Tractability of Multivariate Integration in Hermite Spaces Friedrich - PowerPoint PPT Presentation

Tractability of Multivariate Integration in Hermite Spaces Friedrich Pillichshammer 1 JKU Linz/Austria Joint work with Ch. Irrgeher, P. Kritzer and G. Leobacher (JKU Linz) 1 Supported by the Austrian Science Fund (FWF), Project F5509-N26. Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 1 / 25

Multivariate integration over R s – linear algorithms We study the numerical approximation of integrals � I s ( f ) = R s f ( x ) ϕ s ( x ) d x , where ϕ s is the density of the s -dimensional standard Gaussian measure, and f ∈ H ( K ) (RKHS) with norm � · � K . We use linear algorithms n � A n , s ( f ) = α k f ( t k ) k =1 for α k ∈ R and t k ∈ R s . Linear algorithms are optimal (Smolyak 1965, Bakhvalov 1971) Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 2 / 25

Multivariate integration over R s – worst-case setting Worst-case error : e ( A n , s , K ) = sup | I s ( f ) − A n , s ( f ) | . f ∈H ( K ) � f � K ≤ 1 n th minimal worst-case error : e ( n , s ) = inf A n , s e ( A n , s , K ) . Initial error : For n = 0, we approximate I s ( f ) by zero, and e (0 , s ) = � I s � for all s ∈ N . Information complexity : For ε ∈ (0 , 1), n ( ε, s ) = min { n : e ( n , s ) ≤ ε e (0 , s ) } . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 3 / 25

Smoothness of the problems We study problems with infinite smoothness. It is natural to demand more ◮ of the n th minimal errors e ( n , s ) and ◮ of the information complexity n ( ε, s ) than for those cases where we only have finite smoothness. For problems with unbounded smoothness we are interested in obtaining (uniform) exponential convergence of the minimal errors. Well studied: Korobov spaces of periodic functions over [0 , 1] s with infinite smoothness (Dick, Kritzer, Larcher, Wo´ zniakowski) Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 4 / 25

Exponential convergence Definition Exponential convergence (EXP) if ∃ q ∈ (0 , 1) and functions p , C , C 1 : N → (0 , ∞ ) such that e ( n , s ) ≤ C ( s ) q ( n / C 1 ( s )) p ( s ) for all s , n ∈ N . The largest possible rate of EXP is � p > 0 : ∃ C , C 1 > 0 s.t. ∀ n ∈ N : e ( n , s ) ≤ Cq ( n / C 1 ) p � p ∗ ( s ) = sup . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 5 / 25

Uniform exponential convergence Definition Uniform exponential convergence (UEXP) if ∃ q ∈ (0 , 1) , ∃ p > 0 and functions C , C 1 : N → (0 , ∞ ) such that e ( n , s ) ≤ C ( s ) q ( n / C 1 ( s )) p for all s , n ∈ N . The largest rate of UEXP is � p ∗ = sup p > 0 : ∃ C , C 1 : N → (0 , ∞ ) s.t. ∀ n , s ∈ N : e ( n , s ) ≤ C ( s ) q ( n / C 1 ( s )) p � . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 6 / 25

Exponential convergence UEXP implies � � 1 / p � � log C ( s ) + log ε − 1 n ( ε, s ) ≤ C 1 ( s ) for all s ∈ N , ε ∈ (0 , 1) . log q − 1 �� log ε − 1 � 1 / p � With respect to ε → 0, we need O function values to reduce the initial error by a factor of ε . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 7 / 25

EC-tractability (a) Exponential Convergence-Weak Tractability (EC-WT) if log n ( ε, s ) lim s + log ε − 1 = 0 . s + ε − 1 →∞ (b) Exponential Convergence-Polynomial Tractability (EC-PT) if ∃ c , τ 1 , τ 2 > 0 such that n ( ε, s ) ≤ c s τ 1 (1 + log ε − 1 ) τ 2 for all s ∈ N , ε ∈ (0 , 1) . (c) Exponential Convergence-Strong Polynomial Tractability (EC-SPT) if ∃ c , τ > 0 such that n ( ε, s ) ≤ c (1 + log ε − 1 ) τ for all s ∈ N , ε ∈ (0 , 1) . The exponent τ ∗ of EC-SPT is the infimum of τ for which EC-SPT holds, i.e., τ ∗ = inf { τ ≥ 0 : ∃ c > 0 s.t. n ( ε, s ) ≤ c (1 + log ε − 1 ) τ ∀ s , ε } . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 8 / 25

EC-tractability Proposition 1 EC-SPT ⇒ EC-PT ⇒ UEXP 2 EC-WT ⇒ lim n →∞ n α e ( n , s ) = 0 for all α > 0 3 If we have UEXP ( e ( n , s ) ≤ C ( s ) q ( n / C 1 ( s )) p ), then: ◮ C ( s ) = exp(exp( o ( s ))) and C 1 ( s ) = exp( o ( s )) ⇒ EC-WT ◮ C ( s ) = exp( O ( s τ )) and C 1 ( s ) = O ( s η ) ⇒ EC-PT ◮ C ( s ) = O (1) and C 1 ( s ) = O (1) ⇒ EC-SPT Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 9 / 25

