Lattice rules and sequences for non periodic smooth integrands Dirk - - PowerPoint PPT Presentation

▶
lattice rules and sequences for non periodic smooth
SMART_READER_LITE
LIVE PREVIEW

Lattice rules and sequences for non periodic smooth integrands Dirk - - PowerPoint PPT Presentation

Jan 08, 2024 238 likes •514 views

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens dirk.nuyens@cs.kuleuven.be Department of

slide-1
SLIDE 1

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Lattice rules and sequences for non periodic smooth integrands

Dirk Nuyens

dirk.nuyens@cs.kuleuven.be Department of Computer Science KU Leuven, Belgium Joint work with Josef Dick (UNSW) and Friedrich Pillichshammer (Linz) MCQMC 2012 February 13–17, 2012 Sydney, NSW Australia

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-2
SLIDE 2

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Outline

1

Lattice rules and sequences

2

Basis functions

3

Method I: Symmetrization

4

Method II: Tent transform

5

Numerical examples

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-3
SLIDE 3

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Multivariate integration

Multivariate integration by lattice rules

Approximate the s-dimensional integral I(f ) :=

  • [0,1]s f (x) dx

by an N-point (rank-1) lattice rule Q(f ; g, N) := 1 N

N−1

  • n=0

f gn N

  • with “good” generating vector g ∈ Zs

N.

Aim: Non periodic functions and lattice rules. Result: Function space H(Kcos).

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-4
SLIDE 4

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Preliminaries

Imagery of good lattice rules and sequences

(a) rank-1 rule (b) Fibonacci lattice (c) rank-2 copy rule (d) 33 seq points (e) 64 seq points (f) 34 seq points

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

fixed lattice rules lattice sequence in base 3

slide-5
SLIDE 5

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Preliminaries

Error of integration for lattice rule

For f with Fourier series representation f (x) =

  • h∈Zs

ˆ f (h) exp(2πi h · x),

ˆ f (h) =

  • [0,1]s f (x) exp(−2πi h · x) dx,

we have Q(f ; g, N) − I(f ) =

  • h∈Zs

ˆ f (h) 1 N

N−1

  • n=0

exp(2πi (h · g)n/N) − ˆ f (0) =

  • 0=h∈Zs

h·g≡0 (mod N)

ˆ f (h). The error is given as a sum, h = 0, over the dual lattice: L⊥ := {h ∈ Zs : h · g ≡ 0 (mod N)} ⊆ Zs.

See Sloan & Joe (1994), Sloan & Kachoyan (1987), Niederreiter (1992).

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-6
SLIDE 6

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples The classical Korobov space

A reproducing kernel Hilbert space...

Traditional setting for lattice rules: Korobov space. Absolutely convergent Fourier series representation f (x) =

  • h∈Zs

ˆ f (h) exp(2πi h · x). Then f ∈ H(Kα), for α > 1/2, if f 2

Kα :=

  • h∈Zs

|ˆ f (h)|2 rα(h) < ∞ with rα(h) :=

s

  • j=1

rα(hj), rα(hj) :=

  • 1,

if hj = 0, |hj|−2α,

  • therwise.

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-7
SLIDE 7

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples The classical Korobov space

... and its worst case error

The worst case error of QMC integration using a point set P = {x0, x1, . . . , xN−1} is defined as e(H(K ); P) := sup

f ∈H(K ) f K ≤1

  • [0,1]s f (x) dx − 1

N

N−1

  • n=0

f (xn)

  • .

For Korobov space using a lattice rule P this can be written e(H(Kα); P)2 =

  • 0=h∈Zs

h·g≡0 (mod N)

rα(h). Construction of rules with e(H(Kα); P) = O(N −α) using CBC.

(Korobov, Kuo, Sloan, Joe, Dick, ...)

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-8
SLIDE 8

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples The classical Korobov space

So all is nice?

Are your functions periodic? Thought so... Classical solution: randomly shifted spaces (create shift-invariant kernel); or periodization and symmetrization; or tent transform (bakers transform).

(Kuo, Sloan, Joe, Hickernell (2002), Zaremba, Korobov, ...)

