takashi goda - university of new south wales · richardson extrapolation and higher order qmc...
TRANSCRIPT
Richardson extrapolation and higher order QMC
Takashi Goda
School of Engineering, University of Tokyo
MCM 2019
Partly joint with Josef Dick and Takehito Yoshiki
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 1 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 2 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 3 / 61
Quasi-Monte Carlo
Approximate
I (f ) =
∫[0,1)s
f (x) dx ,
where f : [0, 1)s → R is integrable.
Choose an N-element point set P ⊂ [0, 1)s and approximate I (f ) by
QP(f ) =1
N
∑x∈P
f (x).
This is called a quasi-Monte Carlo (QMC) integration over P.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 4 / 61
Digital nets in base 2
Let m, n ∈ N := {1, 2, . . .}.Let C1, . . . ,Cs ∈ Fn×m
2 (F2 := {0, 1}, the two-element field).
Denote the dyadic expansion of 0 ≤ h < 2m by
h = (ηm−1 . . . η1η0)2.
For 1 ≤ j ≤ s, let
xj = (0.ξ(j)1 . . . ξ
(j)n 00 . . .)2,
where ξ(j)1...
ξ(j)n
= Cj
η0...
ηm−1
∈ Fn2.
This gives a point x = (x1, . . . , xs) ∈ [0, 1)s .
Running through all 0 ≤ h < 2m, we get a 2m-element point set Pcalled a digital net over F2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 5 / 61
Digital nets in base 2
Let m, n ∈ N := {1, 2, . . .}.Let C1, . . . ,Cs ∈ Fn×m
2 (F2 := {0, 1}, the two-element field).
Denote the dyadic expansion of 0 ≤ h < 2m by
h = (ηm−1 . . . η1η0)2.
For 1 ≤ j ≤ s, let
xj = (0.ξ(j)1 . . . ξ
(j)n 00 . . .)2,
where ξ(j)1...
ξ(j)n
= Cj
η0...
ηm−1
∈ Fn2.
This gives a point x = (x1, . . . , xs) ∈ [0, 1)s .
Running through all 0 ≤ h < 2m, we get a 2m-element point set Pcalled a digital net over F2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 5 / 61
Digital nets in base 2
Let m, n ∈ N := {1, 2, . . .}.Let C1, . . . ,Cs ∈ Fn×m
2 (F2 := {0, 1}, the two-element field).
Denote the dyadic expansion of 0 ≤ h < 2m by
h = (ηm−1 . . . η1η0)2.
For 1 ≤ j ≤ s, let
xj = (0.ξ(j)1 . . . ξ
(j)n 00 . . .)2,
where ξ(j)1...
ξ(j)n
= Cj
η0...
ηm−1
∈ Fn2.
This gives a point x = (x1, . . . , xs) ∈ [0, 1)s .
Running through all 0 ≤ h < 2m, we get a 2m-element point set Pcalled a digital net over F2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 5 / 61
Digital nets in base 2
Let m, n ∈ N := {1, 2, . . .}.Let C1, . . . ,Cs ∈ Fn×m
2 (F2 := {0, 1}, the two-element field).
Denote the dyadic expansion of 0 ≤ h < 2m by
h = (ηm−1 . . . η1η0)2.
For 1 ≤ j ≤ s, let
xj = (0.ξ(j)1 . . . ξ
(j)n 00 . . .)2,
where ξ(j)1...
ξ(j)n
= Cj
η0...
ηm−1
∈ Fn2.
This gives a point x = (x1, . . . , xs) ∈ [0, 1)s .
Running through all 0 ≤ h < 2m, we get a 2m-element point set Pcalled a digital net over F2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 5 / 61
Digital sequences in base 2
Let C1, . . . ,Cs ∈ FN×N2 , where every column is assumed to have only
finite non-zero elements.
Denote the dyadic expansion of h ∈ N0 := N ∪ {0} by
h = (. . . η1η0)2.
where all but finite number of ηi are 0. For 1 ≤ j ≤ s, let
xj = (0.ξ(j)1 ξ
(j)2 . . .)2,
where ξ(j)1
ξ(j)2...
= Cj
η0η1...
∈ FN2 .
This gives a point x = (x1, . . . , xs) ∈ [0, 1)s .
This way we get an infinite sequence of points S called a digitalsequence over F2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 6 / 61
Some remarks
“Goodness” (undefined) of digital nets/sequences is determined bygenerating matrices C1, . . . ,Cs .
For nets, m determines total number of points, while n determinesprecision of each point.
For sequences, the assumption that every column has only finitenon-zero elements is to ensure that each point has finite precision.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 7 / 61
Precision?
For classical QMC where “goodness” is measured by discrepancy, it isusual to set n = m (square matrices) for nets, or considerupper-triangular matrices for sequences.
For higher order QMC where “goodness” is measured by theworst-case error for a class of smooth functions, we need higherprecision. Typically, n = αm for nets (α: dominating mixedsmoothness of functions).
Aim of this talk
is to show that Richardson extrapolation can be used as a technique toreduce necessary precision of higher order QMC from αm to m.
Implication
Possibility to try large values of α w/o suffering from round-off error.
Search space for C1, . . . ,Cs is significantly reduced.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 8 / 61
Precision?
For classical QMC where “goodness” is measured by discrepancy, it isusual to set n = m (square matrices) for nets, or considerupper-triangular matrices for sequences.
For higher order QMC where “goodness” is measured by theworst-case error for a class of smooth functions, we need higherprecision. Typically, n = αm for nets (α: dominating mixedsmoothness of functions).
Aim of this talk
is to show that Richardson extrapolation can be used as a technique toreduce necessary precision of higher order QMC from αm to m.
