Quasi-Monte Carlo methods
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1 Quasi-Mote Carlo methods SE Semiar zur Numerik ud Stochastik) Lukas Eikemmer March 29, Itroductio The geeral problem we are iterested i is to umerically compute the itegral I : fx) dx, [0,1] d where f : [0, 1] d R ad d N is large usually d 4). As is show i [3] classical methods of umerical itegratio are ill-suited for such problems. A umber of related methods that alleviates this problem are kow as Mote Carlo itegratio methods. Such methods ca be show to have a probabilistic error boud of order O 1/2) where is the umber of fuctio evaluatios). I this term paper we will itroduce other methods of approximatig the before-metioed itegral, amely quasi-mote Carlo methods, which are supercial similar to Mote Carlo methods but have certai advatages, especially for moderately sized d e.g. d < 20). Mote Carlo methods use a sequece of idepeet) radom umbers to determie the poits at which f is evaluated. Radom umbers, however, have, e.g., the disadvatage that they exhibit clusterig behavior. Quasi- Mote Carlo methods, o the other had, use a fully determiistic sequeces that tries to evely covers the uit cube. This rather vague otio of evely covers will be made more precise i the followig sectio by the so called discrepacy of a sequece. The developmet i this term paper maily follows [1] ad [6]. 2 Quasi-Mote Carlo Itegratio First, let us precisely dee what we mea by the discrepacy of a sequece with elemets. Deitio 2.1. Discrepacy). Suppose E is the set of all rectagles i [0, 1] d a rectagle is assumed to be aliged with respect to the coordiate axes). The, the discrepacy of the sequece x 0,..., x 1 is deed by D : sup #{x i J : i {0,..., 1}} volj. J E Surprisigly a alterative measure of discrepacy, the so called star discrepacy, will prove more useful. Deitio 2.2. Star discrepacy). Suppose E is the set of all rectagles J i [0, 1] d such that 0 is a vertex of the rectagle. The, the star discrepacy of the sequece x 0,..., x 1 is deed by D : sup #{x i J : i {0,..., 1}} volj. J E I additio, we eed the cocept of the variatio of a fuctio i the Hardy-Krause sese). It should be oted that this deitio is ot equivalet to the usual deitio of variatio. It diers i that we cosider variatio at the upper boudary as well.
2 2 Deitio 2.3. Bouded variatio). Suppose f : [0, 1] R is cotiuously dieretiable. The, the variatio of f is deed by 1 V f) : df dx dx. Suppose f : [0, 1] d R is a dieretiable fuctio. The, the variatio of f is deed by V f) : [0,1] d 0 d fx) x 1... x d dx 1... dx d + d V f Ai ), where A i {x [0, 1] d : x i 1}. We say a fuctio f has bouded variatio if V f) exists ad is ite. For umerical itegratio purposes the followig result is of cetral importace. Theorem 2.4. Koksma-Hlawka theorem). Suppose f : [0, 1] d R is a fuctio of bouded variatio ad x i ). The, I I V f)d, where I : 1 fx i ). Proof. Suppose J E i.e. the positio of oe vertex is 0). The, J is completely characterized by the atipodal vertex, deoted by x, of the vertex 0. Therefore, we use the otatio Jx). Set Clearly Rx) : Jx) 1 δy x i ) 1 dy #{x i Jx) : i {1,..., }} Rx) x 1... x d 1 δy x i ) 1 Therefore, the followig computatio gives us a estimate of the approximatio error. volj.
3 I I fx) dx 1 fx i ) [0,1] d [ ] 1 1 δx x i ) fx) dx [0,1] d Rx) fx) dx [0,1] x d 1... x d 1) α Rx) α fx) α Z2 d [0,1] x α 1 x α 0 dx α α 1) α Rx) α fx) α Z2 d [0,1] x α x α 1 dx α ) sup Rx) 1) α α f xα1x) x [0,1] d α Z2 d [0,1] x α dx α α ) sup Rx) α f x α 1x) x [0,1] d α Z2 d [0,1] x α dxα α ) sup Rx) x [0,1] d D V f), V f) where we used the fact that Rx) 0 if ay compoet of x is 0, as well as itegratio by parts. Furthermore, we used the fact that the recursive deitio give i Deitio 2.3 ca be writte as V f) α f x α 1x) x α dxα. α Z d 2 [0,1] α 3 Low discrepacy sequece I the previous sectio we derived a error boud Theorem 2.4) for quasi-mote Carlo itegratio. Sice the variatio of f is xed for a give fuctio our task is to d a sequece that miimizes D. The rst sequece we discuss, the Va der Corput sequece, is a somewhat articial example i a sigle dimesio. However, it forms the basis for a class of importat sequeces i multi-dimesioal space. Theorem 3.1. Every N 0 has a uique digit expasio i a base b N 2 of the form a j )b j, where a j ) {0, 1,..., b 1} for every j 0 ad a j ) 0 for all sucietly large j. j0 Proof. We proceed by iductio. First, 0 has obviously the expasio a j ) 0, j N 0. Suppose we ca expad 1 as 1 a j 1)b j. j0
