Draft Lattice Builder A Software Tool for Constructing Lattice - PowerPoint PPT Presentation

1 Draft Lattice Builder A Software Tool for Constructing Lattice Rules Pierre L’Ecuyer and David Munger Informatique et Recherche Op´ erationnelle, Universit´ e de Montr´ eal

1 Draft Lattice Builder A Software Tool for Constructing Lattice Rules Pierre L’Ecuyer and David Munger Informatique et Recherche Op´ erationnelle, Universit´ e de Montr´ eal Outline: 1. Setting: integration by (randomly-shifted) lattice rules. 2. Want a tool to search for good lattices adapted to problem at hand, for arbitrary dimension, size, figure of merit, ... 3. Selection criteria, construction methods, weights on projections, ... 4. Introducing Lattice Builder .

2 Draft Monte Carlo integration Want to estimate � µ = [0 , 1) s f ( u ) d u = E [ f ( U )] where f : [0 , 1) s → R and U is a uniform r.v. over [0 , 1) s . Standard Monte Carlo: ◮ Generate n independent realizations of U , say U 1 , . . . , U n ; µ n = 1 � n ◮ estimate µ by ˆ i =1 f ( U i ). n

2 Draft Monte Carlo integration Want to estimate � µ = [0 , 1) s f ( u ) d u = E [ f ( U )] where f : [0 , 1) s → R and U is a uniform r.v. over [0 , 1) s . Standard Monte Carlo: ◮ Generate n independent realizations of U , say U 1 , . . . , U n ; µ n = 1 � n ◮ estimate µ by ˆ i =1 f ( U i ). n µ n ] = σ 2 / n where σ 2 = � [0 , 1) s f 2 ( u ) d u − µ . Variance: Var [ˆ Central limit theorem: √ n (ˆ µ n − µ ) /σ ⇒ N (0 , 1) when n → ∞ .

3 Draft Quasi-Monte Carlo (QMC) Replace the random points U i by a set of deterministic points P n = { u 0 , . . . , u n − 1 } that cover [0 , 1) s more evenly. That is, low discrepancy between empirical distribution of P n and uniform distribution.

3 Draft Quasi-Monte Carlo (QMC) Replace the random points U i by a set of deterministic points P n = { u 0 , . . . , u n − 1 } that cover [0 , 1) s more evenly. That is, low discrepancy between empirical distribution of P n and uniform distribution. Main construction methods : lattice rules and digital nets (Korobov, Hammersley, Halton, Sobol’, Faure, Niederreiter, etc.)

4 Draft Randomized quasi-Monte Carlo (RQMC) estimator n − 1 µ n , rqmc = 1 � ˆ f ( U i ) , n i =0 with P n = { U 0 , . . . , U n − 1 } ⊂ (0 , 1) s an RQMC point set: (i) each point U i has the uniform distribution over (0 , 1) s ; (ii) P n as a whole is a low-discrepancy point set. E [ˆ µ n , rqmc ] = µ (unbiased) . Var [ f ( U i )] 2 � Var [ˆ µ n , rqmc ] = + Cov [ f ( U i ) , f ( U j )] . n 2 n i < j We want to make the last sum as negative as possible.

5 Draft Integration lattice   s   � L s =  v = z j v j such that each z j ∈ Z  , j =1 where v 1 , . . . , v s ∈ R s are linearly independent over R and where L s contains Z s . Lattice rule: Take P n = { u 0 , . . . , u n − 1 } = L s ∩ [0 , 1) s .

5 Draft Integration lattice   s   � L s =  v = z j v j such that each z j ∈ Z  , j =1 where v 1 , . . . , v s ∈ R s are linearly independent over R and where L s contains Z s . Lattice rule: Take P n = { u 0 , . . . , u n − 1 } = L s ∩ [0 , 1) s . Lattice rule of rank 1: u i = i v 1 mod 1 for i = 0 , . . . , n − 1. n v 1 = a = ( a 1 , . . . , a s ) ∈ Z s n . Korobov rule: a = (1 , a , a 2 mod n , . . . , a s − 1 mod n ).

5 Draft Integration lattice   s   � L s =  v = z j v j such that each z j ∈ Z  , j =1 where v 1 , . . . , v s ∈ R s are linearly independent over R and where L s contains Z s . Lattice rule: Take P n = { u 0 , . . . , u n − 1 } = L s ∩ [0 , 1) s . Lattice rule of rank 1: u i = i v 1 mod 1 for i = 0 , . . . , n − 1. n v 1 = a = ( a 1 , . . . , a s ) ∈ Z s n . Korobov rule: a = (1 , a , a 2 mod n , . . . , a s − 1 mod n ). For any u ⊂ { 1 , . . . , s } , the projection L s ( u ) of L s is also a lattice, with point set P n ( u ).

5 Draft Integration lattice   s   � L s =  v = z j v j such that each z j ∈ Z  , j =1 where v 1 , . . . , v s ∈ R s are linearly independent over R and where L s contains Z s . Lattice rule: Take P n = { u 0 , . . . , u n − 1 } = L s ∩ [0 , 1) s . Lattice rule of rank 1: u i = i v 1 mod 1 for i = 0 , . . . , n − 1. n v 1 = a = ( a 1 , . . . , a s ) ∈ Z s n . Korobov rule: a = (1 , a , a 2 mod n , . . . , a s − 1 mod n ). For any u ⊂ { 1 , . . . , s } , the projection L s ( u ) of L s is also a lattice, with point set P n ( u ). Random shift modulo 1: generate a single point U uniformly over (0 , 1) s and add it to each point of P n , modulo 1, coordinate-wise: U i = ( u i + U ) mod 1. Each U i is uniformly distributed over [0 , 1) s .

