Quasi-Monte Carlo Algorithms for unbounded, weighted integration problems

Transcription

1 Quasi-Monte Carlo Algorithms for unbounded, weighted integration problems Jürgen Hartinger Reinhold F. Kainhofer Robert F. Tihy Department of Mathematis, Graz University of Tehnology, Steyrergasse 30, A-800 Graz, Austria Abstrat In this artile we investigate Quasi-Monte Carlo methods for multidimensional improper integrals with respet to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish onvergene of the Quasi-Monte Carlo estimator to the value of the improper integral under onditions involving both the integrand and the sequene used. Furthermore, we suggest a modifiation of an approah proposed by Hlawka and Mük for the reation of low-disrepany sequenes with regard to a given density, whih are suited for singular integrands. Key words: Quasi-Monte Carlo integration, weighted integration, non-uniformly distributed low-disrepany sequenes This paper is devoted to Quasi-Monte Carlo (QMC) tehniques for weighted integration problems of the form I = [a,b] f(x)dh(x), () where H denotes a s-dimensional distribution with support K = [a, b] R s and f is a funtion with singularities on the left boundary of K. umerial integration problems of this form frequently our in pratie, e.g. in the field of omputational finane. A typial example is the estimation of the mean of a random variate X with support R s. In ase of variates with This researh was supported by the Austrian Siene Fund Projet S-8308-MAT addresses: hartinger@finanz.math.tu-graz.a.at (Jürgen Hartinger), reinhold@kainhofer.om (Reinhold F. Kainhofer), tihy@tugraz.at (Robert F. Tihy). Preprint submitted to Journal of Complexity 7 February 2003

2 unbounded variane Monte Carlo simulation is inappliable, but (after a transformation) QMC algorithms might be available. Let H(J) denote the probability of J K under H. A sequene ω = (y, y 2,... ) is defined to be H-distributed if for all intervals K K the following ondition holds: lim χ K(y n ) = H( K). For Riemann integrable (thus bounded) funtions it is well known that the integral () an be approximated by K f(x)dh(x) = lim f(y n ), where (y n ) n>0 denotes an H-distributed sequene. The aim is to establish onditions on integrands and sequenes to guarantee the onvergene of QMC tehniques for unbounded weighted problems and to justify the ommonly used strategy of ignoring the singularity. Furthermore tehniques proposed by Hlawka and Mük for bounded, weighted integration problems are adapted for the unbounded ase. Preliminaries and basi definitions Hlawka [] established the first quantitative error bound for QMC-algorithms on the s-dimensional unit ube, where H equals the uniform distribution U. To gain a deep insight in the lassial QMC theory we refer to the monographs of Kuipers and iederreiter [2] or iederreiter [3]. Quasi-Monte Carlo algorithms for (Riemann) improper integrals with regard to the uniform distribution (H = U [0, ] s ) were first investigated by Sobol. In [4] he dealt with integrands, whih are unbounded for x x 2 x s 0. Asymptoti error estimates, as well as numerial examples for some speial funtions were presented by Klinger [5], De Donker and Guan [6]. We onsider integration problems, whih have the following properties The improper integral () on a ompat subinterval of R s exists in the sense that the limes lim f(x)dh(x) (2) a >a [,b] 2

3 exists independent of the, where > a should be understood omponentwise. We will define this limes as the value of the improper integral. If the integration domain is not a ompat subinterval of R s, i.e. at least one oordinate is unbounded (a i = or b i = for a i s), the limes lim f(x)dh(x) a + [,d] has to exist for every finite fixed upper boundary d (a, b), d <. Independently, for any fixed [a, b] as lower integration boundary with > a omponentwise, the limes lim f(x)dh(x) d b [,d] needs to exists. Then we let the lower and upper integration bounds independently tend to the boundaries of the integration domain K and define the value of the improper integral as lim lim d b a + [,d] f(x)dh(x). The integrand f(x) possibly has singularities at the left boundary of the integration domain, i.e. lim xi a i f(x) = for some i {,..., s}. These singularities are the only singularities of f(x). In partiular this means that for all ε > 0 there exists some M <, suh that f(x) < M for all x [a + ε, b]. Before we reall Hlawka s integration error bound (respetively a slight generalization for weighted integration) some more definitions are required: Definition. The H-disrepany of ω = (y, y 2,... ) measures the distribution properties of the sequene. It is defined as D,H (ω) = sup J K A (J, ω) H(J), (3) where A ounts the number of elements in (y,..., y ) falling into the interval J, i.e. A (J, ω) = χ J (y n ). Definition.2 By a partition P of K we mean a set of s finite sequenes (j =,..., s), a j = ν (j) 0 ν (j) ν (j) m j = b j. 3

