[Paper Review] On efficient weighted integration via a change of variables

Peter Kritzer, Friedrich Pillichshammer+2 more|arXiv (Cornell University)|Dec 11, 2018

Mathematical Approximation and Integration11 references4 citations

TL;DR

This paper proposes a novel change of variables for efficient quasi-Monte Carlo (QMC) and sparse grid integration of weighted integrals over unbounded domains such as $\mathbb{R}^d$ or $\mathbb{R}_+^d$. Unlike standard inverse CDF-based transformations that induce boundary singularities, the authors design a transformation $\nu$ that maps the unit cube to the unbounded domain such that the transformed integrand lies in the Sobolev space $W_{d,p}$ with mixed smoothness of order one, ensuring small worst-case error for QMC and sparse grid methods. The key contribution is a sufficient condition on $\nu$ that guarantees finite approximation error constants $C_{d,p}(\nu)$, enabling effective integration for $p \in [1, \infty]$. The method is validated numerically and outperforms standard approaches.

ABSTRACT

In this paper, we study the approximation of $d$-dimensional $ρ$-weighted integrals over unbounded domains $\mathbb{R}_+^d$ or $\mathbb{R}^d$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $L_p$ norm of mixed partial derivatives of first order, where $p\in[1,+\infty].$ The main results give sufficient conditions on the change of variables $ν$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $ρ$ and $p$. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

Motivation & Objective

• To address the inefficacy of standard inverse CDF-based changes of variables in QMC and sparse grid methods for weighted integrals over unbounded domains.
• To identify sufficient conditions on a change of variables $\nu$ such that the transformed integrand belongs to the Sobolev space $W_{d,p}$ with mixed smoothness of order one.
• To ensure the worst-case error of QMC and sparse grid rules remains bounded via a finite constant $C_{d,p}(\nu)$, enabling efficient integration for $p \in [1, \infty]$.
• To demonstrate through numerical experiments that the proposed transformation outperforms the standard inverse CDF method, especially for $p > 1$.

Proposed method

• The method transforms $d$-dimensional $\varrho$-weighted integrals over unbounded domains $D^d$ into standard Lebesgue integrals over the unit cube $B^d$ via a change of variables $x_j = \nu(t_j)$, where $\nu: B \to D$ is a differentiable bijection.
• The transformed integrand is defined as $g_{f,\nu}(\mathbf{t}) = f(\nu(t_1), \dots, \nu(t_d)) \cdot \prod_{j=1}^d \left( \varrho(\nu(t_j)) \cdot \nu'(t_j) \right)$, ensuring measure preservation.
• Sufficient conditions on $\nu$ are derived to guarantee $g_{f,\nu} \in W_{d,p}$ for all $f \in F_{d,p}$, where $F_{d,p}$ is the space of functions with bounded $L_p$ norm of mixed first-order partial derivatives.
• The analysis shows that the standard inverse CDF transformation fails for $p > 1$ because it induces boundary singularities, leading to $C_{d,p}(\nu) = \infty$, whereas the proposed $\nu$ avoids this issue.
• The method is extended to infinite-dimensional settings and $\gamma$-weighted spaces by analyzing the behavior of $C_{d,p}(\nu)$ and its product structure across subsets of variables.
• Numerical experiments use lattice rules with generating vectors from the literature to compare error decay under the proposed $\nu$ versus the standard inverse CDF transformation.

Experimental results

Research questions

• RQ1 Can a change of variables be designed such that the transformed integrand lies in the Sobolev space $W_{d,p}$ with mixed smoothness of order one, even when the original integrand is only of regularity one?
• RQ2 Why does the standard inverse CDF-based transformation fail for $p > 1$ in the context of QMC and sparse grid methods?
• RQ3 What conditions on the transformation $\nu$ ensure that the induced error constant $C_{d,p}(\nu)$ is finite and small, enabling efficient integration?
• RQ4 How does the proposed transformation compare in practice to the standard inverse CDF method in terms of convergence rate and absolute error?
• RQ5 Can the method be extended to infinite-dimensional weighted integration problems with $\gamma$-weights?

Key findings

• The standard inverse CDF-based change of variables leads to $C_{d,p}(\nu) = \infty$ for all $p > 1$, rendering QMC and sparse grid methods ineffective due to boundary singularities in the transformed integrand.
• The proposed change of variables $\nu$ satisfies sufficient conditions that guarantee $g_{f,\nu} \in W_{d,p}$ for all $f \in F_{d,p}$, ensuring finite and small $C_{d,p}(\nu)$, which bounds the worst-case error of QMC and sparse grid rules.
• For the test function $f_d(\mathbf{x}) = \prod_{j=1}^d x_j$ on $\mathbb{R}_+^d$ with $\varrho(x) = e^{-x}$, the proposed transformation with $a = a^* \approx 2.4557$ yields significantly faster error decay than the standard inverse CDF ($a=1$) or suboptimal choices ($a=1.5$).
• With $d=3$, the error for $n=2^{15}$ points drops to $2.05 \times 10^{-7}$ under the optimal $a^*$, compared to $3.26 \times 10^{-3}$ for $a=1$, showing a 3-order-of-magnitude improvement.
• For $d=4$, the error under $a^*$ decreases to $2.60 \times 10^{-6}$ at $n=2^{15}$, while the standard method with $a=1$ still shows errors above $3 \times 10^{-4}$, indicating robustness and superiority of the proposed transformation.
• The method enables efficient approximation of infinite-dimensional weighted integrals with total cost $O(\varepsilon^{-1})$ function evaluations to achieve error $\varepsilon$, as shown in the context of the Multilevel Monte Carlo method.