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APPEARED IN BULLETIN OF THE

arXiv:math/9304216v1 [math.NA] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 308-314RESEARCH ANNOUNCEMENTINTEGRATION AND APPROXIMATIONOF MULTIVARIATE FUNCTIONS:AVERAGE CASE COMPLEXITY WITHISOTROPIC WIENER MEASUREG. W. WASILKOWSKIAbstract.

We study the average case complexity of multivariate integrationand L2 function approximation for the class F = C([0, 1]d) of continuousfunctions of d variables. The class F is endowed with the isotropic Wienermeasure (Brownian motion in Levy’s sense).

Furthermore, for both problems,only function values are used as data.1. IntroductionWe study the integration and function approximation problems for multivariatefunctions f. For the integration problem, we want to approximate the integral of fto within a specified error ε; and for the function approximation problem, we wantto recover f with the L2 error not exceeding ε.

To solve both problems, we wouldlike to use as small a number of function values as possible.Both problems have been extensively studied in the literature (see, [9, 16] forhundreds of references).However, they are mainly addressed in the worst-casesetting. In the worst-case setting the cost and the error of an algorithm are definedby the worst performance with respect to the given class F of functions f. Notsurprisingly, for a number of classes F, the integration and function approximationproblems are intractable (prohibitively expensive) or even unsolvable.

For instance,if F consists of continuous functions that are bounded by 1, no algorithm that uses afinite number of function values can approximate the integral of f, nor can it recoverf with the worst-case error less than 1. Hence, both problems are unsolvable forε < 1.

Assuming that functions f have bounded rth derivative in the sup-norm,the number of function values required for the worst-case error not to exceed ε isof order ε−d/r. Hence, for fixed r, it is exponential in d.Received by the editors December 19, 1991.1991 Mathematics Subject Classification.

Primary 41A50, 41A55, 41A63, 65D15, 65D30.This research was supported in part by the National Science Foundation under Grant CCR-91-14042.c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2G. W. WASILKOWSKIDue to intractability in the worst-case setting, the average-case setting is of inter-est.

In the average-case setting, the class F is equipped with a probability measureµ. The error and the cost of an algorithm are measured by the expectations withrespect to µ.

Then, the average-case complexity (with respect to µ) is defined asthe minimal expected cost needed to compute an approximation with the expectederror not greater than ε.The majority of the average-case results obtained so far (see, [3–6, 9–18, 21]) dealwith scalar functions (d = 1). These results indicate that for a “reasonable” choiceof measure µ, the integration and function approximation problems are significantlyeasier on the average than in the worst-case setting.

Thus, one could hope thatthe intractability (or even noncomputability) of multivariate problems in the worst-case setting can be removed by switching from the worst-case to the average-casesetting.This hope has recently been supported by Wo´zniakowski (see [23, 24]), who an-alyzes integration and function approximation for the class F = C([0, 1]d) endowedwith the Wiener sheet measure µ. He proves that the average-case complexities ofboth problems are only weakly dependent on the number of variables.

Indeed, theaverage-case complexity of computing an ε-approximation is Θ(ε−1(log ε−1)(d−1)/2)for the integration problem and Θ(ε−2(log ε−1)2(d−1)) for the function approxima-tion problem.In this paper we study the average-case complexity of the integration and func-tion approximation problems. However, instead of the Wiener sheet measure, weendow the class F = C([0, 1]d) with the isotropic Wiener measure (or Brown-ian motion in Levy’s sense).

We prove that the average-case complexity equalsΘ(ε−2/(1+1/d)) for the integration problem and Θ(ε−2d) for the function approxi-mation problem. Unlike the Wiener sheet measure, the average-case complexity ofthe function approximation problem depends strongly on d. In particular, for larged this problem is intractable since its complexity Θ(ε−2d) is exponential in d andis huge even for a modest error demand ε.

For large d the average-case complexityof the integration problem is essentially proportional to ε−2, which is the highestpossible average-case complexity of the integration problem. Indeed, for any prob-ability measure with finite expected value of ∥f∥2L2, the average-case complexity isbounded from above by O(ε−2).

Hence, this is again a negative result.Thus, the average-case complexities of integration and function approximationproblems are very different depending on whether µ is the Wiener sheet or isotropicWiener measure. It is interesting to note that both measures are identical whend = 1.

They are different for d > 1; results of [23, 24] and our results indicate howdrastically different they are.The paper is organized as follows. Section 2 provides basic definitions.

The mainresults are presented in §3. In addition to results already mentioned, §3 discussesoptimality of Haber’s [2] modified Monte Carlo quadrature and of a piecewise con-stant function approximation.

It also contains a result relating the average-casecomplexities of the integration and function approximation problems for generalprobability measures. In this paper we omit all proofs because of their substantiallength.2.

