Improved Discrepancy Bounds for Hybrid Sequences Harald - PowerPoint PPT Presentation

Improved Discrepancy Bounds for Hybrid Sequences Harald Niederreiter RICAM Linz and University of Salzburg

— MC vs. QMC methods — Hybrid sequences — The basic sequences — Deterministic discrepancy bounds — The proof techniques

MC vs. QMC methods Monte Carlo (MC) methods are numerical methods based on sampling with random or pseudorandom numbers. Their typical convergence rate is O ( N − 1 / 2 ), where N is the sam- ple size. Being stochastic methods, they allow statistical error estimation. Quasi-Monte Carlo (QMC) methods are deterministic ver- sions of MC methods. They replace sequences of pseu- dorandom numbers by low-discrepancy sequences. Their N − 1 � � typical convergence rate is O up to logarithmic fac- tors.

Hybrid sequences An idea going back to Spanier (1995) is to combine the advantages of MC and QMC methods, i.e., statistical error estimation and faster convergence. Such combined methods use hybrid sequences that are ob- tained by “mixing” different types of sequences, i.e., certain coordinates of the points stem from one type of sequence and the remaining coordinates from another type.

Of greatest practical interest is the combination of low- discrepancy sequences and sequences of pseudorandom num- bers. In view of the Koksma-Hlawka inequality, we want upper bounds on the discrepancy of hybrid sequences. Probabilis- tic discrepancy bounds were given by ¨ Okten (1996), ¨ Okten- Tuffin-Burago (2006), and Gnewuch (2009). The first de- terministic bounds were shown in H.N. (2009). Since then there has been further work by Hofer, Kritzer, Larcher, H.N., and Winterhof. Here I report on recent improve- ments by a new method.

The basic sequences Low-discrepancy sequence: Halton sequence h n = ( φ b 1 ( n ) , . . . , φ b s ( n )) , n = 0 , 1 , . . . , with pairwise coprime bases b 1 , . . . , b s ≥ 2. Sequences of pseudorandom numbers: (i) linear congruential sequence (ii) explicit nonlinear congruential sequence (iii) explicit inversive sequence (iv) digital explicit inversive sequence (v) recursive inversive sequence

Deterministic discrepancy bounds [0 , 1) m m -dim. half-open unit cube ( m ≥ 1 arbitrary), λ m m -dim. Lebesgue measure. J subinterval of [0 , 1) m , c J characteristic function of J . For the first N terms x 0 , x 1 , . . . , x N − 1 ∈ [0 , 1) m of a given hybrid sequence, we define the discrepancy � � N − 1 � � 1 � � � D N = sup c J ( x n ) − λ m ( J ) . � � N J � � n =0 � �

Halton + linear congruential Recall that b 1 , . . . , b s are the bases of the Halton sequence h 0 , h 1 , . . . . Let p ≥ 19 be a prime with gcd( b i , p − 1) = 1 for 1 ≤ i ≤ s . Choose a primitive element a ∈ F p and an initial value z 0 ∈ F ∗ p . Generate the linear congruential sequence z 0 , z 1 , . . . ∈ F p by z n +1 = az n for n = 0 , 1 , . . . . Then define the hybrid sequence h n , z n � � ∈ [0 , 1) s +1 , x n = n = 0 , 1 , . . . . p

Theorem 1 (H.N., 2011). � N − 1 p 1 / 2 (log p ) 2 (log N ) s � D N = O for 2 ≤ N ≤ p − 1, where the implied constant depends only on b 1 , . . . , b s . Remark 1. The earlier discrepancy bound in H.N. (2009) basically had an exponent 1 / ( s + 1) on the new bound, so Theorem 1 yields a substantial improvement. Theorem 1 is in general best possible up to the logarithmic factors.

Remark 2. Theorem 1 can be generalized to the case where the linear congruential sequence is replaced by a se- quence of matrix-method pseudorandom vectors. For a di- mension t ≥ 2, let A be a t × t matrix over F p with a prim- itive characteristic polynomial. Generate z 0 , z 1 , . . . ∈ F t p by z n +1 = z n A for n = 0 , 1 , . . . with z 0 � = 0 . This sequence is periodic with least period p t − 1. The corre- sponding hybrid sequence is x n = ( h n , p − 1 z n ) ∈ [0 , 1) s + t , n = 0 , 1 , . . . .

