Good, low degree, Rank-1 Lattice rules in High Dimensions Tor Srevik - PowerPoint PPT Presentation

Good, low degree, Rank-1 Lattice rules in High Dimensions Tor Sørevik 1 1 joint work with James N. Lyness MCQCM 2012, Sydney Tor Sørevik (UoB) Delta sequences MCQCM-12 1 / 20

Basic definitions An s − dimensional simple rank 1 lattice rule N − 1 Qf = 1 �� j (1 , x 1 , · · · , x s − 1 ) �� � f (1) N N j =0 N s and N ∈ I x = (1 , x 1 , · · · , x s − 1 ) ∈ I N A quadrature rule is of trigonometric degree d iff it integrates exactly all trigonometric polynomials of degree less or equal d . For lattice rules the enhanced trigonometric degree, δ = d + 1 , can be computed as δ ( Q ) = p ∈ Λ ⊥ \{ 0 } || p || 1 min (2) where Λ ⊥ is the dual lattice defined as Λ ⊥ = { p | p T z ∈ Z for all z ∈ Λ } (3) Tor Sørevik (UoB) Delta sequences MCQCM-12 2 / 20

Computing the enhanced degree For rank 1 lattice rules: p ∈ Λ ⊥ ⇔ ∃ λ ∈ Z Z s such that:   s − 1 p T = � λ j x j + λ s N, λ 1 , · · · , λ s − 1 (4)   j =1 Plugging (4) into (2) we get s − 1 s − 1 � � δ ( Q ) = λ ∈ Z s \{ 0 } | min λ j x j + λ s N | + | λ j | (5) j =1 j =1 Restriction to the subset of λ ’s with λ s = 0 gives s − 1 s − 1 � � δ ( Q ) ≤ λ ∈ Z s − 1 \{ 0 } | min λ j x j | + | λ j | ( BC ) j =1 j =1 Tor Sørevik (UoB) Delta sequences MCQCM-12 3 / 20

A simple idea Devide the search into two steps: Basic algorithm 1. For fixed δ , find candidate x -vectors satisfying (BC). (” δ − sequences”) 2. Find the smallest N for x -vectors satisfying (BC). To make this a finite search we need to restrict the set of admissible x -vectors. Tor Sørevik (UoB) Delta sequences MCQCM-12 4 / 20

Limiting the search: Theorem Every rank 1 simple lattice have a symmetric copy which satisfy: 1 < x 1 < · · · < x s ≤ N/ 2; ( Low sequence ) (6) and another satisfying: N/ 2 < x 1 < · · · < x s < N ; ( High sequence ) (7) Tor Sørevik (UoB) Delta sequences MCQCM-12 5 / 20

..and a little refinement Lemma A necessary condition for a low sequence to produce a lattice rule of enhanced trigonometric degree larger or equal to δ is that: x 1 ≥ δ − 1 x j − x j − 1 ≥ δ − 2; 1 < j < s − 1 x s − 1 ≤ ( N − δ ) / 2 + 1 Since N ≥ 2( x s − 1 − 1) + δ we seek δ − sequences with small x s − 1 . Lemma A necessary condition for x = ( x 1 , · · · , x k ) to be a δ − sequence is that ( x 1 , · · · , x k − 1 ) is a δ − sequence. Tor Sørevik (UoB) Delta sequences MCQCM-12 6 / 20

Extension Algorithm for finding δ − sequences extend sequence ( s, δ, k, x (1 : k − 1)) for x k = x k − 1 + δ − 2 , upper ok = check if δ − sequence ( δ, k, x (1 : k ) ) if ok then if ( k = s ) then find minimal N , write result and return else extend sequence ( s, δ, k, x (1 : k )) endif endif end for This computation took us to δ = 5 and s ≤ 10 , δ = 6 and s ≤ 7 . Not good enough! Tor Sørevik (UoB) Delta sequences MCQCM-12 7 / 20

The case δ = 5 James’ last theorem: Theorem Let x s < 2 x 1 − 5 (”high-sequence”) and ε i,j = x i − x j ; i > j Then x 1 , x 2 , · · · , x s is a δ − sequence with δ ≥ 5 if and only if the set { ε i,j ; 1 ≤ j < i ≤ s } (8) contains only distinct elements and ε i,j ≥ 3 . Tor Sørevik (UoB) Delta sequences MCQCM-12 8 / 20

Golomb rulers Definition A set of integers A = { a 1 , a 2 , · · · , a n } a 1 < a 2 < · · · < a n (9) is called a Golomb ruler if a k − a l = a j − a i iff { k, l } = { j, i } (10) If, in addition there exist an p such that that the above hold for all differences modulo p , the Golomb ruler is said to be cyclic modulo p . G ( n ) = a n − a 1 is said to be the length of the Golomb ruler (”Golomb rulers” are equivalent to ”Sidon sets” or ” B 2 sequences”) Tor Sørevik (UoB) Delta sequences MCQCM-12 9 / 20

Properties of Golomb rulers Theorem If the set A = { a 1 , a 2 , · · · , a n } is a Golomb ruler then so is the sets: Translation: A 1 = { a 1 + x, a 2 + x, · · · , a n + x } Multiplication: A 2 = { ca 1 , ca 2 , · · · , ca n } Mirroring: A 3 = { a n − a n , a n − a n − 1 , · · · , a n − a 1 } and for cyclic rulers: Cyclic shift+ Translation: A 4 = { p − a n , a 1 + p − a n , · · · , a n − 1 + p − a n } Tor Sørevik (UoB) Delta sequences MCQCM-12 10 / 20