Basic definitions

INTEGRATION AND APPROXIMATION OF MULTIVARIATE FUNCTIONS3In this paper we consider the following integration and function approximationproblems for multivariate functions.Let F = C(D) be the space of functionsf : D →R where D is a bounded subset of Rd. For simplicity, we take D = [0, 1]das a unit cube.

For every f ∈F we wish to approximate S(f), where S : F →GwithS(f) = Int(f) =ZDf(x) dxandG = Rfor the integration problem,S(f) = App(f) = fandG = L2(D)for the approximation problem.We assume that the functions f are unknown; instead we can compute informa-tion N(f) that consists of a finite number of values of f taken at some points fromD. For a precise definition of N see [16].

Here we stress only thatN(f) = [f(x1), . .

. , f(xn)] ,where the points xi and the number n of them (called the cardinality of N) canbe selected adaptively and/or randomly.That is, for adaptive N, xi’s dependon previously computed values f(x1), .

. .

, f(xi−1), and the cardinality n = n(f)varies with f based on computed values. For randomized N, the points xi and thecardinality n(f) may also depend on an outcome of a random process t. (That is, xiis selected randomly with an arbitrary distribution that may depend on previouslycomputed values of f; the distribution of n(f) may also depend on observed values.

)In such a case, we sometimes write N(f) = Nt(f).An approximation U(f) to S(f) is computed based on N(f). That is,U(f) = φ(N(f)),where φ : N(F) →Gis an arbitrary mapping; φ is called an algorithm that uses N. The algorithm φ canalso be random; in such a case, we sometimes write φ = φt.In the average-case setting, we assume that the space F is endowed with a(Borel) probability measure µ.

Then the average error and the average cost1 of φare defined respectively byeavg(φ, N, S, µ) :=qEµEt(∥S(f) −φt(Nt(f))∥2G),costavg(φ, N, S, µ) :=EµEt(n(f)). (By Eµ and Et we denote the expectations w.r.t.

µ and t, respectively.) Of course,for deterministic N and φ,eavg(φ, N, S, µ) =sZF∥S(f) −φ(N(f))∥2G µ(df),costavg(φ, N, S, µ) =ZFn(f) µ(df).1We measure the cost by the expected number of function values neglecting the combinatorycost of N and of φ.

With the exception of Theorem 3, this is without loss of generality since, asexplained in a number of references (see, e.g., [16]), for Gaussian measures the same results holdfor a more general definition of the average cost, provided that a single arithmetic operation is nomore expensive than a function evaluation.

4G. W. WASILKOWSKIThe average-case complexity is the minimal average cost for solving the problemto within a preassigned error accuracy ε.

That is,compavg(ε, S, µ) := inf {costavg(φ, N, S, µ) : eavg(φ, N, S, µ) ≤ε} . (We stress that the infimum above is taken with respect to all randomized φ andN.

)In this paper we analyze the average-case complexity of the integration andfunction approximation problems (S = Int and S = App) with the assumption thatthe probability µ is the isotropic Wiener measure. This measure is also referred to asthe Brownian motion in Levy’s sense.

For more detailed discussion and propertiesof µ (see [1, 7, 8]). Here we only recall that µ is a zero-mean Gaussian measurewith the correlation functionK(x, y) = ∥x∥+ ∥y∥−∥x −y∥2∀x, y ∈Rd,∥x∥2 =dXi=1x2i .3.

Main resultsTheorem 1. For the integration and function approximation problems,compavg(ε, Int, µ) =Θε−2/(1+1/d),(1)compavg(ε, App, µ) =Θε−2d.

(2)For d = 1, µ equals the classical Wiener measure. Hence, for scalar functionsthis theorem follows from known results (see [12, 13, 19]).We now exhibit algorithms and information that are almost optimal.

Let n =pd. Partition D into n equal-sized cubes Ui, Ui = xi + [−1/(2p), +1/(2p)]d, eachcentered at xi.For the integration problem, consider the following randomizedinformation and algorithm due to Haber (see [2]):N Intn (f) = [f(t1), .

. .

, f(tn)]andφIntn (N Intn (f)) = 1nnXj=1f(tj),(3)where ti’s are uniformly distributed in Ui’s. For the function approximation prob-lem, considerN Appn(f) = [f(x1), .

. .

, f(xn)]andφAppn(N Appn(f)) =nXi=1gi(·)f(xi),(4)with gi being the indicator function for the set Ui.Theorem 2. For every n, the average errors of φIntnand φAppnare respectively equaltoqRDRD ∥x −y∥/2 dxdyn1/2+1/(2d)andqRD ∥x∥/2 dxn1/(2d).

INTEGRATION AND APPROXIMATION OF MULTIVARIATE FUNCTIONS5These algorithms are almost optimal. Indeed, for nInt(ε) and nApp(ε) given bynInt(ε) =&ε−2ZDZD∥x −y∥/2 dxdy1/(d+1)'d,nApp(ε) =ε−2ZD∥x∥/2 dxd,φIntnInt(ε) and φAppnInt(ε) have the average errors less than or equal to ε and their costsare proportional to (1) and (2), respectively.Remark 1.

In the worst-case setting with F = C[0, 1]d, Haber’s modified MonteCarlo algorithm φIntnand the classical Monte Carlo algorithm n−1 Pni=1 f(ti) (withti’s uniformly distributed in D) have (modulo constants) the same errors that areproportional to 1/√n.It can be proven that the average error of the classicalMonte Carlo algorithm equalsqn−1 RDRD ∥x −y∥/2 dxdy. Thus, it is preciselyn1/(2d) times larger than the average error of φIntn .Remark 2.

Although the information N Intnis randomized, the cardinality n is fixed.Thus, the mean value theorem implies the existence of deterministic N ∗n(f) =[f(x∗1), . .

. , f(x∗n)] such that φ∗n(N ∗n(f)) = n−1 Pni=1 f(x∗i ) has the average errornot exceeding the average error of φIntn .

(We do not know the location of the pointsx∗i ; we only know that x∗i ∈Ui, 1 ≤i ≤n.) This and the fact that the algorithmand information given in (4) are deterministic imply that randomization does nothelp for both problems.

The lack of power of randomization holds for more generalproblems. Indeed, randomization does not help for linear S and Gaussian µ (see[20, 22]).The final theorem relates the average-case complexities of the integration andfunction approximation problems for an arbitrary (Borel) probability measure.Theorem 3.

Let ν be an arbitrary probability measure on F. Ifcompavg(ε, Int, ν) = Ωε−pfor some p (obviously, p ≤2 whenever ∥f∥2L2(D) has a finite (ν-) expectation), thencompavg(ε, App, ν) = Ωε−p(1+p/(2−p)).Remark 3. This theorem can easily be extended in a number of ways.

For instance,it holds when the function approximation problem is considered with the L2(D)-norm replaced by ∥f −f ∗∥2 =RD w(x)(f(x)−f ∗(x))2 dx, a weighted norm, for someweight w ≥0 and the integration problem defined by Int(f) =RD f(x)pw(x) dx.It also holds in the worst-case setting with randomization. In this setting, insteadof the expectation Eν, we take the supremum w.r.t.

f ∈F0 (F0 is a given subsetof F) in the definitions of the error and cost (the expectation w.r.t. random tremains).

(For more detailed definitions, see [16]). Hence, again, in the worst-casesetting with randomization, integration is an easier problem than is the functionapproximation problem.We stress that this need not be true if the worst-casedeterministic (without randomization) setting is considered, since for a number of

6G. W. WASILKOWSKIclasses F0 the integration and approximation problems have asymptotically thesame worst-case deterministic complexities (see, [9, 16]).AcknowledgmentThe author wishes to thank David R. Adams, Klaus Ritter, and Henryk Wo´zniakowskifor valuable suggestions.References1.

Z. Ciesielski, On Levy’s Brownian motion with several-dimensional time, Lecture Notes inMath., vol. 472, Springer-Verlag, New York, 1975, pp.

29–56.2. S. Haber, A modified Monte Carlo quadrature, Math.

Comp. 20 (1966), 361–368.3.

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A 32 (1970), 173–180.5. D. Lee, Approximation of linear operators on a Wiener space, Rocky Mountain J.

Math. 16(1986), 641–659.6.

D. Lee and G. W. Wasilkowski, Approximation of linear functionals on a Banach space with aGaussian measure, J. Complexity 2 (1986), 12–43.7.

P. Levy, Processus stochastique et mouvement Brownien, Gauthier-Villars, Paris, 1948.8. G. M. Molchan, On some problems concerning Brownian motion in Levy’s sense, TheoryProbab.

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E. Novak, Deterministic and stochastic error bounds in numerical analysis, Lecture Notes inMath., vol. 1349, Springer-Verlag, Berlin, 1988.10.

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115–136.13. P. Speckman, Lp approximation of autoregressive Gaussian process, report, Department ofStatistics, University of Oregon, Eugene, OR, 1979.14.

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(Russian)16. J. F. Traub, G. W. Wasilkowski, and H. Wo´zniakowski, Information-based complexity, Aca-demic Press, New York, 1988.17.

G. Wahba, On the regression design problem of Sacks and Ylvisaker, Ann. Math.

Stat. 42(1971), 1035–1043.18., Spline models for observational data, SIAM-NSF Regional Conf.

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Math.,vol. 59, SIAM, Philadelphia, PA, 1990.19.

G. W. Wasilkowski, Information of varying cardinality, J. Complexity 2 (1986), 204–228.20., Randomization for continuous problems, J.

Complexity 5 (1989), 195–218.21. G. W. Wasilkowski and F. Gao, On the power of adaptive information for functions with sin-gularities, Math.

Comp. 58 (1992), 285–304.22.

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Amer.Math. Soc.

(N.S. )).Department of Computer Science, University of Kentucky, Lexington, Kentucky40506E-mail address: greg@ms.uky.edu

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