An improvement of Koksma's inequality and convergence rates for the quasi-Monte Carlo method

1. Introduction

A natural way to numerically calculate the integral of a function $f : [0, 1] \to R$ is to take a sequence $x = {x_{n}}_{n = 1}^{\infty} \subseteq [0, 1]$ and use the approximation

(1.1)

\int_{0}^{1} f (t) d t \approx \frac{1}{N} N \sum n = 1 f (x_{n}) .

Introduce the error

(1.2)

E_{N} (f; x) = ∣ ∣ ∣ ∣ \frac{1}{N} N \sum n = 1 f (x_{n}) - \int_{0}^{1} f (t) d t ∣ ∣ ∣ ∣ .

In the Monte Carlo method (MC), one takes $x$ in (1.1) to be a sequence of random numbers sampled uniformly from $[0, 1]$ . The expression inside the absolute value signs of (1.2

) is then a random variable with expected value 0 and standard deviation of the order

1 / \sqrt{N}

N \to \infty

, see e.g. [1].

The quasi-Monte Carlo method (QMC) is based on instead taking a deterministic x in (1.1) with ”good spread” in $[0, 1]$ . This can lead to a better convergence rate of (1.2) than when taking random x. In fact, there exist deterministic $x \subseteq [0, 1]$ such that the rate of decay of $E_{N} (f; x)$ is close to $1 / N$ as $N \to \infty$ (see below).

Useful references for the theory and application of MC and QMC are e.g. [1, 3, 7].

The aim of this note is to discuss error estimates for QMC on $[0, 1]$ . More specifically, we establish an improvement of the elegant Koksma’s inequality. Koksma’s inequality is the main general error estimate for QMC, we need some auxiliary notions in order to formulate it.

Let $x = {x_{n}}_{n = 1}^{\infty} \subseteq [0, 1]$ . The star discrepancy of the set ${x_{1}, x_{2}, . . ., x_{N}}$ (i.e. set of the first $N$ terms of $x$ ) is given by

D_{N}^{*} (x) = max 0 \leq t \leq 1 ∣ ∣ ∣ \frac{♯ ({x_{1}, x_{2}, . . ., x_{N}} \cap [0, t))}{N} - t ∣ ∣ ∣ .

(Here, $♯ (A)$ denotes the cardinality of the set $A$ .) In a sense, the quantity $D_{N}^{*} (x)$ measures how much the distribution of the points of $x$

deviates from the uniform distribution. One can show that for any

x

there holds

D_{N}^{*} (x) \geq 1 / (2 N)

. On the other hand, there is a sequence

x_{C} \subseteq [0, 1]

(called the van der Corput sequence, see [5]) such that

D_{N}^{*} (x_{C}) = O (log (N) / N)

The total $p$ -variation of a function $f : [0, 1] \to R$ is given by

{V a r}_{p} (f) = sup {(\infty \sum j = 1 | f (I_{j}) |^{p})}_{j}^{1 / p},

where the supremum is taken over all non-overlapping collections of intervals ${I_{j}}_{j = 1}^{\infty}$ contained in $[0, 1]$ and $f (I) = f (b) - f (a)$ if $I = [a, b]$ . If ${V a r}_{p} (f) < \infty$ we say that $f$ has bounded $p$ -variation (written $f \in B V_{p}$ ). Note that

B V_{1} ↪ B V_{p} (1 < p < \infty) .

See [4] for a thorough discussion of bounded $p$ -variation and applications.

Koksma’s inequality states that

(1.3)

E_{N} (f; x) \leq D_{N}^{*} (x) {V a r}_{1} (f) .

In other words, the error of QMC is bounded by a product of two factors, the first measuring the ”spread” of the sequence $x$ and the second measuring the variation of the integrand $f$ . An immediate consequence of (1.3) is that we obtain the ”almost optimal” error rate $E_{N} (f; x_{C}) = O (log (N) / N)$ if $f \in B V_{1}$ and $x_{C}$ is the previously mentioned van der Corput sequence.

A drawback of (1.3) is that it provides no error estimate in the case when $f \notin B V_{1}$ . For instance, we were originally interested in finding a general error estimates for $f \in B V_{p} (p > 1)$ (see Corollary 1.2 below). This led us to our main result (Theorem 1.1), which is a sharpening of (1.3) that is effective also when $f \notin B V_{1}$ . In fact, Theorem 1.1 provides an estimate of (1.2) for any function.