Implication
Possibility to try large values of α w/o suffering from round-off error.
Search space for C1, . . . ,Cs is significantly reduced.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 8 / 61
Precision?
For classical QMC where “goodness” is measured by discrepancy, it isusual to set n = m (square matrices) for nets, or considerupper-triangular matrices for sequences.
For higher order QMC where “goodness” is measured by theworst-case error for a class of smooth functions, we need higherprecision. Typically, n = αm for nets (α: dominating mixedsmoothness of functions).
Aim of this talk
is to show that Richardson extrapolation can be used as a technique toreduce necessary precision of higher order QMC from αm to m.
Implication
Possibility to try large values of α w/o suffering from round-off error.
Search space for C1, . . . ,Cs is significantly reduced.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 8 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 9 / 61
Local discrepancy function
For an N-element point set P and y ∈ [0, 1)s , let
∆P(y) :=1
N
∑x∈P
1x∈[0,y) − λ([0, y)),
where λ denotes the Lebesgue measure.
y
0
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 10 / 61
Discrepancy and Koksma-Hlawka inequality
For 1 ≤ p ≤ ∞, the Lp-discrepancy of P is defined by
Lp(P) :=
(∫[0,1)s
|∆P(y)|p dy
)1/p
.
When p = ∞, the obvious modification is needed.
The integration error is bounded by
|QP(f )− I (f )| ≤ VHK(f )L∞(P)
where VHK(f ) is the Hardy-Krause variation of f .
Main focus on classical QMC
How to construct digital nets/sequences with small L∞(P).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 11 / 61
Discrepancy and Koksma-Hlawka inequality
For 1 ≤ p ≤ ∞, the Lp-discrepancy of P is defined by
Lp(P) :=
(∫[0,1)s
|∆P(y)|p dy
)1/p
.
When p = ∞, the obvious modification is needed.
The integration error is bounded by
|QP(f )− I (f )| ≤ VHK(f )L∞(P)
where VHK(f ) is the Hardy-Krause variation of f .
Main focus on classical QMC
How to construct digital nets/sequences with small L∞(P).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 11 / 61
Discrepancy and Koksma-Hlawka inequality
For 1 ≤ p ≤ ∞, the Lp-discrepancy of P is defined by
Lp(P) :=
(∫[0,1)s
|∆P(y)|p dy
)1/p
.
When p = ∞, the obvious modification is needed.
The integration error is bounded by
|QP(f )− I (f )| ≤ VHK(f )L∞(P)
where VHK(f ) is the Hardy-Krause variation of f .
Main focus on classical QMC
How to construct digital nets/sequences with small L∞(P).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 11 / 61
Quality measure: t-value
Let t be an integer such that, for any choice d1, . . . , ds ≥ 0 withd1 + · · ·+ ds = m − t,the first d1 rows of C1...the first ds rows of Cs
are linearly independent over F2. P is called a digital (t,m, s)-net.
...C1 C2 Cs
d1d2 ds
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 12 / 61
Quality measure: t-value
An equi-distribution property of digital (t,m, s)-nets: every dyadicelementary box of the form
s∏j=1
[aj2cj
,aj + 1
2cj
)with
c1, . . . , cs ≥ 0,
c1 + · · ·+ cs = m − t,
0 ≤ aj < 2cj
contains exactly 2t points.
Let t be an integer such that, for any m ≥ t, the first 2m points of Sare a digital (t,m, s)-net. S is called a digital (t, s)-sequence.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 13 / 61
Quality measure: t-value
An equi-distribution property of digital (t,m, s)-nets: every dyadicelementary box of the form
s∏j=1
[aj2cj
,aj + 1
2cj
)with
c1, . . . , cs ≥ 0,
c1 + · · ·+ cs = m − t,
0 ≤ aj < 2cj
contains exactly 2t points.
Let t be an integer such that, for any m ≥ t, the first 2m points of Sare a digital (t,m, s)-net. S is called a digital (t, s)-sequence.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 13 / 61
t-value and star-discrepancy
If P is a digital (t,m, s)-net over F2,
L∞(P) ≤ Cs,t(logN)s−1
Nwith N = 2m.
If P is the first N points of a digital (t, s)-sequence over F2,
L∞(P) ≤ Ds,t(logN)s
N,
for all N ≥ 2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 14 / 61
Explicit construction
For s = 1, C1 can be the identity matrix, which generates the famousdigital (0, 1)-sequence called van der Corput sequence.
For s = 2, C1 and C2 can be
C1 =
1 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
,C2 =
0 · · · 0 10 · · · 1 0...
. . ....
...1 · · · 0 0
,
which generates a digital (0,m, 2)-net, known as the Hammersleypoint set. Not extensible in m.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 15 / 61
Explicit construction
For s = 1, C1 can be the identity matrix, which generates the famousdigital (0, 1)-sequence called van der Corput sequence.
For s = 2, C1 and C2 can be
C1 =
1 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
,C2 =
0 · · · 0 10 · · · 1 0...
. . ....
...1 · · · 0 0
,
which generates a digital (0,m, 2)-net, known as the Hammersleypoint set. Not extensible in m.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 15 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit construction
Figure: Hammersley point set for m = 6
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61
Explicit constructionLet p1, p2, . . . ∈ F2[x ] be a sequence of distinct primitive/irreduciblepolynomials over F2 with e1 ≤ e2 ≤ · · · where ej = deg(pj).
For each j , Cj = (c(j)k,l ) is given by the coefficients of the following
Laurent series:
xej−1
pj(x)=
c(j)1,1
x+
c(j)1,2
x2+ · · ·
...
