DSpace at VNU: N-Widths and epsilon-Dimensions for High-Dimensional Approximations

DSpace at VNU: N-Widths and epsilon-Dimensions for High-Dimensional Approximations tài liệu, giáo án, bài giảng , luận v...

Found Comput Math DOI 10.1007/s10208-013-9149-9 N -Widths and ε-Dimensions for High-Dimensional Approximations ˜ · Tino Ullrich Dinh Dung Received: 27 February 2012 / Revised: 26 November 2012 / Accepted: 25 February 2013 © SFoCM 2013 Abstract In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths dn (W, H γ ), and ε-dimensions nε (W, H γ ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large We are interested in finding the accurate dependence of dn (W, H γ ) and nε (W, H γ ) as a function of two variables n, d and ε, d, respectively Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity However, the parameter d may seriously affect this rate when d is large We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H γ Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths dn (W, H γ ) and ε-dimensions nε (W, H γ ) Some of the received results imply that the curse of dimensionality can be broken in some relevant situations Dedicated to the memory of Professor S.M Nikol’skij Communicated by Wolfgang Dahmen D D˜ung ( ) Information Technology Institute, Vietnam National University, Hanoi, 144 Xuan Thuy, Hanoi, Vietnam e-mail: dinhzung@gmail.com T Ullrich Hausdorff-Center for Mathematics and Institute for Numerical Simulation, 53115 Bonn, Germany Found Comput Math Keywords High-dimensional approximation · Trigonometric hyperbolic cross space · Kolmogorov n-widths · ε-dimensions · Sobolev space · Function classes with anisotropic smoothness Mathematics Subject Classification (2010) 42A10 · 41A25 · 41A63 Introduction In recent decades, there has been increasing interest in solving problems that involve functions depending on a large number d of variables These problems arise from many applications in mathematical finance, chemistry, physics, especially quantum mechanics, and meteorology It is not surprising that these problems can almost never be solved analytically such that one is interested in a proper framework and efficient numerical methods for an approximate treatment Classical methods suffer the “curse of dimensionality” coined by Bellmann [2] In fact, the computation time typically grows exponentially in d, and the problems become intractable already for mild dimensions d without further assumptions A classical model, widely studied in literature, is to impose certain smoothness conditions on the function to be approximated; in particular, it is assumed that mixed derivatives are bounded This is the typical situation for which “hyperbolic crosses” are made for Trigonometric polynomials with frequencies in hyperbolic crosses have been widely used for approximating functions with a bounded mixed derivative or difference These classical trigonometric hyperbolic crosses date back to Babenko [1] Let us also mention “sparse grids” in this context which can be seen as the counterpart of hyperbolic crosses on the spatial domain Sparse grids are discrete point sets consisting of significantly fewer points than a full tensor product grid First considered by Smolyak [37] they turned out to be suitable for sampling recovery of functions and numerical integration For further sources on hyperbolic crosses and sparse grids in this classical context we refer to [12–14, 34, 38, 42] and the references therein Later on, these terminologies were extended to approximations by wavelets [8, 35], to B-splines [15, 36], and even to algebraic polynomials where frequencies are replaced by dyadic scales or the degree of algebraic polynomials [6, 7] Hyperbolic cross and sparse grid techniques have applications in quantum mechanics and PDEs [20, 45–47], finance [18], numerical solution of stochastic PDEs [6, 7, 31, 32], and data mining [17] to mention just a few (see also the surveys [4] and [19] and the references therein) In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths dn (W, H γ ), and ε-dimensions nε (W, H γ ) of d-variate function classes W with anisotropic smoothness properties where d may be large The approximation error is measured in an isotropic Sobolev space H γ , which includes the L2 -metric as a special case We are interested in finding the accurate dependence of dn (W, H γ ) and nε (W, H γ ) as a function of two variables n, d and ε, d, respectively Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity However, the parameter d may seriously affect this rate when d is large Found Comput Math Recall the notion of the Kolmogorov n-widths [23] and linear n-widths introduced by Tikhomirov [39] If X is a normed space and W a subset in X then the Kolmogorov n-width dn (W, X) is given