Tutorial on quasi-monte Carlo methods

Transcription

1 Tutorial on quasi-monte Carlo methods Josef Dick School of Mathematics and Statistics, UNSW, Sydney, Australia

2 Comparison: MCMC, MC, QMC Roughly speaking: Markov chain Monte Carlo and quasi-monte Carlo are for different types of problems; If you have a problem where Monte Carlo does not work, then chances are quasi-monte Carlo will not work as well; If Monte Carlo works, but you want a faster method try (randomized) quasi-monte Carlo (some tweaking might be necessary). Quasi-Monte Carlo is an "experimental design" approach to Monte Carlo simulation; In this talk we shall discuss how quasi-monte Carlo can be faster than Monte Carlo under certain assumptions.

3 Outline DISCREPANCY, REPRODUCING KERNEL HILBERT SPACES AND WORST-CASE ERROR QUASI-MONTE CARLO POINT SETS RANDOMIZATIONS WEIGHTED FUNCTION SPACES AND TRACTABILITY

4 The task is to approximate an integral I s (f ) = f (z) dz [,1] s for some integrand f by some quadrature rule Q N,s (f ) = 1 N 1 f (x n ) N n= at some sample points x,..., x N 1 [, 1] s.

5 In other words: f (x) dx 1 N 1 f (x n ) [,1] N s n= Area under curve = Volume of cube average function value. If x,..., x N 1 [, 1] s are chosen randomly Monte Carlo If x,..., x N 1 [, 1] s chosen deterministically Quasi-Monte Carlo

6 Smooth integrands Integrand f : [, 1] R; say continuously differentiable; We want to study the integration error: Representation: f (x) = f (1) 1 1 f (x) dx 1 N 1 f (x n ). N n= f (t) dt = f (1) 1 x 1 [,t] (x)f (t) dt, where 1 [,t] (x) = { 1 if x [, t], otherwise.

7 Integration error 1 Substitute: 1 f (x)dx = 1 ( = f (1) N 1 1 f (x n ) = 1 N N n= Integration error: f (x) dx 1 N 1 f (x n ) = N n= = f (1) 1 f (1) ( N 1 f (1) n= 1 1 [,t] (x)f (t) dt ) 1 [,t] (x)f (t) dx dt, 1 N 1 1 N n= ( N [,t] (x n ) N n= 1 [,t] (x n )f (t) dt 1 [,t] (x n )f (t) dt. 1 1 [,t] (x) dx dx ) ) f (t) dt.

8 Local discrepancy Integration error: 1 f (x) dx 1 N 1 f (x n ) = N n= 1 ( ) N [,t] (x n ) t f (t) dt. N Let P = {x,..., x N 1 } [, 1]. Define the local discrepancy by n= P (t) = 1 N 1 1 [,t] (x n ) t, t [, 1]. N n= Then 1 f (x) dx 1 N 1 f (x n ) = N n= 1 P (t)f (t) dt.

9 Koksma-Hlawka inequality 1 f (x) dx 1 N 1 1 f (x n ) N = P (t)f (t) dt n= ( ) 1/p ( 1 1 P (t) p dt f (t) q dt where g Lp = ( g p) 1/p. = P Lp f Lq, 1 p + 1 q = 1, ) 1/q

10 Interpretation of discrepancy Let P = {x,..., x N 1 } [, 1]. Recall the definition of the local discrepancy: P (t) = 1 N 1 1 [,t] (x n ) t, t [, 1]. N n= Local discrepancy measures difference between uniform distribution and emperical distribution of quadrature points P = {x,..., x N 1 }. This is the Kolmogorov-Smirnov test for the difference between the empirical distribution of {x,..., x N 1 } and the uniform distribution.

11 Function space Representation Define inner product: f (x) = f (1) 1 x f (t) dt. f, g = f (1)g(1) + 1 f (t)g (t) dt. and norm Function space: f = f (1) f (t) 2 dt. H = {f : [, 1] R : f absolutely continuous and f < }.