4 In onnetion with suh a partition one defines for eah j two operators by j f(x (),..., x (j ), ν (j) i, x (j+),..., x (s) ) = f(x (),.., x (j ), ν (j) i+, x (j+),.., x (s) ) f(x (),.., x (j ), ν (j) i, x (j+),.., x (s) ) for 0 i < m j and jf(x (),..., x (j ),, x (j+),..., x (s) ) = f(x (),.., x (j ), ν (j) i+, x (j+),.., x (s) ) f(x (),.., x (j ), 0, x (j+),.., x (s) ). Definition.3 For a funtion f on K, we set the variation in the sense of Vitali as follows m V (l) m l (f) = sup,...,l f(ν () i,..., ν (l) i l ), P i =0 i l =0 where the supremum is extended over all partitions P of K. The variation of f restrited to an interval [a, b] will be denoted V [a,b] (f). Finally let l s, i < < i l s and V (l) (f; i,..., i l ) be the Vitali variation of the restrition of f to the l-dimensional fae {(u,..., u s ) K : u j = b j for j i,..., i l }. The variation in the sense of Hardy and Krause is defined by V (f) = s l= i < <i l s V (l) (f; i,..., i l ). (4) aturally f is alled of bounded variation on K, when V (f) <. Theorem.4 (Koksma-Hlawka Inequality) Let f be a funtion of bounded variation (in the sense of Hardy and Krause) on K and ω = (y, y 2,... ) a sequene on K. The QMC integration error an be bounded by f(x)dh(x) f(y n ) K V (f)d,h(ω). (5) A proof of this entral theorem an be found e.g. in [2] for the uniform distribution, and in [7] for general distributions H (using a slight speialization to variation in the measure sense). In [4], Sobol looked at Quasi-Monte Carlo integration of singular funtions with respet to the uniform distribution. He gave onditions for its onvergene involving the funtion as well as the sequene used. In partiular, his theorem reads: 4

5 Theorem.5 (Sobol, [4]) Let i {,..., s} and K i be the boundary of [0, ] s where all ( oordinates x ) (i) = for i / i. Let furthermore <, where = min µ x () µ,..., x (s) µ, and Gi the part of K i, where x (i),..., x (ij). If for every i the integral x (i ) x (i j) f (i ) (x (i ) ) dx (i ) (6) K i onverges and D(x i, x 2,... ) G i () f (i ) (x (i ) ) dx (i ) = o(), (7) then lim µ= f(x µ ) = [0,] s f(x)dx. In this paper we will generalize this result to weighted integration, whih means integration with respet to arbitrary densities h(x). We will also investigate several ways to onstrut the sequenes needed for QMC integration w.r.t. arbitrary densities and propose ways to overome some problems ourring in onventional methods. 2 The one-dimensional ase Let us first onsider the one-dimensional ase. The main theorem of this setion generalizes and ombines the Koksma inequality (.4) in one dimension with Sobol s onvergene theorem [4] for QMC integration of singular integrands. Instead of the uniform distribution, we will assume an arbitrary distribution H(x) on [a, b] and show onvergene of the QMC integration. The multidimensional ase will be treated in a similar manner in the next setion, so we present the simpler one-dimensional ase here to give a lear piture of the ideas. Definition 2. For a sequene ω = (y, y 2,... ) let = min n y n be the smallest value of the first elements of the sequene. Theorem 2.2 Let a. If a sequene {y i } i and a differentiable funtion f(x) on [a, b] with a singularity only at the left boundary satisfy the ondition b D,H (ω) f (x) dx = o() (8) as well as a for, then the QMC estimator f(y n ) onverges to the value of the improper integral of f(x) on [a, b]: lim b f(y n ) = f(x)dh(x) (9) a 5

6 Remark 2.3 For non-differentiable funtions the ondition is similar using the variation V [,b)(f) instead of the integral of the derivative. The multidimensional ase will be formulated in suh a more general manner. PROOF. To prove (9) we will approximate b f(x)dh(x) with f(y n ) and show that the remaining terms tend to zero. Without loss of generality we an assume the sequene to be sorted, and we define y 0 and y + so that = y 0 y y y + = b. First, we establish an identity similar to Lemma 5. of [2]: b f(y n ) f(x)dh(x) = H() f(b) b ( A ([, x), ω) H([, x)) ) df(x) (0) The above identity an be proved by inserting terms, applying integration by parts and using the fat that H(x) is a distribution funtion on [a, b]. b f(y n ) f(x)dh(x) = b f(y n ) ( H()) f(b) + [H(x) H()] f(x) b f(x)dh(x) n =f(y + ) n=0 (f(y n+) f(y n )) b f(b) + H()f(b) + [H(x) H()] df(x) yn+ n b = n=0 y n df(x) + H()f(b) + [H(x) H()] df(x) b =H()f(b) ( A ([, x), ω) H([, x)) ) df(x) One this is established, the onvergene is obvious: b f(y n ) f(x)dh(x) a b f(x)dh(x) + H() f(b) + D,H(ω) f (x) dx a Aording to the onvergene of the improper integral on the whole interval, the first and seond terms tend to zero as a, and the ondition of the theorem ensures that the third term is o() and thus also tends to zero. 6

7 Remark 2.4 The minimal element of the one-dimensional Halton-sequene in basis 2 (whih oinides with the Faure and Sobol sequenes) is larger than /2. In arbitrary bases p, the smallest element among the first an be bounded from below by /p log p +. The main drawbak, from a pratial point of view, of the preeding theorem is the lak of sequenes with low H-disrepany. In pratial issues one has to generate these sequenes by transformation from uniform low disrepany sequenes. For distributions, where expliit inverse distribution funtions are available, the transformation of the uniform distributed sequene (x, x 2,... ) by an inversion method y i = H (x i ) is obvious. This transformation preserves the disrepany, i.e D (x, x 2,... ) = D,H (y, y 2,... ). () Unfortunately for most distributions the inverse distribution funtion is not given expliitly and diret numerial methods for the inversion are often ineffiient. In [8], Hlawka and Mük propose a systemati method for onstruting H- distributed sequenes, whih just uses the distribution funtion but not its inverse. We fous on the ase K = [0, ]. Definition 2.5 Let ω = (x, x 2,... ) be a sequene in [0, ) with disrepany D (ω) with regard to the uniform distribution. The sequene ω = (ỹ, ỹ 2,... ) is defined to be the sequene onsisting of the points ỹ k = r= + x k H(x r ) = χ [0,xk ] (H(x r )). (2) r= Lemma 2.6 Let the sequene ω be defined by Eq. (2) and M = sup x [0,] h(x). Then the H-disrepany of ω an be bounded by the following inequality D,H ( ω) ( + M)D (ω). All points onstruted by the Hlawka-Mük method are of the form i/, (i = 0,..., ), in partiular, some elements ỹ k of the transformed sequene ω might assume a value of 0. Sine this is the singularity of f(x), aording to Theorem 2.2 these sequenes are not diretly suited for unbounded problems. Definition 2.7 To overome this problem, we define the sequene ω for i =,..., as follows: ỹ k if ỹ k ȳ k =, (3) if ỹ k = 0. 7