by dn (W, X) := inf sup inf f − g Ln f ∈W g∈Ln X, where the outer inf is taken over all linear manifolds Ln in X of dimension at most n A different worst-case setting is represented by the linear n-width λn (W, X) given by λn (W, X) := inf sup f − Λn (f ) Λn f ∈W X where the inf is taken over all linear operators Λn in X with rank at most n It represents a characterization of the best linear approximation error There is a vast amount of literature on optimal linear approximations and the related Kolmogorov and linear n-widths [30, 40], especially for d-variate function classes [38] In this paper, we are interested in measuring the approximation error in H γ , therefore we can assume X to be a Hilbert space H In this case both concepts coincide, i.e., dn (W, H ) = λn (W, H ) holds true Indeed, orthogonal projections onto a finite-dimensional space in H give the best approximation by its elements Hence, it is sufficient to investigate linear approximations in H γ and the optimality of the approximation in terms of dn (W, H γ ) Let us recall some classical results in this direction For the unit balls U β and U α1 of the periodic d-variate isotropic Sobolev space H β , β > 0, and the space H α1 with mixed smoothness α > 0, the following well-known estimates hold true Note that we have the coincidences H β = H 0,β and H α1 = H α,0 with respect to (2.6) below We have A(β, d)n−β/d ≤ dn U β , L2 ≤ A (β, d)n−β/d , (1.1) and B(α, d)n−α (log n)α(d−1) ≤ dn U α1 , L2 ≤ B (α, d)n−α (log n)α(d−1) (1.2) Here, A(β, d), A (β, d), B(α, d), B (α, d) denote certain constants which are usually not computed explicitly The inequalities (1.1) are a direct generalization of the first result on n-widths proved by Kolmogorov [23] (see also [24, 186–189]) where the exact values of n-widths were obtained for the univariate case The inequalities (1.2) were proved by Babenko [1] already in 1960, where a linear approximation on hyperbolic cross spaces of trigonometric polynomial is used These estimates are quite satisfactory if d, the number of variables, is small In computational mathematics, the so-called ε-dimension nε = nε (W, H ) is used to quantify the problem’s complexity In our setting it is defined as the inverse of dn (W, H ) In fact, the quantity nε (W, H ) is the minimal number nε such that the approximation of W by a suitably chosen nε -dimensional subspace L in H (measured in terms of Kolmogorov n-widths) yields the approximation error ≤ ε (see [10, 11, 16]) Found Comput Math We provide upper and lower bounds of this quantity together with the corresponding n-widths in this paper The quantity nε represents a special case of the information complexity which is defined as the minimal number n(ε, d) of information needed to solve the d-variate problem within error ε (see [26, 4.1.4]) It is the key to study tractability of various multivariate problems We refer the reader to the monographs [26, 29] for surveys and further references in this direction In fact, in high-dimensional settings, i.e., if d is large, it turns out that the smoothness of the isotropic Sobolev class U β is not suitable In (1.1) the curse of dimensionality occurs since here nε ≥ C(β, d)ε −d/β However, the class U α1 is more appropriate for highdimensional problems [4] since we have nε = O(ε −1/α | log ε|d−1 ) In this paper, we extend and refine existing estimates In particular, we give the lower and upper bounds for constants B(α, d), B (α, d) in (1.2) with regards to α, d We are especially concerned with measuring the approximation error in the isotropic smoothness space H γ To motivate this issue let us consider a Galerkin method for approximating the solution of a general elliptic variational problem Let a : H γ × H γ → R be a bilinear symmetric form and f ∈ H −γ , where H γ = H γ (Td ) and Td is the d-dimensional torus Assume that a(u, v) ≤ λ u Hγ v Hγ and a(u, u) ≥ μ u Hγ Then, a(·, ·) generates the so called energy norm equivalent to the norm of H γ Consider the problem of finding an element u ∈ H γ such that a(u, v) = (f, v) for all v ∈ H γ (1.3) In order to get an approximate numerical solution we can consider the same problem on a finite-dimensional subspace Vh in H γ a(uh , v) = (f, v) for all v ∈ Vh (1.4) By the Lax–Milgram theorem [25], the problems (1.3) and (1.4) have unique solutions u∗ and u∗h , respectively, which by Céa’s lemma [5], satisfy the inequality u∗ − u∗h Hγ ≤ (λ/μ) inf u∗ − v v∈Vh Hγ Here a naturally arising question is how to choose optimal n-dimensional subspaces Vh and linear finite element approximation algorithms for the problem (1.4) This certainly leads to the problems of optimal linear approximation in H γ of functions from U and Kolmogorov n-widths dn (U, H γ ), where U is a class of functions u having