Construction Algorithms for (Polynomial) Lattice Points Peter Kritzer Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Joint work with J. Dick, A. Ebert, G. Leobacher, D. Nuyens, O.O. Osisiogu, F . Pillichshammer MCM 2019, Sydney Research supported by the Austrian Science Fund, Project F5506-N26 Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 1
Johann Radon Institute for Computational and Applied Mathematics Introduction and Motivation 1 The reduced CBC construction 2 The reduced fast CBC construction for POD weights 3 4 The successive coordinate search (SCS) construction 5 The reduced fast SCS construction The CBC-DBD construction 6 Conclusion 7 Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 2
Johann Radon Institute for Computational and Applied Mathematics Introduction and Motivation Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 3
Johann Radon Institute for Computational and Applied Mathematics Consider integration of functions on [ 0 , 1 ] d , � I d ( f ) = [ 0 , 1 ] d f ( x ) d x , where f ∈ H , and H is some Banach space. Approximate I d by a quasi-Monte Carlo (QMC) rule, N − 1 � I d ( f ) ≈ Q N , d ( f ) = 1 f ( x k ) , N k = 0 where P N = { x 0 , . . . , x N − 1 } . Both parameters d and N can be (very) large. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 4
Johann Radon Institute for Computational and Applied Mathematics Worst case error in Banach space H with respect to P N = { x 0 , . . . , x N − 1 } : e N , d ( H , P N ) := sup | I d ( f ) − Q N , d ( f ) | . f ∈H , � f �≤ 1 Need P N that makes e N , d ( H , P N ) small: How can we find it? Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 5
Johann Radon Institute for Computational and Applied Mathematics Function space considered here: Weighted Korobov space ( H d ,α, γ , �·� d ,α, γ ) : space of 1-periodic continuous functions f , where � ρ α, γ ( h ) | � � f � 2 f ( h ) | 2 , d ,α, γ = h ∈ Z d where � f ( h ) is the h -th Fourier coefficient of f . The function ρ α, γ ( h ) moderates the decay of the Fourier coefficients. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 6
Johann Radon Institute for Computational and Applied Mathematics Set ρ α, γ ( 0 ) := 1. For h = ( h 1 , . . . , h d ) , let u ⊆ { 1 , . . . , d } be the set of the j with h j � = 0. Then � | h j | α . ρ α, γ ( h ) = γ − 1 u j ∈ u α > 1: “smoothness parameter” (higher α → smoother functions in H d ,α, γ ), γ = ( γ u ) u ⊆{ 1 ,..., d } : coordinate weights. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 7
Johann Radon Institute for Computational and Applied Mathematics Weights? Sloan and Wo´ zniakowski (1998): Assign weights to different groups of coordinates to model their different influence on a problem: γ = ( γ u ) u ⊆{ 1 ,..., d } of positive reals: weights. larger weights ≃ more influence of corresponding variables, smaller weights ≃ less influence of corresponding variables. Suitable weights can help to reduce negative influence of the dimension. Important class of weights: product weights � γ u = γ j j ∈ u for positive reals γ j . In this case, assume 1 = γ 1 , and γ j ≥ γ j + 1 for all j . Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 8
Johann Radon Institute for Computational and Applied Mathematics Here: P N = { x 0 , . . . , x N − 1 } is a rank-1 lattice point set with generating vector z = ( z 1 , . . . , z d ) ∈ { 1 , . . . , N − 1 } d . Points of P N : x n = ( x n , 1 , . . . , x n , d ) with � nz j � x n , j = , N where { t } = t − ⌊ t ⌋ . Note: Given N and d , z fully determines the lattice point set. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 9
Johann Radon Institute for Computational and Applied Mathematics Lattice point set with d = 2, N = 34, and z = ( 1 , 21 ) : Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 10
Johann Radon Institute for Computational and Applied Mathematics Worst-case error of a lattice rule with generating vector z : � ( ρ α, γ ( h )) − 1 . e 2 N , d ,α, γ ( z ) = 0 � = h ∈ Z d h · z ≡ 0 mod N Explicit formula for the (squared) worst-case error for product weights: � �� nz j ��� N − 1 � � d N , d ,α, γ ( z ) = − 1 + 1 e 2 1 + γ j ϕ α , N N n = 0 j = 1 � k � where ϕ α can be computed for all values of k = 0 , . . . , N − 1. N The error formula is easy to implement. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 11
