mjms Meridian Journal of Mathematical Sciences Meridian J. Math. Sci. 2471-3562 2471-3643 Meridian Academic Press 10.58301/sm0mpd26 Optimal error bounds for quasi-Monte Carlo rules via a variational approach https://orcid.org/0015-0000-9922-5079 Sørensen Henrik https://orcid.org/0015-0001-0025-4548 Sørensen Oskar Achebe Oluwaseun Department of Mathematics, Pacific Rim University, Australia Department of Statistics, Han River Institute for Advanced Study, South Korea School of Physics, Hanseatic University of Applied Sciences, Germany Lindqvist Gustav Academic Editor

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

 04 08 2023 9 3 126 141 12 04 2023 15 06 2023 12 07 2023 © 2023 Meridian Academic Press 2023 Meridian Academic Press https://creativecommons.org/licenses/by/4.0/

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.

 numerical analysis random regular graphs probability finite element method partial differential equations convex bodies stability Introduction

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary and let $p \in (1,\infty)$. We consider the boundary value problem1

\begin{aligned} -\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right) &= f \quad \text{in } \Omega,\\ u &= 0 \quad \text{on } \partial\Omega. \end{aligned}

Existence of weak solutions in $W_0^{1,p}(\Omega)$ 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 $f \in L^{q}(\Omega)$ with $q > d/p$. Then there is a constant $C = C(d,p,q)$ such that

\|u\|_{L^\infty(\Omega)} \le C\,|\Omega|^{\frac{1}{d}-\frac{1}{p}+\frac{1}{q}}\,\|f\|_{L^q(\Omega)}^{\frac{1}{p-1}}.

The proof combines a truncation argument with the Sobolev inequality $\|v\|_{p^*} \le S\|\nabla v\|_p$. A second estimate, for the gradient, is given by

\int_\Omega |\nabla u|^p\,dx ;\le; \sum_{k=1}^{\infty} 2^{-k}\,\lambda_k^{-1}\,\|f\|_{q}^{p'}, \qquad p' = \tfrac{p}{p-1}. 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.

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

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.

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