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-9
SLIDE 9

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Building reproducing kernels

Kernels build from orthogonal bases

If {ϕk(x)}k is an ONB w.r.t. a specified inner product, then K (x, y) =

  • k

ϕk(x) ϕk(y), (*) conditions apply to make it a RKHS. E.g. could take {φk(x)}k is an ONB w.r.t. L2([0, 1]) and take Kα(x, y) =

  • k

rα(k) φk(x) φk(y), then (f , g)Kα =

  • k
  • f (k)

g(k) rα(k) , with

  • f (k) =
  • [0,1]

f (x) φk(x) dx.

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-10
SLIDE 10

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Korobov space

Korobov space

What is the basis of the Korobov space of real functions? K (x, y) = 1 +

  • 0=h∈Z

exp(2πi h(x − y)) |h|2α = 1 +

  • 0=h∈Z

exp(2πi hx) |h|α exp(2πi hy) |h|α = 1 +

∞

  • k=1

2 cos(2πkx) cos(2πky) |h|2α +

∞

  • k=1

2 sin(2πkx) sin(2πky) |h|2α So we have, w.r.t. L2([0, 1]):

  • 1

√ 2 cos(2πkx) ∞

k=1

√ 2 sin(2πkx) ∞

k=1

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-11
SLIDE 11

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Korobov space

Basis of Korobov space (periodic)

0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.5 1.0

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-12
SLIDE 12

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Korobov space

A half-period cosine space

Suppose we want to take the L2([0, 1]) ONB:

  • 1

√ 2 cos(πkx) ∞

k=1

and define

  • f (k) :=
  • [0,1]

f (x) ( √ 2)|k|0 cos(πkx) dx, where |k|0 counts the non-zero entries, i.e., |k|0 is 1 if k = 0 and 0 otherwise. Why? Well, for starters: x = 1 2 − 4 π2

∞

  • k=1

k odd

cos(πkx) k2 .

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-13
SLIDE 13

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space

Basis of half-period cosine space (non periodic)

0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.5 1.0

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-14
SLIDE 14

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space

Half-period cosine RKHS

Define rα,β(h) :=      1 if h = 0, |h/2|−2α if h is even, |h/2|−2β if h is odd, Set Kcos(x, y) = 1 +

∞

  • k=1

rα,β(k) 2 cos(πkx) cos(πky), and f 2

Kcos = ∞

  • k=0

| f (k)|2 rα,β(k) =

  • h∈Z

2−|h|0 | f (|h|)|2 rα,β(h) .

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-15
SLIDE 15

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space

Further motivation for the cosine space...

The unanchored Sobolev space of smoothness a = 1 has kernel Ka(x, y) = 1 + B2(|x − y|) + (x − 1

2)(y − 1 2)

= 1 +

∞

  • k=1

1 π2 2 cos(πkx) cos(πky) k2 . Compare with, taking α = β in rα,β, for cos kernel Kcos(x, y) = 1 +

∞

  • k=1

22α 2 cos(πkx) cos(πky) k2α . Now take α = 1 and imagine product weights on Kcos of the form γj ≡ (2π)−2...

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-16
SLIDE 16

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space

In pictures...

− = Ka=1 − Kcos,γ = Ka=1 − Kcos,γ

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-17
SLIDE 17

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Definition

Method I: The symmetrization operation

Define symu(x) = (y1, . . . , ys) with yj =

  • 1 − xj

if j ∈ u, xj if j ∈ u. We define the symmetrized lattice rule Qsym(f ; g, N) := 1 2sN

N−1

  • n=0
  • u⊆1:s

f

  • symu

ng N

  • .

Note that for k ∈ Z and x ∈ R: cos(πkx) + cos(πk(1 − x)) =

  • 2 cos(2πk′x)

if k = 2k′ is even, if k is odd.