1
pj(x)=
c(j)ej ,1
x+
c(j)ej ,2
x2+ · · ·
xej−1
(pj(x))2=
c(j)ej+1,1
x+
c(j)ej+1,2
x2+ · · ·
...
(Sobol’ 1967; Niederreiter 1988; Tezuka 1993)
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 17 / 61
Explicit constructionLet p1, p2, . . . ∈ F2[x ] be a sequence of distinct primitive/irreduciblepolynomials over F2 with e1 ≤ e2 ≤ · · · where ej = deg(pj).
For each j , Cj = (c(j)k,l ) is given by the coefficients of the following
Laurent series:
xej−1
pj(x)=
c(j)1,1
x+
c(j)1,2
x2+ · · ·
...
1
pj(x)=
c(j)ej ,1
x+
c(j)ej ,2
x2+ · · ·
xej−1
(pj(x))2=
c(j)ej+1,1
x+
c(j)ej+1,2
x2+ · · ·
...
(Sobol’ 1967; Niederreiter 1988; Tezuka 1993)
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 17 / 61
Explicit construction
All matrices Cj ∈ FN×N2 are upper triangular and generate a digital
(t, s)-sequence with t = (e1 − 1) + · · ·+ (es − 1).(I used a C implementation for this sequence in at least more than 58636dimensions due to Tomohiko Hironaka.)
Figure: 2D projections of the first 26 points of the Niederreiter sequence.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 18 / 61
Polynomial lattice point sets (Niederreiter, 1992)
Let p ∈ F2[x ] be irreducible with deg(p) = m
Let q = (q1, . . . , qs) ∈ (F2[x ])s with deg(qj) < m.
For each j , Cj isgiven by the square Hankel matrix
Cj =
a(j)1 a
(j)2 · · · a
(j)m
a(j)2 a
(j)3 . . . a
(j)m+1
......
. . ....
a(j)m a
(j)m+1 . . . a
(j)2m−1
∈ Fm×m2
where a(j)1 , a
(j)2 , . . . are the coefficients of the Laurent series
qj(x)
p(x)=
a(j)1
x+
a(j)2
x2+ · · · .
The resulting digital net is called a polynomial lattice point setP(p,q). Usually the vector q is constructed by a (fast) computersearch algorithm (Nuyens & Cools, 2006; P. Kritzer, this afternoon!).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61
Polynomial lattice point sets (Niederreiter, 1992)
Let p ∈ F2[x ] be irreducible with deg(p) = m
Let q = (q1, . . . , qs) ∈ (F2[x ])s with deg(qj) < m. For each j , Cj is
given by the square Hankel matrix
Cj =
a(j)1 a
(j)2 · · · a
(j)m
a(j)2 a
(j)3 . . . a
(j)m+1
......
. . ....
a(j)m a
(j)m+1 . . . a
(j)2m−1
∈ Fm×m2
where a(j)1 , a
(j)2 , . . . are the coefficients of the Laurent series
qj(x)
p(x)=
a(j)1
x+
a(j)2
x2+ · · · .
The resulting digital net is called a polynomial lattice point setP(p,q). Usually the vector q is constructed by a (fast) computersearch algorithm (Nuyens & Cools, 2006; P. Kritzer, this afternoon!).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61
Polynomial lattice point sets (Niederreiter, 1992)
Let p ∈ F2[x ] be irreducible with deg(p) = m
Let q = (q1, . . . , qs) ∈ (F2[x ])s with deg(qj) < m. For each j , Cj is
given by the square Hankel matrix
Cj =
a(j)1 a
(j)2 · · · a
(j)m
a(j)2 a
(j)3 . . . a
(j)m+1
......
. . ....
a(j)m a
(j)m+1 . . . a
(j)2m−1
∈ Fm×m2
where a(j)1 , a
(j)2 , . . . are the coefficients of the Laurent series
qj(x)
p(x)=
a(j)1
x+
a(j)2
x2+ · · · .
The resulting digital net is called a polynomial lattice point setP(p,q). Usually the vector q is constructed by a (fast) computersearch algorithm (Nuyens & Cools, 2006;
P. Kritzer, this afternoon!).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61
Polynomial lattice point sets (Niederreiter, 1992)
Let p ∈ F2[x ] be irreducible with deg(p) = m
Let q = (q1, . . . , qs) ∈ (F2[x ])s with deg(qj) < m. For each j , Cj is
given by the square Hankel matrix
Cj =
a(j)1 a
(j)2 · · · a
(j)m
a(j)2 a
(j)3 . . . a
(j)m+1
......
. . ....
a(j)m a
(j)m+1 . . . a
(j)2m−1
∈ Fm×m2
where a(j)1 , a
(j)2 , . . . are the coefficients of the Laurent series
qj(x)
p(x)=
a(j)1
x+
a(j)2
x2+ · · · .
The resulting digital net is called a polynomial lattice point setP(p,q). Usually the vector q is constructed by a (fast) computersearch algorithm (Nuyens & Cools, 2006; P. Kritzer, this afternoon!).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 20 / 61
What if f is smooth
In some applications, such as PDEs with random coefficients, f canbe smooth. A proper design of QMC point sets enables higher orderconvergence of the integration error than O(1/N) as expected fromthe KH inequality.
So far, QMC point sets achieving higher order convergence fornon-periodic smooth functions are
1 Higher order digital nets/sequences (Dick, 2008; ...), and
2 Tent-transformed lattice point sets (Hickernell, 2002; ...).
In this talk, I will focus on higher order digital nets/sequences.The contents of this section are mostly developed by Dick (2008).Please refer to a recent review by G. & Suzuki (arXiv:1903.12353).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 21 / 61
What if f is smooth
In some applications, such as PDEs with random coefficients, f canbe smooth. A proper design of QMC point sets enables higher orderconvergence of the integration error than O(1/N) as expected fromthe KH inequality.