12 Worst-case error The worst-case error is defined by 1 e(h, P) = sup f (x) dx 1 N 1 f (x n ) N. f H, f 1 n=

13 Reproducing kernel Hilbert space Recall: and f (y) = f (1) f, g = f (1)g(1) + f (x)( 1 [,x] (y)) dx 1 f (x)g (x) dx. Goal: Find set of functions g y H for each y [, 1] such that Conclusions: g y (1) = 1 for all y [, 1]; f, g y = f (y) for all f H. g y(x) = 1 [,x] (y) = Make g continuous such that g H; { 1 if y x, otherwise;

14 Reproducing kernel Hilbert space It follows that The function g y (x) = 1 + min{1 x, 1 y}. K (x, y) := g y (x) = 1 + min{1 x, 1 y}, x, y [, 1] is called reproducing kernel. The space H is called a reproducing kernel Hilbert space (with reproducing kernel K ).

15 Numerical integration in reproducing kernel Hilbert spaces Function representation: 1 f (z) dz = 1 N 1 1 f (x n ) = 1 N N n= f, K (, z) dz = f, 1 K (, z) dz N 1 f, K (, x n ) = f, 1 N 1 K (, x n ) N n= Integration error 1 f (z) dz 1 N 1 f (x n ) = f, N n= 1 = f, h, n= K (, z) dz 1 N 1 K (, x n ) N n= where h(x) = 1 K (x, z) dz 1 N 1 K (x, x n ). N n=

16 Worst-case error in reproducing kernel Hilbert spaces Thus e(h, P) = sup f H, f 1 = sup f H, f =1 = sup = f H,f h, h h 1 f, h h f f, = h, f (z) dz 1 N 1 f (x n ) N since the supremum is attained when choosing f = n= h h H.

17 Worst-case error in reproducing kernel Hilbert spaces e 2 (H, P) = N 1 N 2 n,m= K (x, y) dx dy 2 N K (x n, x m ) N 1 1 n= K (x, x n ) dx

18 Numerical integration in higher dimensions Tensor product space H s = H H. Reproducing kernel K (x, y) = s [1 + min{1 x i, 1 y i }], i=1 where x = (x 1,..., x s ), y = (y 1,..., y s ) [, 1] s. Functions f H s have partial mixed derivatives up to order 1 in each variable u f x u L 2 ([, 1] s ), where u {1,..., s}, x u = (x i ) i u and u denotes the cardinality of u and where u f x u (x u, 1) = for all u {1,..., s}.

19 Worst-case error Again e 2 (H, P) = [,1] s + 1 N 1 N 2 n,m= [,1] s K (x, y) dx dy 2 N K (x n, x m ) N 1 n= [,1] s K (x, x n ) dx and where e 2 (H, P) = P (x) s f (x) dx, [,1] x s P (x) = 1 N 1 1 [,x] (x n ) N n= s x i. i=1

20 Discrepancy in higher dimensions Point set P = {x,..., x N 1 } [, 1] s, t = (t 1,..., t s ) [, 1] s. Local discrepancy: where [, t] = s i=1 [, t i]. P (t) = 1 N 1 1 [,t] (x n ) N n= s t i, i=1

21 Koksma-Hlawka inequality Let f : [, 1] s R with ( f = [,1] s s x f (x) q dx) 1/q, and where u f x u (x u, 1) = for all u {1,..., s}. Then f (x) dx 1 N 1 f (x n ) [,1] N f P Lp, s where 1 p + 1 q = 1. n= Construct points P = {x,..., x N 1 } [, 1] s with small discrepancy L p (P) = P Lp. It is often useful to consider different reproducing kernels yielding different worst-case errors and discrepancies. We will see further examples later.

22 Constructions of low-discrepancy sequences Lattice rules (Bilyk, Brauchart, Cools, D., Hellekalek, Hickernell, Hlawka, Joe, Keller, Korobov, Kritzer, Kuo, Larcher, L Ecuyer, Lemieux, Leobacher, Niederreiter, Nuyens, Pillichshammer, Sinescu, Sloan, Temlyakov, Wang, Woźniakowski,...) Digital nets and sequences (Baldeaux, Bierbrauer, Brauchart, Chen, D., Edel, Faure, Hellekalek, Hofer, Keller, Kritzer, Kuo, Larcher, Leobacher, Niederreiter, Owen, Özbudak, Pillichshammer, Pirsic, Schmid, Skriganov, Sobol, Wang, Xing, Yue,...) Hammersley-Halton sequences (Atanassov, De Clerck, Faure, Halton, Hammersley, Kritzer, Larcher, Lemieux, Pillichshammer, Pirsic, White,...) Kronecker sequences (Beck, Hellekalek, Larcher, Niederreiter, Schoissengeier,...)