For this, we recall the notion of modulus of variation, first introduced in [6] (see also [2]). For any $N \in N$ , we set

ν (f; N) = sup N \sum j = 1 | f (I_{j}) |,

where the supremum is taken over all non-overlapping collections of at most $N$ sub-intervals of $[0, 1]$ . An attractive feature of the modulus of variation is that it is finite for any bounded function. Of course, $f \in B V_{1}$ if and only if $ν (f; N) = O (1)$ as $N \to \infty$ and the growth of $ν (f; N)$ then tells us how ”badly” a function has unbounded 1-variation.

The next result is our main theorem.

Theorem 1.1.

For any function $f$ and $N \in N$ there holds

(1.4)

E_{N} (f; x) \leq 13 D_{N}^{*} (x) ν (f; N) .

The constant 13 in (1.4) is a consequence of our method of proof and certainly not optimal. However, the main point is that we can replace the total variation in (1.3) with a quantity that is finite for all functions. An immediate corollary of (1.4) is a Koksma-type inequality for $B V_{p}$ (which is not possible to derive from (1.3)).

Corollary 1.2.

If $f \in B V_{p} (1 \leq p < \infty)$ , then

(1.5)

E_{N} (f; x) \leq 13 N^{1 - 1 / p} D_{N}^{*} (x) {V a r}_{p} (f) .

In a sense the estimate (1.5) cannot be improved: there is a constant $c > 0$ such that for any $N$ we can find a sequence $x$ and a function $f$ with ${V a r}_{p} (f) = 1$ and

E_{N} (f; x) \geq c N^{1 - 1 / p} D_{N}^{*} (x) .

We shall also discuss error estimates for functions with some continuity properties. Our result here (Corollary 1.3) is known, see [5], but Theorem 1.1 allows us to derive it in a very simple way. Define the modulus of continuity of $f$ by

ω (f; δ) = sup | x - y | \leq δ | f (x) - f (y) | (0 \leq δ \leq 1),

and let $ω : [0, 1] \to [0, \infty)$ be a non-decreasing function with $ω (0) = 0$ , strictly concave and differentiable on (0,1). We denote by $H^{ω}$ the class of functions such that

| f |_{H^{ω}} = sup 0 < δ \leq 1 \frac{ω (f; δ)}{ω (δ)} < \infty .

In particular, if $ω (δ) = δ^{α} (0 < α < 1)$ , then $H^{ω}$ is the space of $α$ -Hölder continuous functions.

Corollary 1.3 (see e.g. [5], p. 146).

If $f \in H^{ω}$ , then

E_{N} (f; x) \leq 13 | f |_{H^{ω}} N ω (\frac{1}{N}) D_{N}^{*} (x) .

Corollaries 1.2 and 1.3 follow from simple estimates of the quantity $ν (f; N)$ (see Proposition 2.3 below).

2. Proofs

We first state a few results that we will use in the proof of Theorem 1.1.

Lemma 2.1.

Let $M \in N$ and $s_{M}$

be the continuous first-order spline interpolating

f

at the knots

0 = x_{0} < x_{1} < . . . < x_{M} = 1

. Then

{V a r}_{1} (s_{M}) \leq ν (f; M) .

Proof.

Let ${x_{n_{k}}}$ be the subset of ${x_{n}}$ consisting of points of local extremum of $s_{M}$ . It is easy to see that

{V a r}_{1} (s_{M}) = \sum k | s_{M} (x_{n_{k + 1}}) - s_{M} (x_{n_{k}}) | = \sum k | f (x_{i_{k + 1}}) - f (x_{i_{k}}) | \leq ν (f; M),

where the last inequality holds since the sum extends over at most $M$ terms. ∎

Lemma 2.2.

Let $f$ be a measurable function and $J \subseteq [0, 1]$ an interval. Then there exists $x_{0} \in J$ such that

\int_{J} | f (t) | d t \leq 2 | J | | f (x_{0}) | .

Proof.

Clearly we cannot have $| f (x) | < (2 | J |)^{- 1} \int_{J} | f (t) | d t$ for all $x \in J$ , since then it would follow

\int_{J} | f (x) | d x < \frac{1}{2} \int_{J} | f (t) | d t,

which is of course a contradiction. ∎

We now prove Theorem 1.1.

Proof of Theorem 1.1.

Without loss of generality, we may assume that

x_{1} < x_{2} < . . . < x_{N} .

