Preasymptotic estimates for approximation of multivariate Sobolev - PowerPoint PPT Presentation

Preasymptotic estimates for approximation of multivariate Sobolev functions Thomas K¨ uhn Universit¨ at Leipzig, Germany ICERM Semester Program ”High-dimensional Approximation” Brown University, Providence, Rhode Island IBC Workshop – 16 September 2014 joint work with S. Mayer (Bonn), W. Sickel (Jena) and T. Ullrich (Bonn) Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 1 / 18

Approximation numbers Approximation numbers of bounded linear operators T : X → Y between two Banach spaces a n ( T : X → Y ) := inf {� T − A � : rank A < n } Interpretation in IBC and Numerical Analysis Every operator A : X → Y of finite rank k can be written as k � Ax = L j ( x ) y j for all x ∈ X j =1 with linear functionals L j ∈ X ∗ and vectors y j ∈ Y . � A is a linear algorithm using k arbitrary linear informations � T − A � = sup � Tx − Ax � = worst-case error of A . � x �≤ 1 Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 2 / 18

For compact operators between Hilbert spaces one has a n ( T ) = s n ( T ) = n -th singular number of T . General problem in functional analysis or approximation theory: Find the asymptotic behaviour of a n ( T ) as n → ∞ . Typical results are of the form c n − α ≤ a n ( T ) ≤ C n − α for all (or for large) n ∈ N , with certain (often unspecified) constants. More relevant for practical issues, for instance in – tractability problems in IBC – error analysis of numerical algorithms is the preasymptotic behaviour of a n ( T ) i.e. estimates for small n Our aims. It is well known that a n ( I d : H s ( T d ) → L 2 ( T d )) ∼ n − s / d . We will give – explicit constants, in particular asymptotic constants – sharp preasymptotic estimates in the range 2 ≤ n ≤ 2 d with special emphasis on the dependence on the dimension d and on the chosen norm. Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 3 / 18

Isotropic Sobolev spaces – integer smoothness T d is the d -dimensional torus = [0 , 2 π ] d with identification of opposite points, equipped with the normalized Lebesgue measure (2 π ) − d dx . Sobolev spaces on T d of integer smoothness m ∈ N H m ( T d ) consists of all f ∈ L 2 ( T d ) such that D α f ∈ L 2 ( T d ) for all multi-indices α ∈ N d 0 with | α | ≤ m . Natural norm (all partial derivatives) � D α f | L 2 ( T d ) � 2 � 1 / 2 � � � f | H m ( T d ) � := | α |≤ m Modified natural norm (only highest derivatives in each coordinate) d � ∂ m f 2 � 1 / 2 � � � � f | H m ( T d ) � ∗ := � � f | L 2 ( T d ) � 2 + � � L 2 ( T d ) � � � ∂ x m � j j =1 Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 4 / 18

Fourier coefficients – equivalent norms Fourier coefficients of f ∈ L 2 ( T d ) 1 � T d f ( x ) e − ikx dx k ∈ Z d c k ( f ) := , (2 π ) d Parseval’s identity and c k ( D α f ) = ( ik ) α c k ( f ) norms in H m ( T d ) can be expressed in terms of c k ( f ) = ⇒ For the natural norm one has equivalence 1 / 2   d | k j | 2 � m � � f | H m ( T d ) � ∼  � � | c k ( f ) | 2 1 +  k ∈ Z d j =1 with equivalence constants independent on d . For the modified natural norm one has even equality 1 / 2   d � f | H m ( T d ) � ∗ = � | k j | 2 m �  � � | c k ( f ) | 2 1 + .  j =1 k ∈ Z d Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 5 / 18

Fractional smoothness s > 0 Idea. Replace, in der Fourier norm, m ∈ N with a real number s > 0. � all norms are weighted ℓ 2 -sums of Fourier coefficients H s , p ( T d ) consists of all f ∈ L 2 ( T d ) such that w s , p ( k ) 2 | c k ( f ) | 2 � 1 / 2 � � � f | H s , p ( T d ) � := < ∞ , k ∈ Z d where the weights are � (1 + | k 1 | p + . . . + | k d | p ) s / p , 0 < p < ∞ w s , p ( k ) = max(1 , | k 1 | , . . . , | k d | ) s , p = ∞ For fixed s > 0 and d ∈ N , all these norms are equivalent. The equivalence constants depend heavily on d , but clearly all spaces H s , p ( T d ), 0 < p ≤ ∞ , coincide as vector spaces. p = 2 ⇐ ⇒ natural norm p = 2 s ⇐ ⇒ modified natural norm Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 6 / 18

New results Existence and computation of the limits n →∞ n s / d a n ( I d : H s , p ( T d ) → L 2 ( T d )) lim for all s > 0, d ∈ N and 0 < p ≤ ∞ Asymptotic behaviour of the constants as d → ∞ Explicit two-sided estimates of a n for large n / small n Similar results for – approximation in the sup-norm – spaces of dominating mixed smoothness − → talk by Winfried Sickel Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 7 / 18

