On the properties and the construction of finite-row ( t , s - - PowerPoint PPT Presentation

On the properties and the construction of finite-row (t, s)-sequences1

Roswitha Hofer2

Institute of Financial Mathematics, University of Linz, Austria

13.02.12, MCQMC12, Sydney, Australia

1Partially joint work with Pirsic and with Larcher 2supported by the Austrian Science Fund (FWF), Project P21943. Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 1 / 19

Overview of my talk

Definition of finite-row (t, s)-sequences Existence and examples of such sequences finite-row (t, s)-sequences and Niederreiter-Halton sequences ... Experiments

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 2 / 19

Definition (digital sequences in base q by Niederreiter 1987)

s ≥ 1, q ∈ P. Let C1, . . . , Cs be N × N-matrices over the finite field Zq. (xn)n≥0, xn =

• x(1)

n , . . . , x(s) n

• x(i)

n

is generated as follows: n = n0 + n1q + n2q2 + · · · Ci · (n0, n1, . . .)⊤ =:

• y(i)

0 , y(i) 1 , . . .

⊤ ∈ ZN

q

and x(i)

n

:= y(i) q + y(i)

1

q2 + y(i)

2

q3 + · · · ∈ [0, 1). If the generator matrices satisfy that each row contains just finitely many nonzero entries ... “finite-row (digital) sequence”.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 3 / 19

digital (t, s)-sequences – condition on the rank structure!

Here the generator matrices fulfill for all m ∈ N and all d1 + . . . + ds = m − t, (di ≥ 0) that

C1 =    

m

gggggg}d1     , . . . , Cs =    

m

gggggg}ds     the matrix

m

• gggggggggg

} d1 . . . } ds has rank m − t

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 4 / 19

Example (van der Corput sequence = finite-row (0, 1)-sequence)

The van der Corput sequence in base q is a finite-row (digital) (0, 1)-sequence, since the generator matrix,        1 . . . 1 . . . 1 . . . 1 . . . . . . . . . . . . . . . ...        ∈ ZN×N

q

, satisfies the condition on the rank structure and is a finite-row matrix.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 5 / 19

Example (digital (0, s)-sequences by Faure 1982)

For prime base q, the Pascal matrices P(i) defined by P(i) :=      1 1

• i1

2

• i2

3

• i3

. . . 1 2

1

• i1

3

1

• i2

. . . 1 3

2

• i1

. . . . . . . . . . . . ... . . .      ∈ ZN×N

q

, i ∈ {0, 1 . . . , q − 1} generate a digital (0, q)-sequence in base q. For q = 2 (Sobol 1967) the matrices are sketched:

1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 6 / 19

Research Question

Let s > 1. Can finite rows satisfy for all m ∈ N and d1, . . . , ds ≥ 0 with d1 + . . . + ds = m

m

• · · · · · · · · · · · ·

} d1 . . . · · · · · · · · · · · · } ds has rank m? Do there exist multi-dimensional finite-row (0, s)-sequences?

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 7 / 19

Lower bounds on the lengths:

For s = 2 it is not so hard to check that

     x . . . x x x . . . x x x x x . . . . . . . . . . . . . . . . . . . . . . . . ...      and      x x . . . x x x x . . . x x x x x x . . . . . . . . . . . . . . . . . . . . . . . . ...     

where the ‘x’ entries are nonzero have “lowest possible row lengths”.

Theorem (Faure & Tezuka 2000)

If

• C1, . . . , Cs ∈ ZN×N

q

generate a digital (0, s)-sequence in prime base q ≥ s and

• M is a NUT matrix in Zq. (“Scrambling Matrix”)

Then the matrices C1M, . . . , CsM generate a digital (0, s)-sequence.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 8 / 19

Idea (Construct a proper NUT scrambling matrix)

       1 . . . 1 . . . 1 . . . 1 . . . . . . . . . . . . . . . ...        ·      1 x x . . . 1 1 x x . . . 1 1 x x . . . . . . . . . . . . . . . . . . . . . ...      =      1 . . . 1 1 . . . 1 1 x . . . . . . . . . . . . . . . . . . . . . ...             1 1 1 1 . . . 1 1 . . . 1 1 . . . 1 . . . . . . . . . . . . . . . ...        ·      1 x x . . . 1 1 x x . . . 1 1 x x . . . . . . . . . . . . . . . . . . . . . ...      =      1 1 . . . 1 1 1 . . . 1 x x . . . . . . . . . . . . . . . . . . . . . ...     