8 Theorem 2.8 The H-disrepany of ω is bounded by ( D,H ( ω) (M + ) D (ω) + ). PROOF. Let ( x, x 2,... ) and ( x, x 2,... ) be the sequenes obtained by x i = H(ȳ i ), resp. x i = H(ỹ i ). By Eq. () it is suffiient to estimate the uniform disrepany of ( x, x 2,... ). For uniform disrepanies the following fat is well known (e.g. iederreiter [3]): Let (u, u 2,... ) and (v, v 2,... ) be two sequenes in [0, ] with disrepanies D and D 2, then D D 2 < ε, whenever max i u i v i < ε. ow, the alulation x k x k x k x k + x k x k ỹk = h(t)dt + ỹk ( h(t)dt M D (x, x 2,... ) + y k ȳ k ). ompletes the proof. Remark 2.9 The disrepany of uniform low disrepany sequenes is typially of the order O( log ) (resp. O( ) for nets). Therefore the additional fator (/) one inherits though the shift (3) does not affet the asymptoti behavior of the integration error. Remark 2.0 Another method to avoid the inversion an be obtained by a suitable integral transformation. With the help of suh a transformation, in Monte Carlo algorithms often referred to as importane sampling, it might even be possible to avoid some singularities. 3 Multivariate singular integration We will now look at arbitrary-dimensional integrals [a,b] f(x)dh(x), (4) where the integration domain is taken as a ompat subinterval [a, b] of R s. Remark 3. This is no restrition: If any of the dimensions is unbounded, we an first arry out the alulation on ompat intervals and then take the limits to infinity as indiated in Setion. 8

9 The notations and operators used in the sequel are defined in Setion. In addition we need the summation symbol : Definition 3.2 Given an expression F depending on variables x (r),..., x (s) and a partition of r,s = {r, r +,..., s} into two subsets L = {x (l ),..., x (lp) } and r,s \ L = {x (l p+),..., x (l s r) }, we use the notation F (L) = F ( x (l ),..., x (lp) ; x (l p+),..., x (l s r) ). The summation operator is defined as the sum over all elements of the set P p = {L r,s : ard(l) = p}, i.e. r,...,s;p F = L P p F (L). The integration by parts formula used in the proof of the previous setion for differentiable one-dimensional funtions an be generalized to arbitrary s-dimensional funtions. It is ommonly referred to as Abel s summation formula: Lemma 3.3 (Abel s summation formula, e.g. [2]) Let ( η (j) 0, η (j) ),..., η m (j) j and ( ξ (j) 0, ξ (j) ),..., ξ m (j) j with j =,..., s be two partitions of the interval [a, b], and f(x) and g(x) two funtions on [a, b]. Then m m s f ( ξ () i +,..., ξ (s) i s+ i =0 i s=0 s = ( ) p m p+,...,s p=0,...,s;p i =0 ),...,s g ( η () i,..., η (s) ) i s m p i p=0 g ( η () i,..., η (p), x (p+),..., x (s)),...,p f ( ξ () i,..., ξ (p) i p, x (p+),..., x (s)). (5) i p For a proof of this important equation we refer to the monograph [2] of Kuipers and iederreiter. Using this summation formula, we an now prove the onvergene of the s-dimensional Quasi-Monte Carlo estimator to the value of the improper integral (4), even though the integrand an be singular on the whole left boundary of the integration area. Conventional methods usually apply the Koksma- Hlawka inequality (5). Here this inequality does not give an upper bound for the integration error, beause the singularity auses the funtion to be of unbounded variation on [a, b]. We only require the funtion to be of bounded variation on every ompat subinterval of (a, b]. The proof will follow the lines of the proof of the Koksma-Hlawka inequality given in [2], so for some parts of the proof we just refer to that book. 9

10 Theorem 3.4 (Convergene of the multidimensional QMC estimator) Let f(x) be a funtion on [a, b] with singularities only at the left boundary of the definition interval (i.e. f(x) ± only if x (j) a j for at least one j), and let furthermore,j = min n y n (j) and a j < j,j. If the improper integral (4) exists in the sense of Setion, and if D,H (ω) V [,b] (f) = o(), (6) then the QMC estimator onverges to the value of the improper integral: lim f (y n ) = f(x)dh(x). (7) [a,b] PROOF. Like in the one-dimensional ase we estimate the integration error on the interval [, b], where the funtion f(x) is regular for any hoie of, and show that the remaining terms vanish as a. Again similar to the proof of Theorem 2.2 we use a funtion g(x) = A([, x), ω) H([, x)) (8) for a given sequene ω = (y n (j) ), n, j s. One has to notie that this funtion is merely the funtion used in the definition of the disrepany, so that sup x [,b] g(x) D,H (ω). Using a double partition j = ξ (j) 0 = η (j) 0 ξ (j) < η (j) < η m (j) j b j (j =,..., s) of the interval [, b] with the additional ondition that the ξ (),..., ξ m (j) j funtion g(x). As argued in [2], the left hand side of Eq. (5), = ξ (j) m j + = ontain at least our sequene ω, we now apply Lemma 3.3 to the m i =0 an be simplified to m s i s=0 LHS = m f(y n ) i =0 f ( ξ () i +,..., ξ (s) i s+ m s i s=0 where denotes the Cartesian produt. ),...s g ( η () i,... η (s) ) i s, f ( ξ () i +,..., ξ (s) i s+),...s H ( s l= ) [ l, η (l) ) i l, (9) For the right hand side we notie that g(x) = 0 if any of the x (j) = j. Also, g(b) = H ([, b)), so that the summand of p = 0 an be simplified as,...,sg ( x (),... x (s) )f( x (),..., x (s)) = g(b)f(b). (20) 0

11 Similarly, for p s, only terms with x (p+) = b p+,..., x (s) = b s ontribute (if any x (p+j) = p+j, the funtion value of g(x) vanishes): s RHS g(b)f(b) + ( ) p m m s g ( η () i,..., η (p) ) i p p=,...,s;p i =0 i s=0,...,p f ( ξ () i,..., ξ (p) ) i p, b p+,..., b s s g(b)f(b) + D,H (ω p+,...,s )V (p) (f(x (),..., x (p ), b p+,..., b s ). p=,...,s;p (2) with ω p+,...,s denoting the projetion of the sequene ω on the upper boundary of [a, b] so that the omponents i p+,..., i s are set to b (i p+),..., b (is), and the disrepany is omputed on the fae of [a, b], in whih ω p+,...,s is ontained. Sine the seond term in Eq. (9) is nothing else but a Riemann-Stieltjes sum to the integral [,b] f(x)dh(x), and the rest is independent of the double partition, we let the mesh size of the double partition tend to zero max max (η (j) j s 0 i m j i+ η (j) i ) 0 to obtain the multidimensional version of Eq. (0): f (y n ) f(x)dh(x) [,b] ( H([, b)))f(b) + D,H(ω) V[,b] (f) (22) The existene of the improper integral in the sense of Setion guarantees that I rest := f(x)dh(x) f(x)dh(x) [a,b] tends to zero as we let a. Thus, we have f (y n ) f(x)dh(x) [a,b] f (y n ) f(x)dh(x) [,b] + I rest [ H([, b))] f(b) + D,H (ω) V [,b] (f) + I rest. (23) [,b] For and so a, the first and third terms obviously tend to zero, while the onditions of the theorem guarantee this also for the seond term. Thus the proof is finished. As in the one-dimensional ase, one faes the problem of generating H-distributed sequenes. Multidimensional versions for inversion methods are well

12 known (e.g. Devroye [9]). Under weak, Lipshitz type onditions, Hlawka and Mük [0] showed the following bound for the H-disrepany of sequenes ω generated by multidimensional inversion methods: D,H ( ω) (D (ω)) /s, (24) where ω denotes the original uniformly distributed sequene. Furthermore in [0] a multidimensional approah similar to Eq. (2) is given. We again fous on distributions on [0, ] s and speialize our analysis to distributions with independent marginals, i.e. H(x) = s i= H i (x (i) ). In this ase one an transform eah dimension separately by one-dimensional inversion methods and improve the bound (24) similar to the one-dimensional ase to D,H ( ω) = D (ω). (25) Hlawka [] suggested to apply the onstrution (2) to every dimension individually to avoid an inversion of H,..., H s and gave a bound on the H- disrepany of sequenes ω generated by this proedure: where M = sup h(x). D,H ( ω) ( + 3sM)D (ω), Similar to one dimension, Hlawka s method might lead to sequenes, whih are not suited for unbounded integrands. Our final theorem shows a modifiation, whih leads to QMC estimators for a wide range of funtions. Before we an state it, we need to reall a lemma, whih will be essential for the proof. Lemma 3.5 (e.g. [8]) Let Ω = (u,..., u ) and Ω 2 = (v,..., v ) be two sequenes in [0, ] s. If the ondition u (j) i v (j) i ε j, holds for all j s and all i, we get the following bound on the differene of the disrepanies s D (Ω ) D (Ω 2 ) ( + 2ε j ). (26) j= Theorem 3.6 Let H be a s-dimensional distribution with independent marginal distributions H,..., H s and M = sup h(x) <. Let furthermore ω = (x, x 2,... ) be a sequene with (uniform) disrepany D (ω) and define the sequene ω = (ỹ, ỹ 2,... ) by ỹ (j) i = + x (j) i H i (x (j) n ), 2