23 Lattice rules Let N N, let g = (g 1,..., g s ) {1,..., N 1} s. Choose the quadrature points as { ng } x n =, for n =,..., N 1, N where {z} = z z for z R +.

24 Fibonacci lattice rules Lattice rule with N = 55 points and generating vector g = (1, 34).

25 Fibonacci lattice rules Lattice rule with N = 89 points and generating vector g = (1, 55).

26 In comparison: Random point set Random set of 64 points generated by Matlab using Mersenne Twister.

27 Lattice rules How to find generating vector g? Reproducing kernel: s K (x, y) = (1 + 2πB α ({x i y i })), i=1 where {x i y i } = (x i y i ) x i y i is the fractional part of x i y i. Reproducing kernel Hilbert space of Fourier series: f (x) = f (h)e 2πih x. h Z s

28 Worst-case error: e 2 α(g) = N N 1 s ( n= i= πB α ({ ngi N })), where B α is the Bernoulli polynomial of order α. For instance B 2 (x) = x 2 x + 1/6.

29 Component-by-component construction (Korobov, Sloan-Reztsov,Sloan-Kuo-Joe, Nuyens-Cools) Set g 1 = 1. For d = 2,..., s assume that we have found g 2,..., g d 1. Then find g d {1,..., N 1} which minimizes e α (g 1,..., g d 1, g) as a function of g, i.e. g d = argmin g {1,...,N 1} e2 α(g 1,..., g d 1, g). Using fast Fourier transform (Nuyens, Cools), a good generating vector g {1,..., N 1} s can be found in O(sN log N) operations.

30 Hammersley-Halton sequence Radical inverse function in base b: Let n N have base b expansion n = n + n 1 b + + n a 1 b a 1. Set ϕ b (n) = n b + n 1 b n a 1 b a [, 1]. Hammersley-Halton sequence: Let P = {p 1, p 2,..., } be the set of prime numbers in increasing order, i.e. p 1 = 2, p 2 = 3, p 3 = 5, p 4 = 7,.... Define quadrature points x, x 1,... by x n = (ϕ p1 (n), ϕ p2 (n),..., ϕ ps (n)) for n =, 1, 2,....

31 Hammersley-Halton sequence Hammerlsey-Halton point set with 64 points.

32 Digital nets Choose prime number b and finite field Z b = {, 1,..., b 1} of order b. Choose C 1,..., C s Z m m b. Let n = n + n 1 b + + n m 1 b m 1 and set n = (n,..., n m 1 ) Z m b. Let y n,i = C i n for 1 i s, n < b m. For y n,i = (y n,i,1,..., y n,i,m ) Z m b let x n,i = y n,i,1 b + + y n,i,m b m. Set x n = (x n,1,..., x n,s ) for n < b m.

33 (t, m, s)-net property Let m, s 1 and b 2 be integers. A point set P = {x,..., x bm 1} is called a (t, m, s)-net in base b, if for all integers d 1,..., d s with d d s = m t the number of points in the elementary intervals s i=1 [ ai b d i, a ) i + 1 b d i where a i < b d i, is b t.

34 If P = {x,..., x bm 1} [, 1] s is a (t, m, s)-net, then local discrepancy function ( a1 P b d,..., a ) s = 1 b ds for all a i < b d i, di, d d s = m t.

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47 Sobol point set The first 64 points of a Sobol sequence which form a (, 6, 2)-net in base 2.

48 Niederreiter-Xing point set The first 64 points of a Niederreiter-Xing sequence.

49 Niederreiter-Xing point set The first 124 points of a Niederreiter-Xing sequence.

50 Kronecker sequence Let α 1,..., α s R be linearly independent over Q. Let x n = ({nα 1 },..., {nα s }) for n =, 1, 2,.... For instance, one can choose α i = p i where p i is the ith prime number.

51 Kronecker sequence The first 64 points of a Kronecker sequence.

52 Discrepancy bounds It is known that for all the constructions above one has (log N) c(s) L p (P) C s, for all N 1, 1 p, N where C s > and c(s) s. Lower bound: For all point sets P consisting of N points we have L p (P) C s (log N) (s 1)/2 N for all N 1, 1 < p. In comparison: random point set ( ) log log N E(L 2 (P)) = O. N

53 For 1 < p <, the exact order of convergence is known to be of order (log N) (s 1)/2. N Explicit constructions of such points were achieved by Chen & Skriganov for p = 2 and Skriganov for 1 < p <. Great open problem: What is the correct order of convergence of min P [,1] s P =N L (P)?

