Problem setting Lattices & tent transform Approximation Numerical results Conclusion Approximation in cosine space using tent transformed lattice rules Dirk Nuyens Department of Computer Science, KU Leuven, Belgium Joint work with Gowri Suryanarayana, Ronald Cools and Frances Kuo (UNSW). ICERM Semester Program on High-dimensional Approximation Information-Based Complexity and Stochastic Computation workshop September 15–19, 2014 Brown University, USA Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 1
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Outline Problem setting Rank-1 lattice & tent transformation Approximation Numerical results Conclusion Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 2
Problem setting Lattices & tent transform Approximation Numerical results Conclusion IBC setup Problem setting ◮ High dimensional multivariate approximation APP d : H d → L 2 ([ 0 , 1 ] d ) � H d the weighted cosine space of smoothness α > 1 / 2, ◮ by linear algorithm using standard information N � A N , d ( f )( x ) = f ( t i ) a i ( x ) i = 1 ◮ using a deterministic point set { t i } N i = 1 � tent transformed lattice points, constructive, ◮ and study the L 2 worst case error. This is a space of non-periodic functions. (Periodic case studied by Kuo, Sloan, Wo´ zniakowski (2006) + lots...) For α = 1 we get the unanchored Sobolev space of smoothness 1. Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 3
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Series expansion Cosine series ◮ Orthonormal basis of L 2 ([ 0 , 1 ]) √ | k | 0 cos ( π kx ) , φ k ( x ) := 2 k ∈ Z + := { 0 , 1 , 2 , . . . } . ◮ For d -dimensions, use tensor product d √ � | k | 0 φ k ( x ) := 2 cos ( π k j x j ) j = 1 where | k | 0 is the number of non-zero elements of k . √ Express f as series: f ( x ) = � | k | 0 � d ˆ f ( k ) 2 j = 1 cos ( π k j x j ) . k ∈ Z d + For d = 1 in Iserles and Nørsett (2008), for multivariate integration studied in Dick, N. and Pillichshammer (2013). → Laplace operator, talk by Art Werschulz on Helmholtz equation, ... Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 4
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Series expansion Function expansion using cosine series A multivariate non-periodic function f ∈ L 2 ([ 0 , 1 ] d ) that is continuously differentiable can be expressed as � ˆ f ( x ) = f ( k ) φ k ( x ) , k ∈ Z d + where ˆ f is the cosine transform of f � ˆ f ( k ) = [ 0 , 1 ] d f ( x ) φ k ( x ) d x . Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 5
Problem setting Lattices & tent transform Approximation Numerical results Conclusion The function space The weighted cosine space We set � f ( k ) | 2 r α, γ ( k ) | ˆ � f � 2 d ,α, γ := k ∈ Z d + and define the (half-period) cosine space by � � f ∈ L 2 ([ 0 , 1 ] d ) : � f � d ,α, γ < ∞ C d ,α, γ := , with α > 1 / 2, weights 1 ≥ γ 1 ≥ γ 2 ≥ · · · > 0 and � d � 1 , if k j = 0 , r α,γ j ( k j ) := r α, γ ( k ) := r α,γ j ( k j ) . k 2 α j /γ j , if k j > 0 , j = 1 (For numerical integration: Dick, N. and Pillichshammer (2013).) The weights control tractability, see Sloan & Wo´ zniakowski, Hickernell, Novak & Wo´ zniakowski, ... Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 6
Problem setting Lattices & tent transform Approximation Numerical results Conclusion The function space Reproducing kernel It is a reproducing kernel Hilbert space with reproducing kernel K d ( x , y ) = � d j = 1 K j ( x j , y j ) , where � r − 1 K j ( x j , y j ) = 1 + 2 α,γ j ( k ) cos ( π kx j ) cos ( π ky j ) . k ≥ 1 For α = 1 and γ j �→ π − 2 γ j we obtain the unanchored Sobolev space of dominating mixed smoothness 1, i.e., for d = 1 � π 2 k 2 cos ( π kx ) cos ( π ky ) = 1 + γ B 1 ( x ) B 1 ( y )+ γ B 2 ( | x − y | ) γ 1 + 2 2 k ≥ 1 (see DNP2013) with inner product given by � 1 � 1 � 1 g ( x ) d x + 1 f ′ ( x ) g ′ ( x ) d x . � f , g � = f ( x ) d x γ 0 0 0 Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 7