Reduction to sequence spaces Commutative diagram I d H s , p ( T d ) L 2 ( T d ) A B D ℓ 2 ( Z d ) ℓ 2 ( Z d ) k ∈ Z d ξ k e ikx B ξ := � with Af := ( w s , p ( k ) c k ( f )) k ∈ Z d , and a diagonal operator D ( ξ k ) := ( ξ k / w s , p ( k )) A and B are unitary operators = ⇒ a n ( I d ) = a n ( D ) = s n ( D ) Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 8 / 18

Diagonal operators and combinatorics Let ( σ n ) n ∈ N be the non-increasing rearrangement of (1 / w s , p ( k )) k ∈ Z d . With this piecewise constant sequence we have a n ( I d : H s , p ( T d ) → L 2 ( T d )) = s n ( D : ℓ 2 ( Z d ) → ℓ 2 ( Z d )) = σ n . The ”sequence” ( w s , p ( k )) k ∈ Z d attains all values (1 + r p ) s / p , r ∈ N , in fact each of them at least 2 d times, for k = ± re 1 , ± re 2 . . . , ± re d . Lemma Let r ∈ N and n = # { k ∈ Z d : � d j =1 | k j | p ≤ r p } . Then a n ( I d : H s , p ( T d ) → L 2 ( T d )) = σ n = (1 + r p ) − s / p . In principle, this gives a n ( I d ) for sufficiently many n ′ s , but to compute these cardinalities exactly is impossible. However, good estimates will be enough. Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 9 / 18

Grid, covering and entropy numbers (Quasi-)norms on R d  j =1 | x j | p � 1 / p �� d , 0 < p < ∞  � x � p := max 1 ≤ j ≤ d | x j | , p = ∞  p := { x ∈ R d : � x � p ≤ 1 } with (closed) unit balls B d Let A ⊆ R d . G ( A ) := #( A ∩ Z d ) Grid number Covering numbers N ε ( A ) := minimal n ∈ N such that there are x 1 , . . . , x n ∈ R d with A ⊆ � n i =1 ( x i + ε B d ∞ ) Entropy numbers ε n ( A ) := { inf ε > 0 : N ε ( A ) ≤ n } Here, covering and entropy numbers are always w.r.t. the sup-norm. Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 10 / 18

Grid numbers vs. covering numbers A subset A ⊂ R d is called solid, if ( x j ) ∈ A and | y j | ≤ | x j | implies ( y j ) ∈ A . Examples: rB d p for all r > 0 and 0 < p ≤ ∞ Lemma Let A ⊆ R d be a solid subset and 0 < ε < 1 / 2 . Then N 1 ( A ) ≤ G ( A ) ≤ N ε ( A ) . Proof. Given x ∈ R d , define k ( x ) = ( k j ) ∈ Z d by k j = sign x j · [ | x j | ]. Then the set { k ( x ) : x ∈ A } is a 1-net for A and, since A is solid, it is equal to A ∩ Z d . This proves N 1 ( A ) ≤ G ( A ). The inequality G ( A ) ≤ N ε ( A ) follows from the fact that each ball of radius ε < 1 / 2 in ℓ d ∞ is a cube of side length 2 ε < 1, whence it contains at most one element of Z d . Therefore every covering of A by ε -balls in ℓ d ∞ must have at least G ( A ) elements. Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 11 / 18

Covering numbers are homogeneous, in the sense of N ε ( A ) = N λε ( λ A ) for all λ, ε > 0 . This is an advantage over grid numbers! For large ℓ p -balls the previous lemma can be improved. Lemma Let 0 < p < ∞ , d ∈ N and r > d 1 / p / 2 . Set ˜ p = min(1 , p ) and p − d ˜ ℓ = ℓ ( r , p , d ) = ( r ˜ p / p / 2 ˜ p ) 1 / ˜ p p + d ˜ L = L ( r , p , d ) = ( r ˜ p / p / 2 ˜ p ) 1 / ˜ p N 1 / 2 ( ℓ B d p ) ≤ G ( r B d p ) ≤ N 1 / 2 ( L B d Then p ) Prooof: By triangle inequality and volume arguments. Note that ℓ ( r , p , d ) ≍ r ≍ L ( r , p , d ) as r → ∞ . Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 12 / 18

Appoximation in H s , p ( T d ) via entropy of ℓ p -balls Recall that we have already shown a n ( I d : H s , p ( T d ) → L 2 ( T d )) = (1 + r p ) − s / p for r ∈ N and n = # { k ∈ Z d : � k � p ≤ r } = G ( rB d p ). Together with the two lemmata this implies the following result. Theorem Let s > 0 , 0 < p ≤ ∞ and d ∈ N. Then, for all n ∈ N , one has � s ≤ a n ( I d : H s , p ( T d ) → L 2 ( T d )) ≤ � s 2 − 1 − 1 / p ε n ( B d 4 ε n ( B d � � p ) p ) n − 1 / d ≤ ε n ( B d ∞ ) ≤ 4 n − 1 / d , For p = ∞ we have for all n ∈ N , For 0 < p < ∞ the entropy numbers ε n ( B d p ) = ε n ( id : ℓ d p → ℓ d ∞ ) are also completely understood. Thomas K¨ uhn (Leipzig) Approximation of Sobolev functions IBC Workshop, ICERM 2014 13 / 18