(Use of symmetrization: Korobov (1963), Genz & Malik (1983))

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-18
SLIDE 18

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Analysis

Error in cos space with symmetrization

Proposition Let f be given as f (x) =

  • k∈{0,1,...}s
  • f (h) (

√ 2)|k|0

s

  • j=1

cos(πkjxj). Then Qsym(f ; g, N) −

  • [0,1]s f (x) dx =
  • 0=h∈Zs

h·g≡0 (mod N)

( √ 2)−|h|0 f (|2h|).

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-19
SLIDE 19

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Analysis

Worst case error in cos space with symmetrization

Worst case error is the same as in the Korobov space with smoothness α. The parameter β does not matter. Corollary e2(H(Kcos); Psym(g, N)) = e2(H(Kα); P(g, N)) =

  • h∈Zs\{0}

h·g≡0 (mod N)

rα,β(2h). We can construct lattice rules by (fast) CBC such that, δ > 0, e(H(Kcos); Psym(g, N)) = e(H(Kα); P(g, N)) ≤ Cα,s,δ N −α+δ ≤ Cα,s,δ 2sαM −α+δ.

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-20
SLIDE 20

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Method II: The tent transform

Define φ(x) := 1 − |2x − 1|, and the tent transformed lattice point set Pφ(g, N) :=

  • φ

ng N

  • : 0 ≤ n < N
  • .

Used by Hickernell (2002) for randomly shifted unanchored Sobolev spaces. NB: Here we don’t shift! Note that for h ∈ Z and x ∈ R: cos(πhφ(x)) = cos(2πhx).

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-21
SLIDE 21

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Error in cos space with tent transform

Proposition Let f be given as f (x) =

  • k∈{0,1,...}s
  • f (k)(

√ 2)|k|0

s

  • j=1

cos(πkjxj), then the error for numerical integration using the tent transformed lattice point set Pφ(g, N) is given by Qφ(f ; g, N) −

  • [0,1]s f (x) dx =
  • 0=h∈Zs

h·g≡0 (mod N)

( √ 2)−|h|0 f (|h|).

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-22
SLIDE 22

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Worst case error in the cos space with tent transform

Corollary e2(H(Kcos); Pφ(g, N)) =

  • 0=h∈Zs

h·g≡0 (mod N)

rα,β(h) and for α′ := min(α, β) and β′ := max(α, β) we have e(H(Kα); P(g, N)) ≤ e(H(Kcos); Pφ(g, N)) ≤ 2sβ′e(H(Kα′); P(g, N)). We can construct lattice rules by (fast) CBC such that, δ > 0, e(H(Kcos); Pφ(g, N)) ≤ Cα,β,s,δN −α′+δ.

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-23
SLIDE 23

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Unweighted

Consider the function f (x) =

s

  • j=1
  • 1 +
  • − 10

21 + 2x2 j − 2x5 j + x6 j

  • in 5 dimensions:

10−16 10−12 10−8 10−4 100 104 relative error relative error 100 101 102 103 104 105 106 number of function evaluations number of function evaluations lat latper latsym latbak Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-24
SLIDE 24

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Unweighted

Consider the function f (x) =

s

  • j=1
  • 1 +
  • − 10

21 + 2x2 j − 2x5 j + x6 j

  • in 10 dimensions:

10−16 10−12 10−8 10−4 100 104 relative error relative error 100 101 102 103 104 105 106 number of function evaluations number of function evaluations lat latper latsym latbak Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-25
SLIDE 25

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Weighted

With weights γj = 0.9j: f (x) =

s

  • j=1
  • 1 + 0.9j

− 10

21 + 2x2 j − 2x5 j + x6 j

  • in 5 dimensions:

10−16 10−12 10−8 10−4 100 104 relative error relative error 100 101 102 103 104 105 106 number of function evaluations number of function evaluations lat latper latsym latbak Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-26
SLIDE 26

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Weighted

With weights γj = 0.9j: f (x) =

s

  • j=1
  • 1 + 0.9j

− 10

21 + 2x2 j − 2x5 j + x6 j

  • in 10 dimensions:

10−16 10−12 10−8 10−4 100 104 relative error relative error 100 101 102 103 104 105 106 number of function evaluations number of function evaluations lat latper latsym latbak Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens

slide-27
SLIDE 27

Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples

Thank you for your attention.

Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens