High-dimensional and infinite-dimensional hyperbolic crosses and their applications in approximation and uncertainty quantification Dinh D˜ ung Vietnam National University, Hanoi, Vietnam Workshop on Information-Based Complexity and Stochastic Computation September 15 – 19, 2014, ICERM, Brown University October 2, 2014 ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 1 / 42 Dinh D˜
This talk is based in the recent joint works: 1 DD and T. Ullrich, N -Widths and ε -dimensions for high-dimensional approximations, Foundations Comp. Math. 13 (2013), 965-1003. 2 A. Chernov and DD, New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness, (2014) http://arxiv.org/abs/1309.5170. 3 DD and M. Griebel, Hyperbolic cross approximation in infinite dimensions and applications in sPDEs, Manuscript (2014). ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 2 / 42 Dinh D˜
The “curse of dimensionality” There has been a great interest in solving numerical problems involving functions of big number s of variables. By classical methods as usually, the computation cost grows exponentially in s . We suffer the “curse of dimensionality” [Bellmann,1957] (“Dimensionality” is referred to the number s of variables). A way to get rid of it is to assume that mixed derivatives of functions are bounded, then to apply hyperbolic cross (HC) approximation. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 3 / 42 Dinh D˜
Infinite-dimensional approximation The efficient approximation of a function of infinitely many variables is important in applications in physics, finance, engineering and statistics. It arises in UQ, computational finance and computational physics and is encountered for stochastic or parametric PDEs. Attempt: Apply infinite-dimensional HC to construct a linear approximation method to the solution of stochastic or parametric PDEs. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 4 / 42 Dinh D˜
Classical hyperbolic crosses Classical HCs Γ( s , T ) are a domain of frequencies of trigonometric polynomials used for approximations of periodic functions having mixed derivative. They are given by s � � k ∈ Z s : � Γ( s , T ) := max( | k i | , 1) ≤ T . i =1 Their cardinality is estimated as C ( s ) T log s − 1 T ≤ | Γ( s , T ) | ≤ C ′ ( s ) T log s − 1 T , where | G | denotes the cardinality of G . ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 5 / 42 Dinh D˜
n -Widths and ε -dimensions Kolmogorov n -widths: d n ( W , H ) := { L n linear subspaces , dim L n ≤ n } sup inf g ∈ L n � f − g � H . inf f ∈ W ε -dimension is the inverse of d n ( W , H ): n ε ( W , H ) := inf { n : ∃ L n : dim L n ≤ n , sup g ∈ L n � f − g � H ≤ ε } . inf f ∈ W n ε ( W , H ) is the necessary dimension of linear subspace for approximation of functions from W with accuracy ε . From the computational view it is more convenient to study n ε ( W , H ) since it is directly related to the computation cost. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 6 / 42 Dinh D˜
High-dimensional approach Sobolev space of mixed smoothness α ∈ N ∂ | k | 1 f 2 � � � f � 2 � mix = 2 , | k | ∞ := max 1 ≤ i ≤ s k i . � � H α ∂ x k 1 1 · · · ∂ x k s � � s | k | ∞ ≤ α U α mix is the unit ball in H α mix . Traditional estimations [Babenko 1960]: A ( α, s ) ε − 1 /α | log ε | s − 1 ≤ n ε ( U α mix , L 2 ) ≤ A ′ ( α, s ) ε − 1 /α | log ε | s − 1 . Our goal: to compute A ( α, s ), A ′ ( α, s ) explicitly in s . The basis for estimation of n ε : Reduction to computation of cardinality of the associated HCs: | Γ( s , 1 /ε ) | − 1 ≤ n ε ( U α mix , L 2 ) ≤ | Γ( s , 1 /ε ) | ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 7 / 42 Dinh D˜
Plan of our talk High-dimensional HC approximation for two models of mixed smoothness: Dyadic version [DD&Ullrich 2013]; Korobov version [Chernov&DD 2014]. Infinite-dimensional HC approximation for two models of regularity: Korobov version [DD&Griebel 2014]; Analytic version [DD&Griebel 2014]. Application of infinite-dimensional HC approximation in stochastic or parametric PDEs. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 8 / 42 Dinh D˜
Dyadic version: decomposition in frequency domain L 2 ( T s ) is the space of periodic functions on the torus T s := [0 , 1] s � equipped with the inner product ( f , g ) := T s f ( x ) g ( x ) d x . Let e k ( x ) := � s j =1 e 2 π i k j x j . For m ∈ Z s + and f ∈ L 2 ( T s ), define the operator: � m := { k ∈ Z s : ⌊ 2 m i − 1 ⌋ ≤ | k i | < 2 m i } , k ∈ � m ˆ δ m ( f ) := � f ( k ) e k , where ˆ f ( k ) is the k th Fourier coefficient. Based on Parseval’s identity � f � 2 + � δ m ( f ) � 2 2 = � 2 , we define the m ∈ Z s space H α mix of mixed smoothness α : s � f � 2 � 2 2 α | m | 1 � δ m ( f ) � 2 � mix := 2 < ∞ , | m | 1 := m j . H α m ∈ Z s j =0 + ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 9 / 42 Dinh D˜
Dyadic version: Fourier HC approximation Step HCs are formed from dyadic boxes � m : � � � � m : m ∈ Z s G step ( s , n ) := + , | m | 1 ≤ n . V s ( n ) – the trigonometric polynomials with frequencies in G step ( s , n ). Linear Fourier operator: � ˆ S n ( f ) := f ( k ) e k . k ∈ G step ( s , n ) Let U α mix be the unit ball in H α mix . For n ∈ N , � f − S n ( f ) � 2 ≤ 2 − n ; sup g ∈ V s ( n ) � f − g � 2 ≤ inf sup f ∈ U α f ∈ U α mix mix We have dim V s ( n ) = | G step ( s , n ) | . ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 10 / 42 Dinh D˜
Dyadic version: n ε and the cardinality of HCs Estimation of n ε is reduced to estimation of | G step ( s , n ) | for ε = 2 − n : | G step ( s , n ) | − 1 ≤ n ε ( U α mix , L 2 ( T s )) ≤ | G step ( s , n ) | , For any n ∈ Z + , � n + s − 1 � � n + s − 1 � 2 n ≤ | G step ( s , n ) | ≤ 2 n +1 . s − 1 s − 1 Theorem (DD&Ullrich 2014) Let α > 0 . Then we have for any 0 < ε ≤ 2 − α s , 2[ α ( s − 1)] − ( s − 1) ≤ n ε ( U α mix , L 2 ( T s )) 1 � α ( s − 1) � − ( s − 1) ε − 1 /α | log ε | s − 1 ≤ 4 . 2 e The ratio decays exponentially fast in s . ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 11 / 42 Dinh D˜
Dyadic version [DD&Ullrich 2013] Estimates in this manner have been proven also for n ε ( U α mix , H γ ( T s )) in energy norm of Sobolev space H γ ( T s ). In the dyadic version, we can prove lower and upper bounds for n ε ( U α mix , L 2 ( T s )) only for very small ε ≤ 2 − α s . The reason: The step HC approximation for the class U α mix involve a whole dyadic block � ˆ δ k ( f ) := f ( m ) e m m ∈ � k with the cardinality | � k | ≥ 2 s . Let us consider another model of mixed smoothness: Korobov-type mixed smoothness. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 12 / 42 Dinh D˜
A modification of Korobov space K r s For r > 0 and k ∈ Z s , define the scalar λ ( k ) by s � λ ( k ) := λ ( k j ) , λ ( k j ) := (1 + | k j | ) , j =1 Korobov function: λ ( k ) − r e k . � κ r s := k ∈ Z s Korobov space K r s : K r s := { f : f = κ r s ∗ g , g ∈ L 2 ( T s ) } with the norm � f � K r := � g � 2 . s ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 13 / 42 Dinh D˜
Hyperbolic crosses for K r s The symmetric continuous HC: s � � k ∈ Z s : � G ( s , T ) := ( | k i | + 1) ≤ T . i =1 U r s is the unit ball in K r s . Using Fourier approximation by trigonometric polynomials with frequencies in HC G ( s , T ) we have | G ( s , T ) | − 1 ≤ n ε ( U r s , L 2 ( T s )) ≤ | G ( s , T ) | , T = ε − 1 / r . ⇒ Estimation of n ε ( U r s , L 2 ( T s )) is reduced to estimation of | G ( s , T ) | . ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 14 / 42 Dinh D˜
New estimates for the cardinality of HCs Theorem (Chernov&DD 2014) For T ≥ 1 , 2 s T (ln T + s ln 2) s | G ( s , T ) | < � , � ( s − 1)! ln T + s ln 2 + s − 1 and for T > (3 / 2) s , 2 s T (ln T − s ln(3 / 2)) s | G ( s , T ) | > ( s − 1)! (ln T − s ln(3 / 2) + s ) For ε > 0, s , L 2 ( T s )) ≤ | G ( s , T ) | , T = ε − 1 / r . | G ( s , T ) | − 1 ≤ n ε ( U r ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 15 / 42 Dinh D˜
New bounds for n ε ( U r s , L 2 ( T s )) ⇒ Theorem (Chernov&DD 2014) Let r > 0 , s ≥ 2 . Then we have for every ε ∈ (0 , 1] , 2 s ε − 1 / r (ln ε − 1 / r + s ln 2) s n ε ( U r s , L 2 ( T s )) ≤ � , ln ε − 1 / r + s ln 2 + s − 1 � ( s − 1)! and for every ε ∈ (0 , [3 / 2] − sr ) , 2 s ε − 1 / r (ln ε − 1 / r − s ln(3 / 2)) s n ε ( U r s , L 2 ( T s )) ≥ ( s − 1)! (ln ε − 1 / r − s ln(3 / 2) + s ) − 1 In traditional estimations, ε − 1 / r | log ε | ( s − 1) / r is a priori split from constants which are a function of dimension parameter s . ⇒ Any high-dimensional estimate based on them leads to a rougher bound. ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 16 / 42 Dinh D˜
Related results [K¨ uhn, Sickel & Ullrich 2014] have established upper and lower bounds explicit in s for large n and small n (preasymptotics), for the approximation number a n ( I s : H α mix → L 2 ( T s )) (given in the talk by Winfried Sickel yesterday). ung (VNU, Hanoi) Hyperbolic cross October 2, 2014 17 / 42 Dinh D˜
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