So far, QMC point sets achieving higher order convergence fornon-periodic smooth functions are
1 Higher order digital nets/sequences (Dick, 2008; ...), and
2 Tent-transformed lattice point sets (Hickernell, 2002; ...).
In this talk, I will focus on higher order digital nets/sequences.The contents of this section are mostly developed by Dick (2008).Please refer to a recent review by G. & Suzuki (arXiv:1903.12353).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 21 / 61
Quality measure: high order t-value
Let α ∈ N.Let t be an integer such that, for any choice1 ≤ dj ,vj < · · · < dj ,1 ≤ αm, 0 ≤ vj ≤ αm, 1 ≤ j ≤ s, with
s∑j=1
min(vj ,α)∑i=1
dj ,i = αm − t,
the d1,v1 , . . . , d1,1-th rows of C1...the ds,vs , . . . , ds,1-th rows of Cs
are linearly independent over F2. P is called an order α digital(t,m, s)-net.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 22 / 61
Quality measure: high order t-value
Order α digital (t,m, s)-nets hold equi-distribution properties: unionof dyadic elementary boxes contains the fair number of points (shownlater visually).
Let t be an integer such that, for any αm ≥ t, the first 2m points ofS are an order α digital (t,m, s)-net. S is called an order α digital(t, s)-sequence.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 23 / 61
Quality measure: high order t-value
Order α digital (t,m, s)-nets hold equi-distribution properties: unionof dyadic elementary boxes contains the fair number of points (shownlater visually).
Let t be an integer such that, for any αm ≥ t, the first 2m points ofS are an order α digital (t,m, s)-net. S is called an order α digital(t, s)-sequence.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 23 / 61
Explicit construction
Define Dα : [0, 1)α → [0, 1) byx1 = (0.ξ
(1)1 ξ
(1)2 ξ
(1)3 . . .)2
x2 = (0.ξ(2)1 ξ
(2)2 ξ
(2)3 . . .)2
...
xα = (0.ξ(α)1 ξ
(α)2 ξ
(α)3 . . .)2
7→ (0. ξ(1)1 ξ
(2)1 . . . ξ
(α)1︸ ︷︷ ︸
α
ξ(1)2 ξ
(2)2 . . . ξ
(α)2︸ ︷︷ ︸
α
. . .)2.
For x ∈ [0, 1)αs , let
Dα(x) = (Dα(x1, . . . , xα),Dα(xα+1, . . . , x2α),
. . . ,Dα(xα(s−1)+1, . . . , xαs)) ∈ [0, 1)s .
For a digital (t,m, αs)-net P, we write
Dα(P) = {Dα(x) | x ∈ P}.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61
Explicit construction
Define Dα : [0, 1)α → [0, 1) byx1 = (0.ξ
(1)1 ξ
(1)2 ξ
(1)3 . . .)2
x2 = (0.ξ(2)1 ξ
(2)2 ξ
(2)3 . . .)2
...
xα = (0.ξ(α)1 ξ
(α)2 ξ
(α)3 . . .)2
7→ (0. ξ(1)1 ξ
(2)1 . . . ξ
(α)1︸ ︷︷ ︸
α
ξ(1)2 ξ
(2)2 . . . ξ
(α)2︸ ︷︷ ︸
α
. . .)2.
For x ∈ [0, 1)αs , let
Dα(x) = (Dα(x1, . . . , xα),Dα(xα+1, . . . , x2α),
. . . ,Dα(xα(s−1)+1, . . . , xαs)) ∈ [0, 1)s .
For a digital (t,m, αs)-net P, we write
Dα(P) = {Dα(x) | x ∈ P}.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61
Explicit construction
Define Dα : [0, 1)α → [0, 1) byx1 = (0.ξ
(1)1 ξ
(1)2 ξ
(1)3 . . .)2
x2 = (0.ξ(2)1 ξ
(2)2 ξ
(2)3 . . .)2
...
xα = (0.ξ(α)1 ξ
(α)2 ξ
(α)3 . . .)2
7→ (0. ξ(1)1 ξ
(2)1 . . . ξ
(α)1︸ ︷︷ ︸
α
ξ(1)2 ξ
(2)2 . . . ξ
(α)2︸ ︷︷ ︸
α
. . .)2.
For x ∈ [0, 1)αs , let
Dα(x) = (Dα(x1, . . . , xα),Dα(xα+1, . . . , x2α),
. . . ,Dα(xα(s−1)+1, . . . , xαs)) ∈ [0, 1)s .
For a digital (t,m, αs)-net P, we write
Dα(P) = {Dα(x) | x ∈ P}.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61
Another look at constructionLet P be a digital (t,m, αs)-net with C1, . . . ,Cαs ∈ Fm×m
2 . We write
C1 =
c(1)1...
c(1)m
,C2 =
c(2)1...
c(2)m
, . . . ,Cα =
c(α)1...
c(α)m
, . . . .
Dα(P) is a digital net with D1, . . . ,Ds ∈ Fαm×m2 where
D1 =
c(1)1
c(2)1...
c(α)1...
c(1)m
c(2)m
...
c(α)m
,D2 =
c(α+1)1
c(α+2)1...
c(2α)1...
c(α+1)m
c(α+2)m
...
c(2α)m
, . . . .
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 25 / 61
Another look at constructionLet P be a digital (t,m, αs)-net with C1, . . . ,Cαs ∈ Fm×m
2 . We write
C1 =
c(1)1...
c(1)m
,C2 =
c(2)1...
c(2)m
, . . . ,Cα =
c(α)1...
c(α)m
, . . . .
Dα(P) is a digital net with D1, . . . ,Ds ∈ Fαm×m2 where
D1 =
c(1)1
c(2)1...
c(α)1...
c(1)m
c(2)m
...
c(α)m
,D2 =
c(α+1)1
c(α+2)1...
c(2α)1...
c(α+1)m
c(α+2)m
...
c(2α)m
, . . . .
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 25 / 61
High order t-value
For a digital (t,m, αs)-net P, Dα(P) is an order α digital(t ′,m, s)-net with
t ′ ≤ αmin
{m, t +
⌊s(α− 1)
2
⌋}.
For a digital (t, αs)-sequences S, Dα(S) is an order α digital(t ′, s)-sequences with
t ′ ≤ αt +sα(α− 1)
2.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 26 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced Sobol’ point sets with α = 2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61
Interlaced polynomial lattice point sets
Let p ∈ F2[x ] be irreducible with deg(p) = m, and letq = (q1, . . . , qαs) ∈ (F2[x ])
αs with deg(qj) < m.
An interlaced polynomial lattice point set is just Dα(P(p,q)).
The vector q can be constructed component by component. In thesimplest case, the necessary construction cost is of O(αsN logN) withO(N) memory (G. & Dick, 2015; G., 2015). In applications to PDEswith random coefficients, the criterion sometimes becomes a bitcomplicated, requiring O(αsN logN + α2s2N) construction cost withO(αsN) memory (Dick, Kuo, Le Gia, Nuyens & Schwab, 2014).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 28 / 61
Interlaced polynomial lattice point sets
Let p ∈ F2[x ] be irreducible with deg(p) = m, and letq = (q1, . . . , qαs) ∈ (F2[x ])
αs with deg(qj) < m.
An interlaced polynomial lattice point set is just Dα(P(p,q)).
The vector q can be constructed component by component. In thesimplest case, the necessary construction cost is of O(αsN logN) withO(N) memory (G. & Dick, 2015; G., 2015). In applications to PDEswith random coefficients, the criterion sometimes becomes a bitcomplicated, requiring O(αsN logN + α2s2N) construction cost withO(αsN) memory (Dick, Kuo, Le Gia, Nuyens & Schwab, 2014).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 28 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 29 / 61
Walsh functions
For k = (. . . κ1κ0)2 ∈ N0, the k-th Walsh function is defined by
walk(x) = (−1)κ0ξ1+κ1ξ2+···,
where x = (0.ξ1ξ2 . . .)2 ∈ [0, 1), unique in the sense that infinitelymany ξi are equal to 0.
For s ≥ 1 and k = (k1, . . . , ks) ∈ Ns0, the k-th Walsh function is
defined by
walk(x) =s∏
j=1
walkj (xj).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 30 / 61
Walsh functions
For k = (. . . κ1κ0)2 ∈ N0, the k-th Walsh function is defined by
walk(x) = (−1)κ0ξ1+κ1ξ2+···,
where x = (0.ξ1ξ2 . . .)2 ∈ [0, 1), unique in the sense that infinitelymany ξi are equal to 0.
For s ≥ 1 and k = (k1, . . . , ks) ∈ Ns0, the k-th Walsh function is
defined by
walk(x) =s∏
j=1
walkj (xj).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 30 / 61
Walsh functions
Figure: The k-th Walsh functions for k = 0, 1, 2, 3
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 31 / 61
Walsh functions
Figure: The k-th Walsh functions for k = 4, 5, 6, 7
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 32 / 61
Walsh functions
Every Walsh function is a piecewise constant function.
Importantly the system of Walsh functions {walk | k ∈ Ns0} is a
complete orthonormal basis in L2([0, 1)s). Thus we have a Walsh
series of f : ∑k∈Ns
0
f̂ (k)walk(x)
where
f̂ (k) =∫[0,1)s
f (x)walk(x) dx .
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 33 / 61
Walsh functions
Every Walsh function is a piecewise constant function.
Importantly the system of Walsh functions {walk | k ∈ Ns0} is a
complete orthonormal basis in L2([0, 1)s). Thus we have a Walsh
series of f : ∑k∈Ns
0
f̂ (k)walk(x)
where
f̂ (k) =∫[0,1)s
f (x)walk(x) dx .
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 33 / 61
Dual nets
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , the dual net is defined by
P⊥ =
k ∈ Ns0
∣∣∣C⊤1
κ1,0...
κ1,n−1
⊕ · · · ⊕ C⊤s
κs,0...
κs,n−1
= 0 ∈ Fm2
,
where we writekj = (. . . κj ,1κj ,0)2.
Trivial fact
k ∈ P⊥ if 2n | k , i.e., 2n | kj for all j = 1, . . . , s.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 34 / 61
Dual nets
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , the dual net is defined by
P⊥ =
k ∈ Ns0
∣∣∣C⊤1
κ1,0...
κ1,n−1
⊕ · · · ⊕ C⊤s
κs,0...
κs,n−1
= 0 ∈ Fm2
,
where we writekj = (. . . κj ,1κj ,0)2.
Trivial fact
k ∈ P⊥ if 2n | k , i.e., 2n | kj for all j = 1, . . . , s.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 34 / 61
Character property
Most of the Walsh functions can be integrated exactly:
1
2m
∑x∈P
walk(x) =
{1 if k ∈ P⊥,
0 otherwise.