13 for j =,..., s and n =,...,. Then the sequene ω ȳ (j) k = ỹ (j) k if ỹ (j) k, if ỹ (j) k = 0 (27) has the following two properties: D,H ( ω) ( + 4M) s D (ω) min j s, i ỹ(j) i PROOF. Let x (j) i = H i x (j) i Using Lemma 3.5 we obtain (ȳ (j) i D ( x, x 2,... ) D (ω) + ). Similar to Theorem 2.8 we get the inequality ( x (j) i M D (ω) + ). ( ( + M D (ω) + )) s. By Eq. (25) and the inequality D (ω) it follows that ( ( D,H ( ω) D (ω) + + M D (ω) + )) s (+4M) s D (ω). 4 Conlusion In this artile we suessfully showed that Quasi-Monte Carlo methods an also be applied to improper s-dimensional integrals, where the integrand funtion f(x) beomes singular at the integration boundary, assuming the funtion and the low-disrepany sequene in use display ertain properties as listed in Theorem 3.4. While similar onditions have long been known [4] for integration with respet to the uniform distribution (i.e. using uniformly distributed lowdisrepany sequenes), we were able to generalize these results to integration with respet to arbitrary multidimensional densities, often alled weighted integration. In [8] and [] Hlawka and Mük proposed a sheme to transform a uniformly distributed low-disrepany to a low-disrepany sequene with given density. However, the new sequene does not in general fulfill the onditions set forth in our onvergene theorems and so annot be used for the QMC integration of singular integrands. We were able to give a slight modifiation of the 3

14 transformed sequene so that it does not loose its low disrepany, but an be used for QMC weighted integration of singular integrands as our onvergene theorems show. We looked at one-dimensional densities and multidimensional densities whih an be fatored. Arbitrary multidimensional densities lead to the problem that even the inversion method does in general not preserve the disrepany of a low-disrepany sequene. Therefore the reation of low-h-disrepany sequenes still poses an open problem, in partiular sequenes suited for singular integrands. Referenes [] E. Hlawka, Funktionen von beshränkter Variation in der Theorie der Gleihverteilung, Ann. Mat. Pura Appl. (4) 54 (96) [2] L. Kuipers, H. iederreiter, Uniform distribution of sequenes, Wiley- Intersiene Publ., ew York, 974, pure and Applied Mathematis. [3] H. iederreiter, Random number generation and quasi-monte Carlo methods, Vol. 63 of SIAM Conf. Ser. Appl. Math., SIAM, Philadelphia, 992. [4] I. M. Sobol, Calulation of improper integrals using uniformly distributed sequenes, Soviet Math. Dokl. 4 (3) (973) [5] B. Klinger, umerial integration of singular integrands using low-disrepany sequenes, Computing 59 (3) (997) [6] E. de Donker, Y. Guan, Error bounds for the integration of singular funtions using equidistributed sequenes, Journal of Complexity, in press. [7] B. Tuffin, Simulation aélérée par les méthodes de monte arlo et quasi-monte arlo: théorie et appliations, Thèse de dotorat, Université de Rennes (997). [8] E. Hlawka, R. Mük, Über eine Transformation von gleihverteilten Folgen. II, Computing 9 (972) [9] L. Devroye, onuniform random variate generation, Springer-Verlag, ew York, 986. [0] E. Hlawka, R. Mük, A transformation of equidistributed sequenes, in: Appliations of number theory to numerial analysis, Aademi Press, ew York, 972, pp [] E. Hlawka, Gleihverteilung und Simulation, Österreih. Akad. Wiss. Math.- atur. Kl. Sitzungsber. II 206 (997)