Set $x_{0} = 0$ and $x_{N + 1} = 1$ and let $s_{N}$ be the continuous first-order spline interpolating $f$ at the knots $x_{0}, x_{1}, . . ., x_{N}, x_{N + 1}$ . Then

\frac{1}{N} N \sum n = 1 f (x_{n}) - \int_{0}^{1} f (t) d t = \frac{1}{N} N \sum n = 1 s_{N} (x_{n}) - \int_{0}^{1} s_{N} (t) d t + \int_{0}^{1} [s_{N} (t) - f (t)] d t

Hence,

E_{N} (f; x) \leq E_{N} (s_{N}, x) + ∥ f - s_{N} ∥_{L^{1} (0, 1)}

By (1.3) and Lemma 2.1, we have

(2.1)

E_{N} (f; x) \leq D_{N}^{*} (x) ν (f; N) + ∥ f - s_{N} ∥_{L^{1} (0, 1)}

We shall estimate $∥ f - s_{N} ∥_{L^{1} (0, 1)}$ . We have

∥ f - s_{N} ∥_{L^{1} (0, 1)} = N \sum n = 0 \int_{x_{n}}^{x_{n + 1}} | f (t) - s_{N} (t) | d t

By Lemma 2.2, there are $y_{n} \in (x_{n}, x_{n + 1})$ for $n = 0, 1, . . ., N$ such that

\int_{x_{n}}^{x_{n + 1}} | f (t) - s_{N} (t) | d t \leq 2 (x_{n + 1} - x_{n}) | f (y_{n}) - s_{N} (y_{n}) | .

Thus,

(2.2)

∥ f - s_{N} ∥_{L^{1} (0, 1)} \leq 2 δ_{N} (x) N \sum n = 0 | f (y_{n}) - s_{N} (y_{n}) |,

where $δ_{N} (x) = {max}_{0 \leq n \leq N} (x_{n + 1} - x_{n})$ . We shall first prove that

(2.3)

δ_{N} (x) \leq 4 D_{N}^{*} (x) .

Indeed, the discrepancy of $x$ is defined as

D_{N} (x) = sup J \subset [0, 1] ∣ ∣ ∣ \frac{♯ (J \cap {x_{1}, x_{2}, . . ., x_{N}})}{N} - | J | ∣ ∣ ∣ .

It is well-known (see [5], p. 91) that

D_{N} (x) \geq \frac{1}{N} a n d D_{N} (x) \leq 2 D_{N}^{*} (x) .

Note that if $J_{n} = [x_{n}, x_{n + 1}) (0 \leq n \leq N)$ we have

J_{n} \cap {x_{0}, x_{1}, . . ., x_{N + 1}} = {x_{n}}

Thus, for $1 \leq n \leq N$ we have

$\| J_{n} \|$	$=$	$∣ ∣ ∣ \frac{♯ (J \cap {x_{0}, x_{1}, . . ., x_{N}, x_{N + 1}})}{N} - \frac{1}{N} - \| J_{n} \| ∣ ∣ ∣$
	$\leq$	$\frac{1}{N} + ∣ ∣ ∣ \frac{♯ (J \cap {x_{1}, . . ., x_{N}})}{N} - \| J_{n} \| ∣ ∣ ∣$
	$\leq$	$\frac{1}{N} + D_{N} (x) \leq 2 D_{N} (x)$
	$\leq$	$4 D_{N}^{*} (x) .$

A similar inequality holds for $J_{0} = [x_{0}, x_{1})$ . This proves (2.3). Hence, by (2.2), we have

∥ f - s_{N} ∥_{L^{1} (0, 1)} \leq 4 D_{N}^{*} (x) N \sum n = 0 | f (y_{n}) - s_{N} (y_{n}) | .

Furthermore,

	$\| f (y_{n}) - s_{N} (y_{n}) \|$	$\leq$	$\| f (y_{n}) - f (x_{n}) \| + \| f (x_{n}) - s_{N} (y_{n}) \|$
		$=$	$\| f (y_{n}) - f (x_{n}) \| + \| s_{N} (x_{n}) - s_{N} (y_{n}) \|$

and it follows that

$N \sum n = 0 \| f (y_{n}) - s_{N} (y_{n}) \|$	$\leq$	$N \sum n = 0 \| f (y_{n}) - f (x_{n}) \| + N \sum n = 0 \| s_{N} (x_{n}) - s_{N} (y_{n}) \|$
	$\leq$	$ν (f; N + 1) + {V a r}_{1} (s_{N})$
	$\leq$	$ν (f; N + 1) + ν (f; N)$
	$\leq$	$3 ν (f; N) .$

Consequently,

∥ f - s_{N} ∥_{L^{1} (0, 1)} \leq 12 D_{N}^{*} (x) ν (f; N) .

and by (2.1) we obtain

E_{N} (f; x) \leq 13 D_{N}^{*} (x) ν (f; N) .