Problem setting Lattices & tent transform Approximation Numerical results Conclusion The algorithm The approximation algorithm Truncate cosine series expansion to k ∈ H d ,α, γ , then T approximate those cosine coefficients by an N -point tent transformed rank-1 lattice rule with points Λ ϕ ( z , N ) : � 1 � � � A N , d , T ( f )( x ) := f ( t ) φ k ( t ) φ k ( x ) . N k ∈ H d ,α, γ t ∈ Λ ϕ ( z , N ) T Here H d ,α, γ is a weighted hyperbolic cross, for T ≥ 0, T � � H d ,α, γ k ∈ Z d := + : r α, γ ( k ) ≤ T , T and, for α > 1 / 2 and positive weights γ j , and as before � d � 1 , if k j = 0 , r α,γ j ( k j ) = r α, γ ( k ) = r α,γ j ( k j ) . k 2 α j /γ j , if k j > 0 , j = 1 Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 8
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Worst case error Worst case error The approximation error at a point x is then � ˆ ( f − A N , d , T ( f ))( x ) = f ( k ) φ k ( x ) k �∈ H d ,α, γ T � � � � f ( k ) − 1 ˆ + f ( t ) φ k ( t ) φ k ( x ) . N k ∈ H d ,α, γ t ∈ Λ ϕ ( z , N ) T The worst case error of the algorithm A N , d , T is then given as e wor n , d ( A N , d , T ; C d ,α, γ ) := sup � f − A N , d , T ( f ) � L 2 ([ 0 , 1 ] d ) . f ∈ C d ,α, γ � f � d ,α, γ ≤ 1 Many references from people here present... Too many to list! Same setup as in Kuo, Sloan, Wo´ zniakowski (2006) for Fourier series. Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 9
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Rank- 1 lattice Lattice points For given number of points N ≥ 1 and z ∈ Z d N a rank-1 lattice is defined as � z k � Λ( z , N ) := N mod 1 : k = 0 , 1 , . . . , N − 1 , and z is called the generating vector. (Sloan & Joe, Korobov, Temlyakov, ... Mostly for integration.) Rank-1 lattice points with N = 34 , z = [ 1 , 21 ] 1 0 . 8 0 . 6 0 . 4 0 . 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 10
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Tent transform Tent transformation Tent transformed lattice points Λ ϕ ( z , N ) (a multiset): ◮ Tent transformation, ϕ : [ 0 , 1 ] → [ 0 , 1 ] is given by ϕ ( x ) := 1 − | 2 x − 1 | . ◮ Applied to each coordinate separately. ◮ Brings symmetry to the sampling points. ◮ Then for any k ∈ Z d + : √ √ | k | 0 � d | k | 0 � d 2 j = 1 cos ( π k j ϕ ( x j )) = 2 j = 1 cos ( 2 π k j x j ) . This is the “baker’s transform” in Hickernell (2000). Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 11
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Tent transform Tent transformation Rank-1 lattice points with N = 34 , z = [ 1 , 21 ] Tent transformed lattice points (18 points) 1 1 0 . 8 0 . 8 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 12
Problem setting Lattices & tent transform Approximation Numerical results Conclusion Two cases Approximation for two cases We consider two different cases: 1. Find generating vector which “approximates exactly” for f with frequency support on a hyperbolic cross of “degree” T . 2. Find generating vector which minimises L 2 worst case error. Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 13
Problem setting Lattices & tent transform Approximation Numerical results Conclusion 1. Hyperbolic cross support Functions with support on a hyperbolic cross Suppose f has frequency support limited to a hyperbolic cross H d , β = H d , 1 / 2 , β ⊆ Z d + then T T � � ˆ ˆ f ( x ) = f ( k ) φ k ( x ) = f ( k ) φ k ( x ) . k ∈ Z d k ∈ H d , β + T H 2 , { 1 , 1 } 20 20 15 10 5 0 0 5 10 15 20 Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 14
Problem setting Lattices & tent transform Approximation Numerical results Conclusion 1. Hyperbolic cross support Functions with support on a hyperbolic cross We have a weighted hyperbolic cross given by � � � � d � 1 , k j H d , β = H d , 1 / 2 , β k ∈ Z d := + : max ≤ T T T β j j = 1 with T ≥ 0, α = 1 / 2 (which is allowed as we now have finite sums), and weights β 1 ≥ β 2 ≥ · · · > 0. 2 , { 1 , 1 2 } H 20 10 8 6 4 2 0 0 5 10 15 20 “Degrees of exactness” for integration from Cools, Kuo, N. (2000). Picked up by Kämmerer et al for approximation on hyperbolic cross. Approximation in cosine space using tent transformed lattice rules / Dirk Nuyens 15
Recommend
More recommend
