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

Abstract

We study the numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic 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 evaluations that depend on the number of active variables of the chosen sample points, and we study two classes of weights, namely product and order-dependent weights and the newly introduced finite projective dimension weights. 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.

Appendix

Here we provide a detailed proof of Lemma 6.

Lemma 8

Let $r \!>\! 1$ be a real number, and define the POD weights $\gamma _u \!=\! \Gamma _{|u|} \prod _{j\in u} j^{-r}$ for $u \in {\mathcal {U}}$. Then there is a constant $c_r > 0$ such that

$$\begin{aligned} \sum _{u \in {\mathcal {U}}} \gamma _u \ge \Gamma _0 + c_r \sum _{k=1}^\infty \frac{\Gamma _k}{(k!)^{2 \lceil r/2 \rceil }} k^{-\lceil r/2 \rceil } \left( \frac{\pi }{2 \lceil r/2 \rceil \sin \pi / (2 \lceil r/2 \rceil ) } \right) ^{rk}. \end{aligned}$$

(61)

If $r \ge 2$, then there is a constant $C_r > 0$ such that

$$\begin{aligned} \sum _{u \in {\mathcal {U}}} \gamma _u \le \Gamma _0 + C_r \sum _{k=1}^\infty \frac{\Gamma _k}{(k!)^r} k^{-r/2} \left( \frac{\pi }{2 \lfloor r/2 \rfloor \sin \pi / (2 \lfloor r/2 \rfloor ) }\right) ^{rk}. \end{aligned}$$

(62)

Note that $\sin x < x$ for $x > 0$; thus, $\sin \pi /r < \pi /r$, which implies

$$\begin{aligned} 1 < \frac{\pi }{r \sin \pi /r}. \end{aligned}$$

Proof

We have

$$\begin{aligned}&\sum _{u \in {\mathcal {U}}} \gamma _u = \sum _{k=0}^\infty \Gamma _{k} \sum _{\mathop {\scriptstyle {|u| = k}}\limits ^{\scriptstyle {u \in {\mathcal {U}}}}} \prod _{j\in u} j^{-r} = \Gamma _0 + \sum _{k=1}^\infty \Gamma _{k} \sum _{1 \le j_1 < j_2 < \cdots < j_k} \prod _{i=1}^k j_i^{-r} \\&\quad = \Gamma _0 + \sum _{k=1}^\infty \Gamma _k \zeta (\underbrace{r,\ldots , r}_{k \text{ times }}), \end{aligned}$$

where $\zeta (\underbrace{r,\ldots , r}_{k \text{ times }})$ is the multiple Hurwitz zeta function.

The general behavior of the multiple Hurwitz zeta function is given in [8, Eq. (48)]. From [8, p. 8] it is known that if $r \ge 2$ is an even integer, then

$$\begin{aligned} \zeta (\underbrace{r,\ldots , r}_{k \text{ times }}) = \frac{r (2\pi )^{rk}}{(rk + r/2)!} \left( \frac{1}{2 \sin \pi /r} \right) ^{rk+r/2} \left( 1 + \sum _{j=2}^{N_{r}} R_{r,j}^{rk+r/2} \right) , \end{aligned}$$

where $R_{r,j}$ are some numbers with $|R_{r,j}| < 1$ and $N_{r}$ is a positive integer satisfying $N_r < 2^{r/2}/r$. From Stirling’s formula we obtain

$$\begin{aligned} \frac{(k!)^r}{(rk)!} \asymp _k \frac{\sqrt{2\pi }}{\mathrm {e}} \frac{k^{kr} \mathrm {e}^{-rk}}{(rk)^{rk} \mathrm {e}^{-rk}} = \frac{\sqrt{2\pi }}{\mathrm {e}} r^{-rk}, \end{aligned}$$

where $f(k) \asymp _k g(k)$ means that there are constants $C,c> 0$ independent of $k$ such that $c g(k) \le f(k) \le C g(k)$. Thus,

$$\begin{aligned} (k!)^r \zeta (\underbrace{r,\ldots , r}_{k \text{ times }})&= \frac{(k!)^r r (2\pi )^{rk}}{(rk + r/2)!} \left( \frac{1}{2 \sin \pi /r} \right) ^{rk+r/2} \left( 1 + \sum _{j=2}^{N_{r}} R_{r,j}^{rk+r/2} \right) \\&\quad \asymp _k \frac{\sqrt{2\pi } r}{\mathrm {e}} \left( \frac{1}{2\sin \pi /r} \right) ^{r/2} \frac{1}{(rk+r/2)^{r/2}} \left( \frac{\pi }{r \sin \pi /r} \right) ^{rk}\\&\quad \times \, \left( 1 + \sum _{j=2}^{N_{r}} R_{r,j}^{rk+r/2} \right) \asymp _k \frac{1}{k^{r/2}} \left( \frac{\pi }{r \sin \pi /r} \right) ^{rk}. \end{aligned}$$