∎

The next result proves the corollaries 1.2 and 1.3.

Proposition 2.3.

Let $N \in N$ , then we have

(2.4)

ν (f; N) \leq N^{1 - 1 / p} {V a r}_{p} (f) (1 \leq p < \infty),

and

(2.5)

ν (f; N) \leq | f |_{H^{ω}} N ω (\frac{1}{N}) .

Proof.

The inequality (2.4) follows immediately from Hölder’s inequality. For (2.5), take $N$ intervals ${I_{j}}_{j = 1}^{N}$ , then clearly

N \sum j = 1 | f (I_{j}) | \leq | f |_{H^{ω}} N \sum j = 1 ω (| I_{j} |) .

Define

(2.6)

M_{ω} (N) = max (N \sum j = 1 ω (t_{j})) s u b j e c t t o N \sum j = 1 t_{j} = 1, t_{j} > 0.

Since $\sum_{j = 1}^{N} | I_{j} | \leq 1$ and $ω$ is non-decreasing, we clearly have

ν (f; N) \leq | f |_{H^{ω}} M_{ω} (N) .

To calculate $M_{ω} (N)$ , we use Lagrange multipliers. The critical point $(t_{1}, t_{2}, . . ., t_{N}, λ)$ of the Lagrangian function solves

ω^{'} (t_{j}) - λ = 0 (1 \leq j \leq N) a n d N \sum j = 1 t_{j} - 1 = 0.

By the strict concavity of $ω$ , the above system has the unique solution $t_{1} = t_{2} = . . . = t_{N}$ . Hence, the maximum (2.6) is

M_{ω} (N) = N ω (\frac{1}{N}) .

∎

References

[1] R. E. Caflisch, ”Monte Carlo and quasi-Monte Carlo methods”, Acta Numerica 7 (1998), 1–49.
[2] Z. A. Chanturiya, ”The modulus of variation of a function and its application in the theory of Fourier series”, Dokl. Akad. Nauk SSSR 214 (1974), 63–66.
[3] J. Dick and F. Pillichshammer, Digital Nets and Sequences: Discrepancy Theory and Quasi–Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.
[4] R. M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and $p$ -Variation, Wwth the collaboration of Jinghua Qian. Lecture Notes in Mathematics, 1703. Springer-Verlag, Berlin, 1999.
[5] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley and Sons, New York, 1974.
[6] R. Lagrange, ”Sur les oscillations d’ordre supérieur d’une fonction numérique”, Ann. Scient. Ecole Norm. Sup. 82 (1965) 101-130.
[7] A. B. Owen, Monte Carlo theory, methods and examples, preprint, Stanford University, 2013.

	$\| f (y_{n}) - s_{N} (y_{n}) \|$	$\leq$	$\| f (y_{n}) - f (x_{n}) \| + \| f (x_{n}) - s_{N} (y_{n}) \|$
		$=$	$\| f (y_{n}) - f (x_{n}) \| + \| s_{N} (x_{n}) - s_{N} (y_{n}) \|$

An improvement of Koksma's inequality and convergence rates for the quasi-Monte Carlo method

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1. Introduction

Theorem 1.1.

Corollary 1.2.

Corollary 1.3 (see e.g. [5], p. 146).

2. Proofs

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Proof of Theorem 1.1.

Proposition 2.3.

Proof.

References

An improvement of Koksma's inequality and convergence rates for the quasi-Monte Carlo method

Related Research

On Quasi-Monte Carlo Methods in Weighted ANOVA Spaces

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Stability of doubly-intractable distributions

Integration of bounded monotone functions: Revisiting the nonsequential case, with a focus on unbiased Monte Carlo (randomized) methods

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1. Introduction

Theorem 1.1.

Corollary 1.2.

Corollary 1.3 (see e.g. [5], p. 146).

2. Proofs

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Proof of Theorem 1.1.

Proposition 2.3.

Proof.

References