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
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Outline Lattice rules and sequences 1 Basis functions 2 Method I: Symmetrization 3 Method II: Tent transform 4 Numerical examples 5 Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 ) d x by an N -point (rank-1) lattice rule N − 1 �� g n �� � Q ( f ; g , N ) := 1 f N N n = 0 with “good” generating vector g ∈ Z s N . Aim: Non periodic functions and lattice rules. Result: Function space H ( K cos ) . Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Preliminaries Imagery of good lattice rules and sequences lattice rules fixed (a) rank-1 rule (b) Fibonacci lattice (c) rank-2 copy rule lattice sequence in base 3 (d) 3 3 seq points (f) 3 4 seq points (e) 64 seq points Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 � ˆ � f ( x ) = f ( h ) exp ( 2 π i h · x ) , ˆ f ( h ) = [ 0 , 1 ] s f ( x ) exp ( − 2 π i h · x ) d x , h ∈ Z s we have N − 1 � � f ( h ) 1 ˆ exp ( 2 π i ( h · g ) n / N ) − ˆ Q ( f ; g , N ) − I ( f ) = f ( 0 ) N h ∈ Z s n = 0 � ˆ = f ( h ) . 0 � = h ∈ Z s h · g ≡ 0 ( mod N ) The error is given as a sum, h � = 0 , over the dual lattice: L ⊥ := { h ∈ Z s : h · g ≡ 0 ( mod N ) } ⊆ Z s . See Sloan & Joe (1994), Sloan & Kachoyan (1987), Niederreiter (1992). Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 � ˆ f ( x ) = f ( h ) exp ( 2 π i h · x ) . h ∈ Z s Then f ∈ H ( K α ) , for α > 1 / 2, if � | ˆ f ( h ) | 2 � f � 2 K α := r α ( h ) < ∞ h ∈ Z s with � s � 1 , if h j = 0 , r α ( h ) := r α ( h j ) , r α ( h j ) := | h j | − 2 α , otherwise . j = 1 Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 = { x 0 , x 1 , . . . , x N − 1 } is defined as � � � � � N − 1 � [ 0 , 1 ] s f ( x ) d x − 1 � � e ( H ( K ); P ) := f ( x n ) � . sup � � � N f ∈H ( K ) n = 0 � f � K ≤ 1 For Korobov space using a lattice rule P this can be written � e ( H ( K α ); P ) 2 = r α ( h ) . 0 � = h ∈ Z s h · g ≡ 0 ( mod N ) Construction of rules with e ( H ( K α ); P ) = O ( N − α ) using CBC. (Korobov, Kuo, Sloan, Joe, Dick, ...) Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Building reproducing kernels Kernels build from orthogonal bases If { ϕ k ( x ) } k is an ONB w.r.t. a specified inner product, then � K ( x , y ) = ϕ k ( x ) ϕ k ( y ) , k (*) conditions apply to make it a RKHS. E.g. could take { φ k ( x ) } k is an ONB w.r.t. L 2 ([ 0 , 1 ]) and take � K α ( x , y ) = r α ( k ) φ k ( x ) φ k ( y ) , k then � � � f ( k ) � g ( k ) � ( f , g ) K α = , with f ( k ) = f ( x ) φ k ( x ) d x . r α ( k ) [ 0 , 1 ] k Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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? � exp ( 2 π i h ( x − y )) K ( x , y ) = 1 + | h | 2 α 0 � = h ∈ Z � exp ( 2 π i hx ) exp ( 2 π i hy ) = 1 + | h | α | h | α 0 � = h ∈ Z ∞ ∞ � � 2 cos ( 2 π kx ) cos ( 2 π ky ) 2 sin ( 2 π kx ) sin ( 2 π ky ) = 1 + + | h | 2 α | h | 2 α k = 1 k = 1 So we have, w.r.t. L 2 ([ 0 , 1 ]) : � � � � √ � ∞ � � √ � ∞ 1 2 cos ( 2 π kx ) 2 sin ( 2 π kx ) k = 1 k = 1 Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Korobov space Basis of Korobov space (periodic) 1.0 0.5 0.2 0.4 0.6 0.8 1.0 � 0.5 � 1.0 Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 L 2 ([ 0 , 1 ]) ONB: � � � � √ � ∞ 1 2 cos ( π kx ) k = 1 and define � √ � 2 ) | k | 0 cos ( π kx ) d x , f ( k ) := f ( x ) ( [ 0 , 1 ] 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 cos ( π kx ) . π 2 k 2 k = 1 k odd Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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) 1.0 0.5 0.2 0.4 0.6 0.8 1.0 � 0.5 � 1.0 Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space Half-period cosine RKHS Define 1 if h = 0 , | h / 2 | − 2 α r α,β ( h ) := if h is even , | h / 2 | − 2 β if h is odd , Set ∞ � K cos ( x , y ) = 1 + r α,β ( k ) 2 cos ( π kx ) cos ( π ky ) , k = 1 and ∞ � | � � 2 −| h | 0 | � f ( k ) | 2 f ( | h | ) | 2 � f � 2 K cos = r α,β ( k ) = r α,β ( h ) . k = 0 h ∈ Z Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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 K a ( x , y ) = 1 + B 2 ( | x − y | ) + ( x − 1 2 )( y − 1 2 ) ∞ � 1 2 cos ( π kx ) cos ( π ky ) = 1 + . π 2 k 2 k = 1 Compare with, taking α = β in r α,β , for cos kernel ∞ � 2 2 α 2 cos ( π kx ) cos ( π ky ) K cos ( x , y ) = 1 + . k 2 α k = 1 Now take α = 1 and imagine product weights on K cos of the form γ j ≡ ( 2 π ) − 2 ... Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Half-period cosine space In pictures... − = K a = 1 − K cos ,γ = K a = 1 − K cos ,γ Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
Lattice rules and sequences Basis functions Method I: Symmetrization Method II: Tent transform Numerical examples Definition Method I: The symmetrization operation Define � 1 − x j if j ∈ u , sym u ( x ) = ( y 1 , . . . , y s ) with y j = x j if j �∈ u . We define the symmetrized lattice rule � �� n g ��� N − 1 � � 1 Q sym ( f ; g , N ) := f sym u . 2 s N N n = 0 u ⊆ 1 : s Note that for k ∈ Z and x ∈ R : � if k = 2 k ′ is even , 2 cos ( 2 π k ′ x ) cos ( π kx ) + cos ( π k ( 1 − x )) = 0 if k is odd . (Use of symmetrization: Korobov (1963), Genz & Malik (1983)) Lattice rules and sequences for non periodic smooth integrands Dirk Nuyens
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