4 Basis i Table 3.1: Va der Corput sequece. The, 1 + a 0 1)) + a j 1)b j. If 1 + a 0 < b 1 we are doe. Otherwise, 1 + a 0 b ad we write a 1 1)) + a j 1)b j. This procedure termiates after a ite umber of steps, as desired. Deitio 3.2. Radical-iverse fuctio). j1 j2 φ b :N 0 [0, 1) a j )b j 1 Deitio 3.3. Va der Corput sequece). Suppose b N 2. The, the sequece x i ) i0 deed by j0 x i φ b i) for every i 0 is called the va der Corput Sequece i base b. Example 3.4. The rst seve elemets of the va der Corput sequece i base b 2, 3, 4 are give i table 3.1. The va der Corput sequece ca be exteded i a obvious way to a sequece i [0, 1) d. Deitio 3.5. Halto sequece). Suppose b 1,..., b d N 2. The, the sequece x i ) i0 deed by φ b1 i) x i. φ bd i) is called the Halto sequece i the bases b 1,..., b d. Obviously, for d 1 the Halto sequece reduces to the va der Corput sequece i base b 1 ). Example 3.6. A compariso betwee radom umbers ad the Halto sequece i bases 2, 3 is show i gure 1 ad gure 2. Theorem 3.7. If x 0,..., x 1 are the rst elemets of the Halto sequece i the bases b 1,..., b d, where b 1,..., b d are pairwise coprime. The, D < d + 1 d bi 1 log + b ) i log b i 2 Therefore, D Ab 1,..., b d ) logd + O ) log d 1 with Ab 1,..., b d ) d b i 1 2 log b i.
5 5 Figure 1: Halto sequece ad radom umbers 100 poits each). Figure 2: Halto sequece ad radom umbers poits each).
6 6 Figure 3: Halto sequece i 15 ad 50 dimesios. Proof. See e.g. [6, p ]. The questio immediately arises i which way b 1,..., b d has to be chose such as to miimize Ab 1,..., b d ). Sice b 1,..., b d are pairwise coprime the obvious aswer is give by Colollary 3.8. Corollary 3.8. If b 1,..., b d is chose such that b 1 p 1,..., b d p d, where p i is the i-th prime umber, the Ab 1,..., b d ) A d d p i 1 2 log p i is the uique) global miimum uder the costraits that b 1,..., b d are pairwise coprime. Proof. Sice b i 1 2 log b i is mootoically decreasig, we must choose b i as small as possible. umbers are the uique choice. Give these costraits the rst d prime It is widely believed that o sequece with better asymptotic properties with respect to ) tha the Halto sequece ca be foud see e.g. [6, p. 32]). Therefore, we make the followig deitio. Deitio 3.9. Low-discrepacy sequece). Suppose x 0,..., x 1 are the rst elemets of a sequece x i ) i0. We call x i) i0 a low-discrepacy sequece if ) D log d O, as. are pairwise coprime, is a low- Theorem The Halto sequece for ay base b 1,..., b d, where b 1,..., b d discrepacy sequece. Proof. Follows from The eed for additioal low-discrepacy sequeces The Halto sequece discussed i the previous sectio has a aw that reders it, almost, useless if the umber of dimesios is large. This result ca be aticipated from gure 3. The precise statemet is give by Theorem 4.1.
7 Theorem 4.1. Suppose x 0,..., x 1 are the rst elemets of a Halto sequece. The, log A d 1. Proof. The prime umber theorem states that see e.g. [5]) x πx) x/ log x 1, where π #{p x : p prime}. Applyig the logarithm we get ) πx) 0 log x x/ log x log x x Therefore, which implies log πx) log x ) log log x 1 +. log x log πx) 0 x log x 1 + log log x ) ) log πx) log x x log x 1, log πx) x log x 1. Suppose p i is the i-th prime umber. The πp i ) i. Thus, for x p i we get This result ca be applied to 1 x x πx) log x x x πx) log πx) i p i i log i. d pi 1 log A d log 2 log p i sice for sucietly large d) I additio, d d [logi log i 1) log2 logi log i))] d d log d d d log2 logi log i)) d log2 logd log d)) log2 logd log d)) 0. d log d d logi log i 1) d logd log d) logd 1+ɛ ) 1 + ɛ, d log d for every ɛ > 0. Furthermore, sice logi log i 1) is mootoically icreasig d d logi log i 1) logi1 ɛ ) for every ɛ > 0 ad sucietly large d. Thus, as desired. 1 ɛ) d log A d 1, d d logd + 1) + log d d d log d d+1 1 ɛ) logx) dx 1 d log d 1 ɛ, logi log i 1), d log d The previous theorem implies that A d grows superexpoetially. Therefore, for large d, the costat A d becomes extremely large ad reders the Halto sequece uusable for most practical applicatios. The ext sectio discusses a geeral theory of costructig low-discrepacy sequeces that tries to overcome this diculty.