I will also use the following fact:
1
2n
2n−1∑i=0
walk
(i
2n
)=
1
2n
∑ξ0,...,ξn−1∈{0,1}
n∏i=1
(−1)κi−1ξi
=1
2n
n∏i=1
∑ξi∈{0,1}
(−1)κi−1ξi
=
{1 if κ0 = . . . = κn−1 = 0, i.e., if 2n | k,0 otherwise.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 35 / 61
Character property
Most of the Walsh functions can be integrated exactly:
1
2m
∑x∈P
walk(x) =
{1 if k ∈ P⊥,
0 otherwise.
I will also use the following fact:
1
2n
2n−1∑i=0
walk
(i
2n
)=
1
2n
∑ξ0,...,ξn−1∈{0,1}
n∏i=1
(−1)κi−1ξi
=1
2n
n∏i=1
∑ξi∈{0,1}
(−1)κi−1ξi
=
{1 if κ0 = . . . = κn−1 = 0, i.e., if 2n | k,0 otherwise.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 35 / 61
Decomposition of integration error
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , we have
QP(f )− I (f ) =1
2m
∑x∈P
∑k∈Ns
0
f̂ (k)walk(x)− f̂ (0)
=∑k∈Ns
0
f̂ (k)1
2m
∑x∈P
walk(x)− f̂ (0)
=∑
k∈P⊥\{0}
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈P⊥\{0}2n|k
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 36 / 61
Decomposition of integration error
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , we have
QP(f )− I (f ) =1
2m
∑x∈P
∑k∈Ns
0
f̂ (k)walk(x)− f̂ (0)
=∑k∈Ns
0
f̂ (k)1
2m
∑x∈P
walk(x)− f̂ (0)
=∑
k∈P⊥\{0}
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈P⊥\{0}2n|k
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 36 / 61
Decomposition of integration error
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , we have
QP(f )− I (f ) =1
2m
∑x∈P
∑k∈Ns
0
f̂ (k)walk(x)− f̂ (0)
=∑k∈Ns
0
f̂ (k)1
2m
∑x∈P
walk(x)− f̂ (0)
=∑
k∈P⊥\{0}
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈P⊥\{0}2n|k
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 36 / 61
Decomposition of integration error
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , we have
QP(f )− I (f ) =1
2m
∑x∈P
∑k∈Ns
0
f̂ (k)walk(x)− f̂ (0)
=∑k∈Ns
0
f̂ (k)1
2m
∑x∈P
walk(x)− f̂ (0)
=∑
k∈P⊥\{0}
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈P⊥\{0}2n|k
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 36 / 61
Decomposition of integration error
For a digital net P with C1, . . . ,Cs ∈ Fn×m2 , we have
QP(f )− I (f ) =1
2m
∑x∈P
∑k∈Ns
0
f̂ (k)walk(x)− f̂ (0)
=∑k∈Ns
0
f̂ (k)1
2m
∑x∈P
walk(x)− f̂ (0)
=∑
k∈P⊥\{0}
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈P⊥\{0}2n|k
f̂ (k)
=∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 36 / 61
What is the second term?
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k)
The second term is...
∑k∈Ns
0\{0}2n|k
f̂ (k) =∑k∈Ns
0
f̂ (k)1
2ns
2n−1∑i1,...,is=0
walk
(i12n
, . . . ,is2n
)− f̂ (0)
=1
2ns
2n−1∑i1,...,is=0
∑k∈Ns
0
f̂ (k)walk
(i12n
, . . . ,is2n
)− I (f )
=1
2ns
2n−1∑i1,...,is=0
f
(i12n
, . . . ,is2n
)− I (f )
the integration error for a regular grid!
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 37 / 61
What is the second term?
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +∑
k∈Ns0\{0}
2n|k
f̂ (k)
The second term is...∑k∈Ns
0\{0}2n|k
f̂ (k) =∑k∈Ns
0
f̂ (k)1
2ns
2n−1∑i1,...,is=0
walk
(i12n
, . . . ,is2n
)− f̂ (0)
=1
2ns
2n−1∑i1,...,is=0
∑k∈Ns
0
f̂ (k)walk
(i12n
, . . . ,is2n
)− I (f )
=1
2ns
2n−1∑i1,...,is=0
f
(i12n
, . . . ,is2n
)− I (f )
the integration error for a regular grid!
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 37 / 61
Digital net as a subset of regular grid
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +
1
2ns
2n−1∑i1,...,is=0
f
(i12n
, . . . ,is2n
)− I (f )
Figure: Hammersley point set for m = n = 6 with regular grid
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 38 / 61
Euler-Maclaurin
If f has partial mixed derivatives up to order α in each variable,
1
2ns
2n−1∑i1,...,is=0
f
(i12n
, . . . ,is2n
)− I (f )
=c1(f )
2n+ · · ·+ cα−1(f )
2(α−1)n+ O(2−αn),
where
cτ (f ) =∑
τ1,...,τs≥0τ1+···+τs=τ
I (f (τ1,...,τs))s∏
j=1τj>0
bτj
with bi denoting the i-th Bernoulli number divided by i !.
For a detailed proof of this claim, see Dick, G. & Yoshiki (2019).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 39 / 61
Euler-Maclaurin
Using this result, we have
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +c1(f )
2n+ · · ·+ cα−1(f )
2(α−1)n+ O(2−αn).
Consider high precision, say n = αm. Noting that N = 2m, the firstterm becomes dominant since
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +c1(f )
Nα+ · · ·+ cα−1(f )
Nα(α−1)+ O(N−α2
).
Remark: For functions with dominating mixed smoothness α, theintegration error cannot be better than O(N−α) in general.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 40 / 61
Euler-Maclaurin
Using this result, we have
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +c1(f )
2n+ · · ·+ cα−1(f )
2(α−1)n+ O(2−αn).
