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

• n 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 typical convergence rate is O

• N−1

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 hn = (φb1(n), . . . , φbs(n)), n = 0, 1, . . . , with pairwise coprime bases b1, . . . , bs ≥ 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), λm m-dim. Lebesgue measure. J subinterval of [0, 1)m, cJ characteristic function of J. For the first N terms x0, x1, . . . , xN−1 ∈ [0, 1)m of a given hybrid sequence, we define the discrepancy DN = sup

J

• 1

N

N−1

• n=0

cJ(xn) − λm(J)

• .

Halton + linear congruential Recall that b1, . . . , bs are the bases of the Halton sequence h0, h1, . . . . Let p ≥ 19 be a prime with gcd(bi, p − 1) = 1 for 1 ≤ i ≤ s. Choose a primitive element a ∈ Fp and an initial value z0 ∈ F∗

• p. Generate the linear congruential sequence

z0, z1, . . . ∈ Fp by zn+1 = azn for n = 0, 1, . . . . Then define the hybrid sequence xn =

• hn, zn

p

• ∈ [0, 1)s+1,

n = 0, 1, . . . .

Theorem 1 (H.N., 2011). DN = O

• N−1p1/2(log p)2(log N)s

for 2 ≤ N ≤ p − 1, where the implied constant depends

• nly on b1, . . . , bs.

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 Fp with a prim- itive characteristic polynomial. Generate z0, z1, . . . ∈ Ft

p

by zn+1 = znA for n = 0, 1, . . . with z0 = 0. This sequence is periodic with least period pt − 1. The corre- sponding hybrid sequence is xn = (hn, p−1zn) ∈ [0, 1)s+t, n = 0, 1, . . . .

Halton + explicit nonlinear congruential Let p ≥ 3 be a prime with gcd(bi, p) = 1 for 1 ≤ i ≤ s. Choose g1, . . . , gt ∈ Fp[X] of distinct degrees with 2 ≤ deg(gj) < p for 1 ≤ j ≤ t. Then define the hybrid sequence xn =

• hn, g1(n)

p , . . . , gt(n) p

• ∈ [0, 1)s+t, n = 0, 1, . . . .

Theorem 2 (H.N., to appear). DN = O

• N−1Gp1/2(log p)t+1(log N)s

for 2 ≤ N ≤ p, where G = max1≤j≤t deg(gj) and the implied constant depends only on b1, . . . , bs, 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(bi, p) = 1 for 1 ≤ i ≤ s. Choose a1, . . . , at ∈ F∗

p and d1, . . . , dt ∈ Fp such that

d1/a1, . . . , dt/at are distinct elements of Fp. For 1 ≤ j ≤ t and n = 0, 1, . . ., put e(j)

n = (ajn + dj)p−2 ∈ Fp.

Then define the hybrid sequence xn =

• hn, e(1)

n

p , . . . , e(t)

n

p

• ∈ [0, 1)s+t, n = 0, 1, . . . .

Theorem 3 (H.N., to appear). DN = O

• N−1p1/2(log p)t+1(log N)s

for 2 ≤ N ≤ p, where the implied constant depends only

• n b1, . . . , bs, 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 Fq be the finite field of order q = pk, p prime, k ≥ 1. Choose α, β, γ ∈ F∗

q with γ of order T ≥ 2 in the cyclic

group F∗

• q. Put ̺n = (αγn + β)q−2 ∈ Fq for n = 0, 1, . . ..

For an ordered basis {β1, . . . , βk} of Fq over Fp, we can write ̺n = k

l=1 cn,l βl with all cn,l ∈ Fp being unique.

Then a digital explicit inversive sequence is defined by yn =

k

• l=1

cn,l p−l ∈ [0, 1) for n = 0, 1, . . . .

Assume gcd(bi, T) = 1 for 1 ≤ i ≤ s and choose integers 0 ≤ i1 < i2 < · · · < it < T. Then define the hybrid sequence xn = (hn, yn+i1, . . . , yn+it) ∈ [0, 1)s+t for n = 0, 1, . . . . Theorem 4 (H.N., to appear). DN = O

• N−1q1/2(log q)t(log N)s log T
• for 2 ≤ N ≤ T, where the implied constant depends only
• n b1, . . . , bs, t.

Halton + recursive inversive For a prime p ≥ 3, let r0, r1, . . . ∈ Fp be a recursive inversive generator in the sense of H.N.-Rivat (2008) with least period T ≤ p + 1. Then define the hybrid sequence xn =

• hn, rn

p

• ∈ [0, 1)s+1,

n = 0, 1, . . . . Theorem 5 (H.N., to appear). DN = O

• N−1/2p1/4(log N)s log p
• for 2 ≤ N ≤ T, where the implied constant depends only
• n b1, . . . , bs.

The proof techniques The crucial auxiliary result provides a criterion for hn to fall into a certain type of subinterval of [0, 1)s. Lemma 1 (H.N., 2011). Let v1, . . . , vs, f1, . . . , fs ∈ N with vi ≤ bfi

i for 1 ≤ i ≤ s. Then for any n ≥ 0, we have

hn ∈ J =

s

• i=1

[0, vib−fi

i

) iff n ∈ ⊔M

e=1Re, where 1 ≤ M ≤ b1 · · · bsf1 · · · fs and

R1, . . . , RM are disjoint residue classes in Z depending

• nly on J.

If xn = (hn, yn) ∈ [0, 1)s+t is a point of a hybrid sequence, then xn ∈ J × K ⊆ [0, 1)s+t with J as in Lemma 1 iff n ∈ Re for some e and yn ∈ K. We are thus led to studying subsequences of (yn)∞

n=0, with n running through

an arithmetic progression, and bounding the discrepancy

• f such subsequences. In the cases we have considered, the

techniques for bounding the discrepancy of (yn) can be applied also to these subsequences. We also employ techniques for passing from intervals J to arbitrary subintervals of [0, 1)s.