Hermite polynomials Univariate Hermite polynomials H k ( x ) = ( − 1) k d k x 2 d x k e − x 2 √ for k ∈ N 0 , x ∈ R e 2 2 k ! � E.g. H 0 ( x ) = 1, H 1 ( x ) = x , H 2 ( x ) = x 2 2 , H 3 ( x ) = x 3 1 3 2 − 3 − 2 x √ √ √ Multivariate Hermite polynomials s � for k ∈ N s 0 , x ∈ R s H k ( x ) = H k j ( x j ) j =1 0 is an ONB of L 2 ( R s , ϕ s ) ( H k ) k ∈ N s Hermite expansion of f ∈ L 2 ( R s , ϕ s ): � � f ( x ) ∼ f ( k ) H k ( x ) k ∈ N s 0 � with k th Hermite coefficient � f ( k ) = R s f ( x ) H k ( x ) ϕ s ( x ) d x Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 10 / 25

Hermite spaces 0 → R + be a summable function. Let r : N s Define a kernel � for x , y ∈ R s , K r ( x , y ) = r ( k ) H k ( x ) H k ( y ) k ∈ N s 0 and a inner product � 1 � � f , g � K r := f ( k ) � g ( k ) . r ( k ) k ∈ N s 0 Let � f � 2 K r = � f , f � K r . We call the RKHS H ( K r ) a Hermite space . Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 11 / 25

The Hermite space H ( K s , a , b ,ω ) Let a = { a j } j ≥ 1 and b = { b j } j ≥ 1 , where we assume that 1 ≤ a 1 ≤ a 2 ≤ a 3 ≤ . . . and 1 ≤ b 1 ≤ b 2 ≤ b 3 ≤ . . . . Fix ω ∈ (0 , 1). For a vector k = ( k 1 , . . . , k s ) ∈ N s 0 , consider bj � s j =1 a j k j . r ( k ) = ω We modify the notation for the kernel function to � bj � s j =1 a j k j H k ( x ) H k ( y ) . K s , a , b ,ω ( x , y ) := ω k ∈ N s 0 Proposition f ∈ H ( K s , a , b ,ω ) ⇒ f is analytic R [ x ] ⊂ H ( K s , a , b ,ω ) f ( x ) = exp( λ · x ) belongs to H ( K s , a , b ,ω ) for suitable a , b e (0 , s ) = 1 Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 12 / 25

The main result Main Theorem 1 EXP holds for all a and b and s � p ∗ ( s ) = 1 / B ( s ) b − 1 with B ( s ) := . j j =1 2 Let B := � ∞ j =1 b − 1 . Then j B < ∞ ⇔ UEXP ⇔ EC-PT ⇔ EC-SPT and p ∗ = 1 / B and the exponent τ ∗ of EC-SPT is B . 3 EC-WT ⇒ lim j →∞ a j 2 b j = ∞ 4 lim j →∞ a j = ∞ ⇒ EC-WT Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 13 / 25

Remarks on the theorem We always have EXP, independent of a and b . The best rate p ∗ ( s ) is 1 / B ( s ), which decreases for growing s . A necessary and sufficient condition for UEXP, EC-PT and EC-SPT is that ∞ � b − 1 B = < ∞ j j =1 with no extra conditions on a and ω . The best rate p ∗ is 1 / B < 1. Small B implies a large p ∗ . a and ω have no influence on UEXP, EC-PT and EC-SPT. There is a gap between the necessary and sufficient condition for EC-WT. The results for EXP, UEXP, EC-PT and EC-SPT are constructive. Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 14 / 25

Gauss-Hermite rules – one-dimensional case A Gauss-Hermite rule of order n is a linear integration rule A n of the form n � A n ( f ) = α i f ( x i ) i =1 that is exact for all p ∈ R [ x ] with deg( p ) < 2 n , i.e. � p ( x ) ϕ ( x ) d x = A n ( p ) ∀ p ∈ R [ x ] with deg( p ) < 2 n . R The nodes x 1 , . . . , x n ∈ R are exactly the zeros of the n th Hermite polynomial H n and the weights are given by 1 α i = n − 1 ( x i ) . nH 2 Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 15 / 25

Gauss-Hermite rules – multivariate case For j = 1 , 2 , . . . , s let m j � A ( j ) α ( j ) i f ( x ( j ) m j ( f ) = ) i i =1 Let n = m 1 m 2 · · · m s and set A n , s = A (1) m 1 ⊗ · · · ⊗ A ( s ) m s , i.e., for f ∈ H ( K s , a , b ,ω ) m 1 m s � � α (1) i 1 · · · α ( s ) i s f ( x (1) i 1 , . . . , x ( s ) A n , s ( f ) = . . . i s ) . i 1 =1 i s =1 Proposition � � √ s � 8 π 1 + ω a j (2 m j ) bj e 2 ( A n , s , K s , a , b ,ω ) ≤ − 1 + . 1 − ω 2 j =1 Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 16 / 25

Gauss-Hermite rules – multivariate case Theorem For s ∈ N , let B ( s ) := � s j =1 b − 1 . For ε ∈ (0 , 1), let j  B ( s )  � √ �   8 π s log  1  1 − ω 2 log(1+ ε 2 )   m = max .    log ω − 1  a j j =1 , 2 ,..., s   and define m j := ⌊ m 1 / ( B ( s ) · b j ) ⌋ . Then � � 1 + 1 n ( ε, s ) ≪ s log B ( s ) e ( A n , s , K s , a , b ,ω ) ≤ ε and . ε This implies EXP. Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 17 / 25