54 Randomized quasi-monte Carlo Introduce random element to deterministic point set. Has several advantages: yields an unbiased estimator; there is a statistical error estimate; better rate of convergence of the random-case error compared to worst-case error; performs similar to Monte Carlo for L 2 functions but has better rate of convergence for smooth functions;

55 Randomly shifted lattice rules Lattice rules can be randomized using a random shift: Lattice point set: {ng/n}, n =, 1,..., N 1. Shifted lattice rule: choose σ [, 1] s uniformly distributed; then the shifted lattice point set is given by {ng/n + σ}, n =, 1,..., N 1.

56 Lattice point set

57 Shifted lattice point set

58 Owen s scrambling Let Z b = {, 1,..., b 1} and let x = x 1 b + x 2 b 2 +, x 1, x 2,... Z b. Randomly choose permutations σ, σ x1, σ x1,x 2,... : Z b Z b. Let y 1 = σ(x 1 ) y 2 = σ x1 (x 2 ) y 3 = σ x1,x 2 (x 3 ) Set y = y 1 b + y 2 b 2 +.

59 Scrambled Sobol point set The first 124 points of a Sobol sequence (left) and a scrambled Sobol sequence (right).

60 Scrambled net variance Let Then e(r) = f (x) dx 1 [,1] N s E(e(R)) = b m 1 n= f (y n ). ( ) (log N) s Var(e(R)) = O N 2α+1 for integrands with smoothness α 1.

61 Dependence on the dimension Although the convergence rate is better for quasi-monte Carlo, there is a stronger dependence on the dimension: Monte Carlo: error = O(N 1/2 ); Quasi-Monte Carlo: error = O ( (log N) s 1 N ) ; Notice that g(n) := N 1 (log N) s 1 is an increasing function for N e s 1. So if s is large, say s 3, then N 1 (log N) 29 increases for N 1 12.

62 Intractable For integration error in H with reproducing kernel s K (x, y) = min{1 x i, 1 y i } i=1 it is known that e 2 (H, P) ( 1 N ( ) s ) 8 e 2 (H, ). 9 Error can only decrease if N > constant ( ) s 9. 8

63 Weighted function spaces (Sloan and Woźniakowski) Study weighted function spaces: Introduce γ 1, γ 2,..., > and define K (x, y) = Then if s (1 + γ i min{1 x i, 1 y i }). i=1 γ i < we have e(h s,γ, P) CN δ, where C > is independent of the dimension s and δ < 1. i=1

64 Tractability Minimal error over all methods using N points: Inverse of the error: for ε > ; Strong tractability: e N(H) = inf e(h, P); P: P =N N (s, ε) = min{n N : e N(H) εe (H)} for some constant C > ; N (s, ε) Cε β Sloan and Woźniakowski: For the function space considered above, we get strong tractability if γ i <. i=1

65 Tractability of star-discrepancy Recall the local discrepancy function P (t) = 1 N 1 1 [,t) (x n ) t 1 t s, where P = {x,..., x N 1 }. N n= The star-discrepancy is given by D N(P) = sup P (t). t [,1] s Result by Heinrich, Novak, Woźniakowski, Wasilkowski: s Minimal star discrepancy D (s, N) = inf P D N(P) C N for all s, N N. This implies that N (s, ε) Csε 2.

66 Extensions, open problems and future research Quasi-Monte Carlo methods which achieve convergence rates of order N α (log N) αs, α > 1 for sufficiently smooth integrands; Construction of point sets whose star-discrepancy is tractable; Completely uniformly distributed point sets to speed up convergence in Markov chain Monte Carlo algorithms; Infinite dimensional integration: Integrands have infinitely many variables; Connections to codes, orthogonal arrays and experimental designs; Quasi-Monte Carlo for function approximation; Choosing the weights in applications; Quasi-Monte Carlo on the sphere;

67 Book

68 Thank You! Special thanks to Dirk Nuyens for creating many of the pictures in this talk; Friedrich Pillichshammer for letting me use a picture of his daughters; Fred Hickernell, Peter Kritzer, Art Owen, and Friedrich Pillichshammer for helpful comments on the slides; All colleagues who worked on quasi-monte Carlo methods over the years.