Thus, for any fixed positive even integer $r$ we have

$$\begin{aligned} \sum _{k=1}^\infty \Gamma _k \zeta (\underbrace{r,\ldots , r}_{k \text{ times }}) = \sum _{k=1}^\infty \frac{\Gamma _k}{(k!)^r} (k!)^r \zeta (\underbrace{r,\ldots , r}_{k \text{ times }}) \asymp \sum _{k=1}^\infty \frac{\Gamma _k}{(k!)^r} k^{-r/2} \left( \frac{\pi }{r \sin \pi /r}\right) ^{rk}. \end{aligned}$$

Therefore, (61) follows since decreasing $r$ only increases the sum $\sum _{u \in {\mathcal {U}}} \gamma _u$, and the result holds for all even integers $r \ge 2$, as shown earlier.

Now assume that $r \ge 2$. For $1/r < \lambda \le 1$ we have by Jensen’s inequality that

$$\begin{aligned}{}[\zeta (r,\ldots , r)]^\lambda = \left[ \sum _{1\le j_1 < \cdots < j_k} \prod _{i=1}^k j_i^{-r} \right] ^\lambda \le \sum _{1 \le j_1 < \cdots < j_k} \prod _{i=1}^k j_i^{-r \lambda } = \zeta (r\lambda ,\ldots , r\lambda ). \end{aligned}$$

Choose $1/r < \lambda \le 1$ such that $\lambda r$ is the largest even integer less than or equal to $r$. Then

$$\begin{aligned} (k!)^r \zeta (r,\ldots , r) \le \left[ (k!)^{\lambda r} \zeta (r\lambda ,\ldots , r\lambda )\right] ^{1/\lambda } \le C_r \frac{1}{k^{r/2}} \left( \frac{\pi }{\lambda r \sin \pi / (\lambda r)} \right) ^{rk} \end{aligned}$$

for some constant $C_r > 0$. Thus,

$$\begin{aligned} \sum _{u \in {\mathcal {U}}} \gamma _u \le \Gamma _0 + C_r \sum _{k=1}^\infty \frac{\Gamma _k}{(k!)^r} k^{-r/2} \left( \frac{\pi }{\lambda r \sin \pi / (\lambda r)} \right) ^{rk}, \end{aligned}$$

from which (62) follows. $\square$

Corollary 8

Let $\varvec{\gamma }= (\gamma _u)_{u \in {\mathcal {U}}}$ be POD weights with $\gamma _u = \Gamma _{|u|} \prod _{j\in u} \gamma _j$. Let $p^*:={{\mathrm{decay}}}_{\varvec{\gamma },1} < \infty$. Further, let $c, c_0 > 0$ be constants such that

$$\begin{aligned} c_0 j^{-p^*} \le \gamma _j \le c j^{-p^*} \quad \text{ for } \text{ all } j \ge 1. \end{aligned}$$

If for some $q \le p^*/2$ we have

$$\begin{aligned} \sum _{k=1}^\infty \frac{c^{k/q} \Gamma _k^{1/q}}{(k!)^{p^*/q}} k^{-p^*/ (2q)} \left( \frac{\pi }{2 \lfloor p^*/(2q) \rfloor \sin \pi /(2 \lfloor p^*/ (2q) \rfloor } \right) ^{k p^*/q} < \infty , \end{aligned}$$

(63)

then $\mathrm {decay}_{\varvec{\gamma },\infty } \ge q$.

On the other hand, if for $q < p^*$ we have

$$\begin{aligned} \sum _{k=1}^\infty \frac{c_0^{k/q} \Gamma _k^{1/q}}{(k!)^{2 \lceil p^*/(2q) \rceil }} k^{-\lceil p^*/(2q) \rceil } \left( \frac{\pi }{2 \lceil p^*/(2q) \rceil \sin \pi / (2 \lceil p^*/(2q) \rceil ) } \right) ^{k p^*/q} = \infty , \end{aligned}$$

(64)

then $\mathrm {decay}_{\varvec{\gamma },\infty } \le q$.