Consider high precision, say n = αm. Noting that N = 2m, the firstterm becomes dominant since
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +c1(f )
Nα+ · · ·+ cα−1(f )
Nα(α−1)+ O(N−α2
).
Remark: For functions with dominating mixed smoothness α, theintegration error cannot be better than O(N−α) in general.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 40 / 61
Richardson extrapolation
Consider the case n = m.
Let us simplify the situation for a moment and consider the followingexpansions for successive values of m:
I(1)m−α+1 := c0 +
c12m−α+1
+c2
22(m−α+1)+ · · ·+ cα−1
2(α−1)(m−α+1)
...
I(1)m−1 := c0 +
c12m−1
+c2
22(m−1)+ · · ·+ cα−1
2(α−1)(m−1)
I(1)m := c0 +
c12m
+c222m
+ · · ·+ cα−1
2(α−1)m
Problem: Estimate c0 from values of I(0)m−α+1, . . . , I
(0)m as precisely as
possible (without knowing c1, . . . , cα−1).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 41 / 61
Richardson extrapolation
Consider the case n = m.
Let us simplify the situation for a moment and consider the followingexpansions for successive values of m:
I(1)m−α+1 := c0 +
c12m−α+1
+c2
22(m−α+1)+ · · ·+ cα−1
2(α−1)(m−α+1)
...
I(1)m−1 := c0 +
c12m−1
+c2
22(m−1)+ · · ·+ cα−1
2(α−1)(m−1)
I(1)m := c0 +
c12m
+c222m
+ · · ·+ cα−1
2(α−1)m
Problem: Estimate c0 from values of I(0)m−α+1, . . . , I
(0)m as precisely as
possible (without knowing c1, . . . , cα−1).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 41 / 61
Richardson extrapolation
Do the following recursion: For τ = 1, . . . , α− 1, compute
I(τ+1)n :=
2τ I(τ)n − I
(τ)n−1
2τ − 1for n = m − α+ τ + 1, . . . ,m
Schematically this recursion goes like
I(1)m−α+1 I
(1)m−α+2 · · · I
(1)m−1 I
(1)m
↘ ↓ ↘ · · · ↓ ↘ ↓I(2)m−α+2 · · · I
(2)m−1 I
(2)m
↘ · · · ↓ ↘ ↓. . .
...
I(α−1)m−1 I
(α−1)m
↘ ↓I(α)m
We will end up with I(α)m = c0. In reality, we have other terms...
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 42 / 61
Use a set of digital nets
For m − α < n ≤ m, let Pn be a digital net with C1, . . . ,Cs ∈ Fn×n2 .
Here we do NOT require Pm−α+1 ⊂ Pm−α+2 ⊂ · · · ⊂ Pm.
Richardson extrapolation can push out the intermediate terms of theexpansion
QPn(f )− I (f ) =∑
k∈P⊥n \{0}2n∤k
f̂ (k) +c1(f )
2n+ · · ·+ cα−1(f )
2(α−1)n+ O(2−αn)
without knowing c1(f ), . . . , cα−1(f ).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 43 / 61
Use a set of digital nets
This way we arrive at an extrapolated QMC rule which satisfies
m∑n=m−α+1
wnQPn(f )− I (f ) =m∑
n=m−α+1
wn
∑k∈P⊥
n \{0}2n∤k
f̂ (k) + O(2−αm),
where wn’s depend only on α,
m∑n=m−α+1
wn = 1 andm∑
n=m−α+1
|wn| ≤α−1∏i=1
2i + 1
2i − 1.
NOTE: The total number of points is 2m−α+1 + · · ·+ 2m < 2m+1.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 44 / 61
What we want
m∑n=m−α+1
wnQPn(f )− I (f ) =m∑
n=m−α+1
wn
∑k∈P⊥
n \{0}2n∤k
f̂ (k) + O(2−αm),
If we have digital nets with C1, . . . ,Cs ∈ Fn×n2 which satisfy∑
k∈P⊥n \{0}2n∤k
f̂ (k) = O(2−(α−ε)n) with arbitrarily small ε > 0,
extrapolated QMC rule achieves the almost optimal rate ofconvergence for smooth functions.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 45 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 46 / 61
Truncation operator
For x = (0.ξ1ξ2 . . .)2 ∈ [0, 1), define the map
trm(x) := (0.ξ1ξ2 . . . ξm00 . . .)2,
For a point set P ⊂ [0, 1)s , we write
trm(P) = {trm(x) | x ∈ P},
where trm is applied componentwise.
If P is a digital net with C1, . . . ,Cs ∈ Fn×m2 for n ≥ m, trm(P) is a
digital net with upper m ×m submatrices of C1, . . . ,Cs .
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 47 / 61
What G. (2019+) proves
Theorem
Let P be an order α digital (t,m, s)-net. For functions with dominatingmixed smoothness α ≥ 2, we have∑
k∈(trm(P))⊥\{0}2m∤k
f̂ (k) = O
((logN)αs
Nα
),
with N = 2m.
Compare:∑k∈P⊥\{0}
f̂ (k) = O
((logN)αs
Nα
)but
∑k∈(trm(P))⊥\{0}
f̂ (k) = O
(1
N
)
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 48 / 61
Truncation of higher order digital nets
Figure: Interlaced Sobol’ point sets: original (left) and after truncation (right)
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 49 / 61
Numerical results: original higher order QMCUnivariate test function:
f (x) = x3 (log x + 0.25) .
Interlaced Sobol’ sequences: If αm > 52, tr52 is applied.
