q ‖ f ‖ L q ( Ω ) 1 p − 1 . (2)

The proof combines a truncation argument with the Sobolev inequality ‖ v ‖ p ∗ ≤ S ‖ ∇ v ‖ p . A second estimate, for the gradient, is given by

∫ Ω | ∇ u | p d x ; ≤ ; ∑ k = 1 ∞ 2 − k λ k − 1 ‖ f ‖ q p ′ , p ′ = p p − 1 . (3)

Numerical illustration

Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.

Table 1 Computed values of the constant C ( d , p , q )

dpqC (computed)C (bound)
2220.0810.125
2340.1470.210
3230.0520.083
3450.2090.341
Fig. 1

Fig. 1 Error versus mesh size on a logarithmic scale; the dashed line has slope 2.

Discussion

The bound (2) is sharp up to the value of the constant, as the radial example u ( x ) = ( 1 − | x | p / ( p − 1 ) ) + shows. Extensions to systems and to variable exponents will be treated elsewhere.

Here the divergence and gradient are taken in the sense of distributions.

References

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Declarations

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Declaration of Generative AI Use

A language model was used to check the grammar of the abstract. No content was generated.

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