Johann Radon Institute for Computational and Applied Mathematics Question: is the Korobov space interesting? Yes, first reason: it is a model function space that makes it easier to understand how lattice rules work. Yes, second reason: Bounds on the worst-case error of lattice rules in the Korobov space with α = 2 immediately yield bounds on the worst-case error of slightly modified lattice rules in certain unanchored Sobolev spaces. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 12
Johann Radon Institute for Computational and Applied Mathematics All that remains is to find “good” z ∈ { 1 , . . . , N − 1 } d . Huge search space of size ( N − 1 ) d . (e.g., N = 10 000 and d = 100). Component by component (CBC) construction: greedy algorithm to construct z j one at a time. Size of search space is N − 1 per component. (Almost) optimal convergence of error bounds (Kuo, 2003). Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 13
Johann Radon Institute for Computational and Applied Mathematics Algorithm 1 (CBC construction) Let N be given. Construct z = ( z 1 , . . . , z d ) as follows. Set z 1 = 1 . For s ∈ { 2 , . . . , d } assume that z 1 , . . . , z s − 1 have already been found. Now choose z s ∈ { 1 , . . . , N − 1 } such that e 2 N , s ,α,γ (( z 1 , . . . , z s − 1 , z s )) is minimized as a function of z s . Increase s and repeat the second step until ( z 1 , . . . , z d ) is found. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 14
Johann Radon Institute for Computational and Applied Mathematics Can do fast CBC (Cools, Nuyens, 2006): computation cost of O ( dN log N ) . Computation cost of O ( dN log N ) can still be demanding for big N , d . Might want to have big N , d simultaneously. → Can we speed up the search? Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 15
Johann Radon Institute for Computational and Applied Mathematics The reduced CBC construction (joint work with J. Dick, G. Leobacher, F. Pillichshammer) Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 16
Johann Radon Institute for Computational and Applied Mathematics Idea: small weights for later components → make search space smaller for later components z j . Let N be a prime power, N = b m , b prime, m ∈ N . Let w 1 , w 2 , . . . ∈ N 0 with 0 = w 1 ≤ w 2 ≤ · · · . Consider the sequence of reduced search spaces � { 1 ≤ z < b m − w j : gcd ( z , N ) = 1 } if w j < m , Z N , w j := { 1 } if w j ≥ m . Note that � b m − w j − 1 ( b − 1 ) if w j < m , |Z N , w j | := w j ≥ m . 1 if write Y j := b w j . Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 17
Johann Radon Institute for Computational and Applied Mathematics Algorithm 2 (Reduced CBC construction) Let N, w 1 , . . . , w d , and Y 1 , . . . , Y d be as above. Construct z = ( Y 1 z 1 , . . . , Y d z d ) as follows. Set z 1 = 1 . For s ∈ { 2 , . . . , d } assume that z 1 , . . . , z s − 1 have already been found. Now choose z s ∈ Z N , w s such that e 2 N , s ,α,γ (( Y 1 z 1 , . . . , Y s − 1 z s − 1 , Y s z s )) is minimized as a function of z s . Increase s and repeat the second step until ( Y 1 z 1 , . . . , Y d z d ) is found. Usual CBC construction: w j = 0 and Y j = 1 for all j . Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 18
Johann Radon Institute for Computational and Applied Mathematics Theorem 3 (Dick/K./Leobacher/Pillichshammer, 2015) Let z = ( Y 1 z 1 , . . . , Y d z d ) ∈ Z d be constructed according to the reduced CBC algorithm. Then, e N , d ,α, γ (( Y 1 z 1 , . . . , Y d z d )) ≤ α/ 2 − δ � � � � d � 1 ≤ N − α/ 2 + δ 2 α b w j 1 + γ α − 2 δ 2 ζ j α − 2 δ j = 1 � � 0 , α − 1 for all δ ∈ , where ζ is the Riemann zeta function. 2 Theorem formulated for product weights, similar result holds for general weights. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 19
Johann Radon Institute for Computational and Applied Mathematics If ∞ � 1 b w j < ∞ , α − 2 δ B := γ j j = 1 then the error bound is independent of the dimension. Peter Kritzer Construction Algorithms for (Polynomial) Lattice Points 20
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