8 5 Geeral theory of low-discrepacy sequeces Now we tur to a property that dees a class of low-discrepacy sequeces. Although rather techical at rst sight, the followig deitios ca be uderstood by thikig of a sequece that covers the space at a give degree of eess before movig to a higher degree. Deitio 5.1. A iterval E [0, 1] d of the form E d [a i b ei, a i + 1)b ei ) with a i, e i Z ad e i 0, 0 a i < b ei for 1 i d is called a elemetary iterval i base b. Deitio 5.2. Suppose m, t Z ad 0 t m. A t, m, d)-et i base b is a set P with #P ) b m such that for every elemetary iterval of volume b t m it holds that #P E) b t. Deitio 5.3. Suppose t Z, where t 0. A sequece x i ) i0 is a t, d)-sequece i base b if, for all k Z, k 0 ad m > t the set {x i : kb m i k + 1)b m } is a t, m, d)-et i base b. Theorem 5.4. Suppose x i ) i0 is a t, d)-sequece i base b. The, D t log N)d b t log N) d 1 ) B d b + O, N N where if either d 2, or b 2 ad d 3, 4. Otherwise, Proof. See e.g. [6, p ]. B d 1 d B d 1 b 1 d! 2 b/2 ) d b 1, 2 log b ) d b/2. log b Various methods have bee proposed to costruct such sequeces. The most promiet beig Niederreiter ad geeralized Niederreiter sequeces, which iclude may other sequeces e.g. Sobol sequeces or Faure sequeces) as a special case. From the estimate i Theorem 5.4 it is obvious that sequeces with a small value of t are sought. It ca be show, e.g., that Niederreiter sequeces are optimal i this regard. However, a detailed treatmet is beyod the scope of this term paper. The iterested reader is referred to [6, Chap ] or [2]. 6 Browia bridge discretizatio As already discussed, quasi-mote Carlo methods oer a improvemet over Mote Carlo methods oly if the umber if dimesios is moderate. We will ow discuss a method to solve the heat equatio diusio equatio, imagiary time Schrödiger equatio) by usig quasi-mote Carlo methods, eve though the umber of dimesios is large. Theorem 6.1. Feyma-Kac formula). Suppose u C 1,2 [0, T ] R d, R) is bouded ad satises where f : R d R ad V : R d R. The t ut, x) E where X t is a Browia motio with EX t ) x. u t 1 u + V u, 2 u0, x) fx), 0 ) fx s )e s 0 V Xσ) dσ ds, 1)
9 9 Proof. See e.g. [4, p. 5-7]. The Feyma-Kac formula ca be stated i a way that does't it its applicatio to simple heat equatios with a diusio term. However, for our developmet the theorem stated above is suciet. The rst step is to approximate the itegral i equatio 1. The most basic method is to use a uiform grid. Deitio 6.2. Uiform grid discretizatio with m odes). t 0 fx s )e s 0 V Xσ) dσ ds 1 m m j1 [ fy j )e 1 j j k1 V Y k) ], where Y 0 x ad Y j Y j 1 + N 0, t/m). To compute the expected value i equatio 1 we eed to evaluate a m d dimesioal itegral. For a good approximatio we choose m as large as computatioal costraits permit. To this itegral, we could easily apply Mote Carlo methods. However, due to the large umber of dimesios usually eeded i such a approximatio quasi-mote Carlo methods would lose most of their advatage. Therefore, we use the followig approximatio. Deitio 6.3. Browia bridge discretizatio with m odes). t 0 fx s )e s 0 V Xσ) dσ ds 1 m m j1 [ fy j )e 1 j j k1 V Y k) ]. Suppose m 2 l for l N 1. Give Y 0 x ad Y m x + N 0, t) we use the followig costructio Y m/2 1 2 Y Y m + N 0, a m/2), Y m/4 1 2 Y Y m/2 + N 0, a m/4), Y 3m/4 1 2 Y m/ Y m + N 0, a m/4). Similar costructios are possible for m 2 l See e.g. [1, p. 39]). For Mote Carlo methods the above discretizatio does't chage the error boud sice the overall variace is still the same. However, i the Browia bridge discretizatio the large time steps are lled i rst, resultig i a umber of dimesios with large variace ad a umber of dimesios with smaller variace. The basic idea is to use quasi-mote Carlo itegratio o say the rst 10 dimesios, which have large variace. I this case the error scales close to O 1 ). For the remaiig dimesios Mote Carlo itegratio is employed. I this case the error scales as O 1/2 ). Refereces [1] Russel E. Caisch. Mote Carlo ad quasi-mote Carlo methods. Acta Numerica, pages 149, http: //dsec.pku.edu.c/~tieli/otes/umer_aal/mcqmc_caflisch.pdf. [2] Josef Dick ad Harald Niederreiter. O the exact t-value of iederreiter ad sobol' sequeces [3] Lukas Eikemmer. Mote Carlo methods, Uiversity of Isbruck bachelor thesis), [4] Nikola Kamburov. The feyma-kac formula [5] Joh Kopfmacher. Ecyclopaedia of mathematics [6] Harald Niederreiter. Radom Number Geeratio ad Quasi-Mote Carlo Methods. Society for Idustrial Mathematics, 1992.
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