Proof

We have

$$\begin{aligned} \mathrm {decay}_{\varvec{\gamma },\infty } = \sup \left\{ q \in \mathbb {R}: \sum _{u \in {\mathcal {U}}} \gamma _u^{1/q} < \infty \right\} . \end{aligned}$$

Thus, we have for some $q \le p^*/2$

$$\begin{aligned}&\sum _{u \in {\mathcal {U}}} \gamma _u^{1/q} \le \Gamma _0^{1/q} + C_{p^*/q} \sum _{k=1}^\infty \frac{c^{k/q} \Gamma _k^{1/q}}{(k!)^{p^*/q}} k^{-p^*/ (2q)}\\&\quad \left( \frac{\pi }{2 \lfloor p^*/(2q) \rfloor \sin \pi /(2 \lfloor p^*/ (2q) \rfloor } \right) ^{k p^*/q} \end{aligned}$$

that the right-hand side is finite, then $\mathrm {decay}_{\varvec{\gamma },\infty } \ge q$.

On the other hand, for $q < p^*$ we have

$$\begin{aligned}&\sum _{u \in {\mathcal {U}}} \gamma _u^{1/q} \ge \Gamma _0^{1/q} + c_{p^*/q} \sum _{k=1}^\infty \frac{c_0^{k/q} \Gamma _k^{1/q}}{(k!)^{2 \lceil p^*/(2q) \rceil }} k^{-\lceil p^*/(2q) \rceil } \\&\quad \left( \frac{\pi }{2 \lceil p^*/(2q) \rceil \sin \pi / (2 \lceil p^*/(2q) \rceil ) } \right) ^{p^*k/q}. \end{aligned}$$

If the right-hand side is infinite for some $q < p^*$, then $\mathrm {decay}_{\varvec{\gamma },\infty } \le q$.

We suspect that the condition $q \le p^*/2$ in the preceding Corollary can be replaced by $q \le p^*$.

The preceding corollary allows us to construct an example of POD weights where

$$\begin{aligned} 1 \le \mathrm {decay}_{\varvec{\gamma },\infty } < \mathrm {decay}_{\varvec{\gamma },1}. \end{aligned}$$

For instance, let $\gamma _j = j^{-p^*}$. Thus, $\mathrm {decay}_{\varvec{\gamma },1} = p^*$ and $c_0 = c = 1$ in the preceding corollary. Let $q^*$ be such that $p^*/(2q^*) \in \mathbb {N}$. For $k \in \mathbb {N}_0$ let

$$\begin{aligned} \Gamma _k = (k!)^{p^*} k^{p^*/2-q^*} \left( \frac{(p^*/q^*) \sin (q^*\pi /p^*)}{\pi } \right) ^{k p^*}. \end{aligned}$$

Then we have for $q=q^*$ that (64) is of the same form as (63), which is

$$\begin{aligned} \sum _{k=1}^\infty \frac{\Gamma _k^{1/q}}{(k!)^{p^*/q}} k^{-p^*/(2q)} \left( \frac{\pi }{2 \lfloor p^*/(2q) \rfloor \sin \pi / (2 \lfloor p^*/(2q) \rfloor ) } \right) ^{k p^*/q} = \sum ^\infty _{k=1} k^{-1}= \infty .\nonumber \\ \end{aligned}$$

(65)

Due to (64), we have ${{\mathrm{decay}}}_{\varvec{\gamma },\infty } \le q^*$.

Now let $q<q^*$ such that $\lfloor p^*/2q \rfloor = p^*/2q^*$. For this $q$ the left-hand side of (63) is

$$\begin{aligned} \sum ^\infty _{k=1} \frac{\Gamma _k^{1/q}}{(k!)^{p^*/q}} k^{-p^*/2q} \left( \frac{\pi }{(p^*/q^*) \sin (q^*\pi /p^*)} \right) ^{kp^*/q} = \sum ^\infty _{k=1} k^{-q^*/q} <\infty . \end{aligned}$$

Thus, (63) gives us ${{\mathrm{decay}}}_{\varvec{\gamma },\infty } \ge q$.

Together with Lemma 3, this establishes Lemma 6.