Length of Golomb rulers G ( n ) > n 2 − 2 n √ n + √ n − 2 Proved by Lindstr¨ om -69. G ( n ) < n 2 Conjectured by Erd¨ os and Turan -41. This conjecture is proved for n prime and computational verified for n ≤ 65000 . A crude and pessimistic upper bound for lattice rules with δ = 5 based on Golomb rulers are then N < 4(2 s − 3) 2 . Compared to the lower bound based on the moment equations: N ≥ N ME = 2 s 2 + 2 s + 1 (Mysovskikh -87). Tor Sørevik (UoB) Delta sequences MCQCM-12 11 / 20

Construction of good Golomb rulers Theorem Let p be a prime number and g a primitive element of the multiplicative group Z ∗ p . The following sequence is a cyclic Golomb ruler. R ( p, g ) = pk + ( p − 1) g k mod ( p ( p − 1)); for 1 ≤ k ≤ p − 1 (11) [Rusza -93] Theorem Let q = p m be a prime power and θ a primitive element in the Galois field GF ( q 2 ) . Then the q integers d 1 , · · · , d q = { a | 1 ≤ a < q 2 and θ a − θ ∈ GF ( q ) } (12) have distinct pairwise modulo differences modulo q 2 − 1 . [Bose and Chowla -62] Tor Sørevik (UoB) Delta sequences MCQCM-12 12 / 20

Creating δ − sequences from a Golomb ruler Remove element(s) to satisfy the ε ij ≥ 3 condition. If shorter δ − sequences are wanted, remove more elements. Let A = { 0 , a 1 , ..., a s − 1 } be the standard form of the Golomb ruler. For a cyclic GR modulo p we can do cyclic shifts + translations to get new GRs in standard form: C ( A − a 1 ) = { 0 , a 2 − a 1 , ..., a s − 1 − a 1 , p − a 1 } . For each GR in standard form we may use the translation property A = A + c ; c ∈ I N to produce multiple candidates. Bottom line: 1 Golomb ruler makes many related δ − sequences. Tor Sørevik (UoB) Delta sequences MCQCM-12 13 / 20

Computing the optimal N for a fixed Delta-sequence For fixed x find minimal N of the set: s − 1 s − 1 � � | λ j | ≥ δ ; λ ∈ Z s \{ 0 }} Ω( N ) = { N : | λ j x j + λ s N | + j =1 j =1 The second sum implies that we only need to consider λ where � s − 1 j =1 | λ j | ≤ δ − 1 . For fixed λ 1 , · · · λ s − 1 the λ s which minimize the expression is: − � s − 1 j =1 λ j x j λ s = nint (13) N Tor Sørevik (UoB) Delta sequences MCQCM-12 14 / 20

Computing the minimal N (cont.) Let H ∈ Z r × ( s − 1) contain all s − 1 -dim λ ’s such that � s − 1 j =1 | λ j | < δ then for all admissible λ we compute s − 1 � N r ; l i = q = Hx and l ∈ I | h ij | j =1 then � q i � ( x , N ) of degree δ ⇔ min i =1 ,...,r | q i − nint N | + l i ≥ δ N Note: To check for multiple N , only the last step needs to be repeated. Tor Sørevik (UoB) Delta sequences MCQCM-12 15 / 20

Computing the minimal N for related x’s Creating related x n by translation e T = (1 , 1 , · · · , 1) x n = x 0 + n e ; (14) we have q n = Hx n = q 0 + n ∆q ; q 0 = Hx 0 ; ∆q = He (15) Thus for multiple vectors created by translation only the first need a full matrix-vector multiply. Tor Sørevik (UoB) Delta sequences MCQCM-12 16 / 20

Computational complexity for δ = 5 The number of rows in H is � s − 2 + δ � r ≈ 2 3 (16) s − 1 For δ = 5 and s ≫ 5 we have r ≈ ( s + 2) 4 / 3 N is bounded above by a N-best-so-far and from below by max( x s − 1 + 3 , N ME ) . In typical computation we test ∼ 2 s 2 values of N for each x . The cyclic shift gives us s possible x 0 for each Golomb ruler, and each of these have ∼ s 2 translates. Tor Sørevik (UoB) Delta sequences MCQCM-12 17 / 20

Computational results By the exhaustive search we were able to compute optimal rank-1 lattice for δ = 5 s ≤ 10 and for δ = 6 s ≤ 7 Using Golomb rulers we have (so far) computed good rank-1 lattices of enhanced trig. degree up to s ≤ 27 . Tor Sørevik (UoB) Delta sequences MCQCM-12 18 / 20

Quality of results Number of lattice points, δ = 5 80 Lower bound 70 Optimal rank−1 Best Golumb ruler Upper bound 60 50 N 1/2 40 30 20 10 0 0 5 10 15 20 25 30 s, Dimension Tor Sørevik (UoB) Delta sequences MCQCM-12 19 / 20

Conclusions The exhaustive search by the extension algorithm produce to many delta sequences, thus we have to many candidates to examine. For δ = 5 Golomb rulers produce good δ − sequences almost free of cost. There do exist rank 1 lattice rules with δ = 5 having N = O ( s 2 ) . The bottleneck is the computation of optimal N for fixed x . Preliminary results: New rank 1 lattice rules for δ = 5 and s < 28 (so far...). Tor Sørevik (UoB) Delta sequences MCQCM-12 20 / 20