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 9 / 19

Theorem (H.& Larcher 2009)

Let s ∈ N and q ∈ P. For all generator matrices C1, . . . , Cs ∈ ZN×N

q

• f a

digital (0, s)-sequence in base q there exists a NUT matrix M ∈ ZN×N

q

such that C1M, . . . , CsM ∈ ZN×N

q

are generator matrices of a finite-row (0, s)-sequence in base q and they have even lowest possible row lengths.

Figure: The Pascal matrices in base 2 and the modified matrices with lowest possible row lengths.

1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 10 / 19

Figure: The Pascal matrices in base 5:

1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50

Figure: The modified matrices in base 5:

1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 11 / 19

A formula for the scrambling matrix?

Theorem (H. & Pirsic 2011)

Let q ∈ P. Then the following matrix is a suitable scrambling matrix for the Pascal matrices in base q. S =

• r

j − 1

• j≥1,r≥0

, where n

m

• is the Karamata notation for the unsigned Stirling numbers of

the first kind. Furthermore the new generator matrices P(0)S, P(1)S, . . . , P(q−1)S, satisfy P(i)S = SQi , with Q =      1 −1 · · · 1 −2 · · · 1 −3 · · · . . . ... ... ... · · ·      .

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 12 / 19

Further results on finite-row (t, s)-sequences

(H.& Pirsic, unp.) A formula for the scrambling matrix which goes along with the generator matrices of classical Niederreiter sequences. (H., 2012) Explicit construction of finite-row (0, s)-sequences

• ver finite fields Fq.

(H., unp.) Explicit construction of finite-row (t, s)-sequences

• ver finite fields Fq.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 13 / 19

Motivation of finite-row sequences

...Koksma-Hlawka Inequality.

• [0,1]s f (x)dx − 1

N

N−1

• n=0

f (xn)

• ≤ V (f )D∗

N

Need “low-discrepancy sequence”.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 14 / 19

Examples of low-discrepancy sequences

digital (t, s)-sequences (Sobol sequences, Faure sequences, Niederreiter sequences, ...) Halton sequences: ...

Definition (Halton-sequence, Halton 1960)

Take s different primes q1, . . . , qs and juxtapose the van der Corput sequences (a digital (0, 1)-sequence) in the different bases q1, . . . , qs.

Observation

We take s one-dimensional low-discrepancy sequences and get an s-dimensional low-discrepancy sequence!

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 15 / 19

Question (H., Kritzer, Larcher & Pillichshammer 2009)

What happens if digital sequences in different prime bases are juxtaposed ?

Question (Faure, 1994)

What happens if, for example, the Faure sequences in bases 2 and 3 are juxtaposed?

Theorem (H. & Larcher 2010)

The discrepancy of this 5-dimensional “Faure-Halton” sequence satisfies NDN ≥ cNlog4 3 = cN0.79... >> c′ log5 N for all N ∈ N.

Figure: The generator matrices in bases 2 and 3:

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 16 / 19

Question

What happens if, for example, the finite-row sequences in bases 2 and 3 are juxtaposed?

Figure: The generator matrices in bases 2 and 3:

Theorem (H., Kritzer, Larcher & Pillichshammer 2009)

General discrepancy bound for juxtaposed finite-row digital sequences which depends on the row lengths.

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 17 / 19

BUT: The bound for the finite-row generator matrices,

is also valid for other generator matrices, e.g.,

... NEED NEW TECHNIQUES!!!!

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 18 / 19

Experiments

Applied different QMC-rule for some examples of f : [0, 1]5 → R ...

• [0,1]5 f (x)dx − 1

N

N−1

• n=0

f (xn)

• ≤ V (f )D∗

N.

Figure: f : (x1, . . . , x5) →

((5

i=1 xi)2−1)2

(5

i=1 xi)2−(5 i=1 x2 i )+1: 100 104 106 105 104 0.001 0.01 0.1 1 10 MCrule TrivFiniterow 2,3 Faure 2,3 Finiterow 2,3 Halton 2,3,5,7,11 Faure 5

Roswitha Hofer (Linz) finite-row (t, s)-sequences MCQMC12 19 / 19