6 Draft Randomly-shifted lattice: Two-dim. example 1 u i , 1 u i , 2 0 1

6 Draft Randomly-shifted lattice: Two-dim. example 1 u i , 1 u i , 2 0 1

6 Draft Randomly-shifted lattice: Two-dim. example 1 u i , 1 U u i , 2 0 1

6 Draft Randomly-shifted lattice: Two-dim. example 1 u i , 1 u i , 2 0 1

6 Draft Randomly-shifted lattice: Two-dim. example 1 u i , 1 u i , 2 0 1

7 Draft Example of a poor lattice: s = 2 , n = 101 , a = 51 1 u i , 1 u i , 2 0 1 Good uniformity in one dimension, but not in two!

8 Draft Variance expression Suppose f has Fourier expansion f ( h ) e 2 π √− 1 h t u . � ˆ f ( u ) = h ∈ Z s For a randomly shifted lattice, the exact variance is � | ˆ f ( h ) | 2 , Var [ˆ µ n , rqmc ] = 0 � = h ∈ L ∗ s s = { h ∈ R s : h t v ∈ Z for all v ∈ L s } ⊆ Z s is the dual lattice. where L ∗ From the viewpoint of variance reduction, an optimal lattice for f minimizes D 2 f ( P n ) = Var [ˆ µ n , rqmc ]. But not a practical criterion...

9 Draft Periodic smooth functions If f has square-integrable mixed partial derivatives up to order α/ 2, and the periodic continuations of its derivatives up to order α/ 2 − 1 are continuous across the unit cube boundaries, then f ( h ) | 2 = O ((max(1 , | h 1 | ) · · · max(1 , | h s | )) − α ) . | ˆ Moreover, there is a vector v 1 = v 1 ( n ) such that (max(1 , | h 1 | ) · · · max(1 , | h s | )) − α = O ( n − α + δ ) . def � P α = 0 � = h ∈ L ∗ s This P α has been proposed long ago as a figure of merit, often with α = 2. It is the variance for a worst-case f having f ( h ) | 2 = (max(1 , | h 1 | ) · · · max(1 , | h s | )) − α . | ˆ

10 Draft If α/ 2 is a positive integer, this worst-case f is (2 π ) α/ 2 � � f ( u ) = ( α/ 2)! B α/ 2 ( u j ) . u ⊆{ 1 ,..., s } j ∈ u where B α/ 2 is the Bernoulli polynomial of degree α/ 2. This worst-case function is not necessarily representative of what happens in applications. Moreover, the hidden factor in O increases quickly with s , so this result is not very useful for large s . To get a bound that is uniform in s , the Fourier coefficients must decrease faster with the dimension and “size” of vectors h ; that is, f must be “smoother” in high-dimensional projections. This is typically what happens in applications where RQMC is really effective! [cf., tractability theory; W´ ozniakowski, Novak, Sloan, Dick, etc.]

11 Draft A very general weighted P α P α can be generalized by giving different weights w ( h ) to the vectors h : def ˜ � w ( h )(max(1 , | h 1 | ) · · · max(1 , | h s | )) − α . P α = 0 � = h ∈ L ∗ s But how do we choose these weights? There are too many! The optimal weights to minimize the variance are: w ( h ) = (max(1 , | h 1 | ) · · · max(1 , | h s | )) α | ˆ f ( h ) | 2 .

12 Draft ANOVA decomposition A cruder expansion of f ( u ) = f ( u 1 , . . . , u s ): s s � � � f ( u ) = f u ( u ) = µ + f { i } ( u i ) + f { i , j } ( u i , u j ) + · · · u ⊆{ 1 ,..., s } i =1 i , j =1 where � � f u ( u ) = u | f ( u ) d u ¯ u − f v ( u v ) . [0 , 1) | ¯ v ⊂ u The Monte Carlo variance decomposes as σ 2 = � σ 2 where σ 2 u , u = Var [ f u ( U )] . u ⊆{ 1 ,..., s } The σ 2 u ’s can be estimated by MC or RQMC. Heuristic intuition: Make sure the projections P n ( u ) are very uniform for the important subsets u (i.e., with larger σ 2 u ).

13 Draft Weighted P γ,α with projection-dependent weights γ u Denote u ( h ) = u ( h 1 , . . . , h s ) the set of indices j for which h j � = 0. � γ u ( h ) (max(1 , | h 1 | ) · · · max(1 , | h s | )) − α . P γ,α = 0 � = h ∈ L ∗ s For α/ 2 integer > 0, if u i = ( u i , 1 , . . . , u i , s ) = i v 1 mod 1, n − 1 � | u | � � − ( − 4 π 2 ) α/ 2 1 � � P γ,α = γ u B α ( u i , j ) ( α )! n ∅� = u ⊆{ 1 ,..., s } i =0 j ∈ u (finite sum) and the corresponding variation is 2 ∂ α | u | / 2 � � 1 � � V 2 � � γ ( f ) = ∂ u α/ 2 f u ( u ) d u , � � γ u (4 π 2 ) α | u | / 2 [0 , 1] | u | � � ∅� = u ⊆{ 1 ,..., s } for f : [0 , 1) s → R smooth enough. Then, � µ n , rqmc ( f u )] ≤ V 2 Var [ˆ µ n , rqmc ] = Var [ˆ γ ( f ) P γ,α . u ⊆{ 1 ,..., s }