Halton + explicit nonlinear congruential Let p ≥ 3 be a prime with gcd( b i , p ) = 1 for 1 ≤ i ≤ s . Choose g 1 , . . . , g t ∈ F p [ X ] of distinct degrees with 2 ≤ deg( g j ) < p for 1 ≤ j ≤ t . Then define the hybrid sequence h n , g 1 ( n ) , . . . , g t ( n ) � � ∈ [0 , 1) s + t , n = 0 , 1 , . . . . x n = p p

Theorem 2 (H.N., to appear). � N − 1 Gp 1 / 2 (log p ) t +1 (log N ) s � D N = O for 2 ≤ N ≤ p , where G = max 1 ≤ j ≤ t deg( g j ) and the implied constant depends only on b 1 , . . . , b s , t . Remark 3. The earlier discrepancy bound in H.N. (2010) basically had an exponent 1 / ( s + 1) on the new bound, so Theorem 2 yields a substantial improvement. Even in the case t = 1, Theorem 2 is in general best possible up to the logarithmic factors.

Halton + explicit inversive Let p ≥ 5 be a prime with gcd( b i , p ) = 1 for 1 ≤ i ≤ s . Choose a 1 , . . . , a t ∈ F ∗ p and d 1 , . . . , d t ∈ F p such that d 1 /a 1 , . . . , d t /a t are distinct elements of F p . For 1 ≤ j ≤ t and n = 0 , 1 , . . . , put e ( j ) n = ( a j n + d j ) p − 2 ∈ F p . Then define the hybrid sequence h n , e (1) p , . . . , e ( t ) � n n � ∈ [0 , 1) s + t , n = 0 , 1 , . . . . x n = p

Theorem 3 (H.N., to appear). � N − 1 p 1 / 2 (log p ) t +1 (log N ) s � D N = O for 2 ≤ N ≤ p , where the implied constant depends only on b 1 , . . . , b s , t . Remark 4. This is again a substantial improvement on an earlier discrepancy bound in H.N. (2010). Theorem 3 is in general best possible up to the logarithmic factors.

Halton + digital explicit inversive Let F q be the finite field of order q = p k , p prime, k ≥ 1. Choose α, β, γ ∈ F ∗ q with γ of order T ≥ 2 in the cyclic q . Put ̺ n = ( αγ n + β ) q − 2 ∈ F q for n = 0 , 1 , . . . . group F ∗ For an ordered basis { β 1 , . . . , β k } of F q over F p , we can write ̺ n = � k l =1 c n,l β l with all c n,l ∈ F p being unique. Then a digital explicit inversive sequence is defined by k c n,l p − l ∈ [0 , 1) � y n = for n = 0 , 1 , . . . . l =1

Assume gcd( b i , T ) = 1 for 1 ≤ i ≤ s and choose integers 0 ≤ i 1 < i 2 < · · · < i t < T . Then define the hybrid sequence x n = ( h n , y n + i 1 , . . . , y n + i t ) ∈ [0 , 1) s + t for n = 0 , 1 , . . . . Theorem 4 (H.N., to appear). � N − 1 q 1 / 2 (log q ) t (log N ) s log T � D N = O for 2 ≤ N ≤ T , where the implied constant depends only on b 1 , . . . , b s , t .

Halton + recursive inversive For a prime p ≥ 3, let r 0 , r 1 , . . . ∈ F p be a recursive inversive generator in the sense of H.N.-Rivat (2008) with least period T ≤ p + 1. Then define the hybrid sequence h n , r n � � ∈ [0 , 1) s +1 , n = 0 , 1 , . . . . x n = p Theorem 5 (H.N., to appear). N − 1 / 2 p 1 / 4 (log N ) s log p � � D N = O for 2 ≤ N ≤ T , where the implied constant depends only on b 1 , . . . , b s .

The proof techniques The crucial auxiliary result provides a criterion for h n to fall into a certain type of subinterval of [0 , 1) s . Lemma 1 (H.N., 2011). Let v 1 , . . . , v s , f 1 , . . . , f s ∈ N with v i ≤ b f i i for 1 ≤ i ≤ s . Then for any n ≥ 0, we have s [0 , v i b − f i � h n ∈ J = ) i i =1 iff n ∈ ⊔ M e =1 R e , where 1 ≤ M ≤ b 1 · · · b s f 1 · · · f s and R 1 , . . . , R M are disjoint residue classes in Z depending only on J .

If x n = ( h n , y n ) ∈ [0 , 1) s + t is a point of a hybrid sequence, then x n ∈ J × K ⊆ [0 , 1) s + t with J as in Lemma 1 iff n ∈ R e for some e and y n ∈ K . We are thus led to studying subsequences of ( y n ) ∞ n =0 , with n running through an arithmetic progression, and bounding the discrepancy of such subsequences. In the cases we have considered, the techniques for bounding the discrepancy of ( y n ) can be applied also to these subsequences. We also employ techniques for passing from intervals J to arbitrary subintervals of [0 , 1) s .