0 5 10 15 20 25log
2N
-70
-60
-50
-40
-30
-20
-10
0lo
g 2(abs
olut
e in
tegr
atio
n er
ror)
Order 2 Sobol'Order 3 Sobol'N-2
N-3
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 50 / 61
Numerical results: extrapolation
QP(f )− I (f ) =∑
k∈P⊥\{0}2n∤k
f̂ (k) +c1(f )
2n+ · · ·+ cα−1(f )
2(α−1)n+ O(2−αn).
Extrapolation applied to truncated interlaced Sobol’ sequences: order2 (left) and order 3 (right)
0 5 10 15 20 25log
2N
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
log 2(a
bsol
ute
inte
grat
ion
erro
r)
Im(1)
Im(2)
N-1
N-2
0 5 10 15 20 25log
2N
-70
-60
-50
-40
-30
-20
-10
0
log 2(a
bsol
ute
inte
grat
ion
erro
r)
Im(1)
Im(2)
Im(3)
N-1
N-2
N-3
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 51 / 61
Numerical results: comparison for large sTest function:
f (x) =s∏
j=1
[1 + γj
(xcj − 1
c + 1
)],
with s = 100, c = 1.3 and γj = j−2.
0 5 10 15 20log
2N
-40
-35
-30
-25
-20
-15
-10
-5
log 2(a
bsol
ute
inte
grat
ion
erro
r)
Im(2)
Jm(2)
Order 2 Sobol'N-2
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 52 / 61
Outline
QMC and digital nets/sequences
Classical QMC
Higher order QMC
Richardson extrapolation and QMC
Application 1: Truncation of higher order nets and sequences
Application 2: Extrapolated polynomial lattice rules
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 53 / 61
What Dick, G. & Yoshiki (2019) prove
Theorem
Let p ∈ F2[x ] be irreducible with deg(p) = m. Component-by-componentalgorithm based on a computable criterion can find good q ∈ (F2[x ])
s
which satisfies∑k∈P(p,q)⊥\{0}
2m∤k
f̂ (k) = O(2−(α−ε)m) with arbitrarily small ε > 0,
for functions with dominating mixed smoothness α.
Here “a computable criterion” has been provided by Baldeaux, Dick,Leobacher, Nuyens & Pillichshammer (2012), which stems from thedecay of Walsh coefficients for smooth functions.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 54 / 61
Construction cost
Since we do NOT require Pm−α+1 ⊂ Pm−α+2 ⊂ · · · ⊂ Pm, polynomiallattice point set of each size can be constructed independently.
Construction cost of each is of O((s + α)n2n). In total, the cost willbe of order
m∑n=m−α+1
(s + α)n2n ≤ (s + α)N log2N,
where N = 2m−α+1 + · · ·+ 2m. This is better than interlacedpolynomial lattice point sets for which we need O(αsN logN).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 55 / 61
Fast QMC matrix-vector product
In some applications, we want to estimate the integral of the form∫[0,1)s
f (Ax) dx .
In terms of computational cost, computing Ax can be dominant.
Even if A does not have any special structure, this computation canbe made fast by using (polynomial) lattice point sets (Dick, Kuo, LeGia, Schwab, 2015).
This strategy does not work for interlaced polynomial lattice rules,whereas it does work for extrapolated polynomial lattice rules.
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 56 / 61
Numerical results (α = 2)Test function:
f (x) =s∏
j=1
[1 +
γj1 + γjx2j
],
with s = 100 and γj = j−2.Extrapolated rule with α = 2 (red), interlaced rule with α = 2 (green)
4 6 8 10 12 14 16
log2N
10-12
10-10
10-8
10-6
10-4
10-2
100
Ab
solu
te e
rro
r
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 57 / 61
Numerical results (α = 3)Test function:
f (x) =s∏
j=1
[1 +
γj1 + γjx2j
],
with s = 100 and γj = j−2.Extrapolated rule with α = 3 (red), interlaced rule with α = 3 (green)
4 6 8 10 12 14 16
log2N
10-12
10-10
10-8
10-6
10-4
10-2
100
Ab
solu
te e
rro
r
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 58 / 61
Conclusions
As far as I know, this is the first attempt to apply an extrapolationtechnique to QMC. Roughly speaking, Richardson extrapolation isshown useful for reducing necessary precision significantly.
Two results are obtained so far1 truncation of higher order digital nets/sequences: high-order
convergence can be recovered by applying Richardson extrapolation.2 introduction of extrapolated polynomial lattice rules: without relying on
interlacing, we can construct high-order convergent QMC-based rules.
Do we have another good application of extrapolation? What aboutlattice rules?
Can we randomize extrapolated QMC rule? Partially yes, if the samerandomly chosen digital shift is applied. What about scrambling?
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 59 / 61
Conclusions
As far as I know, this is the first attempt to apply an extrapolationtechnique to QMC. Roughly speaking, Richardson extrapolation isshown useful for reducing necessary precision significantly.
Two results are obtained so far1 truncation of higher order digital nets/sequences: high-order
convergence can be recovered by applying Richardson extrapolation.2 introduction of extrapolated polynomial lattice rules: without relying on
interlacing, we can construct high-order convergent QMC-based rules.
Do we have another good application of extrapolation? What aboutlattice rules?
Can we randomize extrapolated QMC rule? Partially yes, if the samerandomly chosen digital shift is applied. What about scrambling?
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 59 / 61
If you are interested, please refer to
J. Dick, TG, T. Yoshiki: Richardson extrapolation of polynomiallattice rules, SIAM Journal on Numerical Analysis, 57(1):44-69, 2019.
TG: Richardson extrapolation allows truncation of higher order digitalnets and sequences, IMA Journal of Numerical Analysis, online(arXiv:1805.09490).
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 60 / 61
Thank you for your attention!
Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 61 / 61