in some sense more regularity than the class H γ The regularity of the class U (in high-dimensional settings) is usually measured by L2 -boundedness of mixed derivatives of higher order or other anisotropic derivatives (a mixed derivative is sometimes referred to as an anisotropic derivative) Finite element approximation spaces based on hyperbolic cross frequency domains are suitable for this framework It is well-known that the cost of approximately solving Poisson’s equation in d dimensions in the Sobolev space H is exponentially growing in d Standard finite element methods lead to a cost nε = O(ε −d ) If we know in advance that the solution belongs to a space of functions with dominating mixed first derivative, and if we use Found Comput Math hyperbolic cross spaces for finite element methods, then this requires the cost of nε ≤ C(d)ε −1 | log ε|d−1 Here and below, C(d, ) are various constants depending on d and other parameters In [3] it was shown how to get rid of the additional logarithmic term by the use of a subspace of the hyperbolic cross spaces This results in energy norm-based hyperbolic cross spaces and H -norm approximation of functions with dominating mixed second derivative Then the total cost for the solution of Poisson’s equation is of the order nε ≤ C(d)ε −1 , see also [41] for a generalization In [21, 22] Griebel and Knapek generalized the construction of [3] to the elliptic variational problem (1.3) By use of tensor-product biorthogonal wavelet bases, they constructed for finite element methods so-called optimized sparse grid subspaces of lower dimension than the standard full-grid spaces These subspaces preserve the approximation order of the standard full-grid spaces, provided that the solution possesses H α,β -regularity To this end, the authors measured the approximation error in the energy H γ -norm and estimated it from above by terms involving the H α,β -norm of the solution The smoothness of spaces H α,β is a “hybrid” of isotropic smoothness β and mixed smoothness α [22, Definition 2.1] It turns out that the necessary dimension nε of the optimized sparse grid space for the approximation with accuracy ε does not exceed C(d, α, γ , β) ε −(α+β−γ ) if α > γ − β > Due to the construction, the optimized sparse grid spaces can be considered as an extension of hyperbolic cross spaces The curse of dimensionality is not sufficiently clarified unless “constants” such as B(α, d), B (α, d) in (1.2) for dn or C(d) and C(d, α, γ , β) in the above inequalities for nε are not completely determined We are interested, so far possible, in explicitly determining these constants The aim of the present paper is to compute dn (U, H γ ) α,β and nε (U, H γ ) where U is the unit ball U α,β in H α,β or its subsets U∗ and the α,β α,β below characterized class Uν for α > γ − β ≥ The function class U∗ is the set of all functions f ∈ U α,β such that fˆ(s) = whenever dj =0 sj = In [21, 22], the α,β authors considered a counterpart of the class U∗ defined via a biorthogonal wavelet decomposition, see Sect in the present paper They investigated the approximation of functions from this class by optimized sparse grid spaces We complement their investigations by establishing sharp lower and upper bounds in an explicit form of all relevant components depending on α, β, γ and d, n, ν This includes the case (1.2) and its modifications when α > γ = β = In contrast to [21, 22] we also obtain lower bounds and prove therefore that trigonometric hyperbolic cross approximations are optimal in terms of Kolmogorov n-widths For the case α > γ − β > 0, we prove that the hyperbolic cross approximation spaces from [21, 22] are optimal for α,β dn (U∗ , H γ ) Moreover, the modifications given in the present paper are optimal for α,β dn (U α,β , H γ ) and dn (Uν , H γ ) In the case α > γ − β = 0, we prove that classical hyperbolic cross spaces (see, e.g., [38]) and their modifications in this paper are α,β α,β optimal for dn (U∗ , H γ ), dn (U α,β , H γ ) and dn (Uν , H γ ) It seems that smoothness is not enough for ridding the curse of dimensionality However, by imposing some additional restrictions on functions in U α,β this is posα,β sible In fact, Uν is the set of all functions f ∈ U α,β actually depending on at most ν (unknown) variables by formally being a d-variate function For this function class, the curse of dimensionality is broken For instance, in Theorem 4.7 in Sect 4, for the Found Comput Math case α > γ − β > 0, we obtain the relations νδ + ρ+3δ d ν(2ρ/δ − 1) δν n−δ ≤ dn Uνα,β , H γ ≤ α δ δ 22ρ+δ ν δ + d 2ρ/δ − δν n−δ , if n ≥ αδ ν2ν(2α/δ+1) (1 + d/(2ρ/δ − 1))ν , where δ := α + β − γ and ρ := γ − β A corresponding result for the ε-dimension nε (see Theorem 4.8 in Sect 4) states α,β that the number nε (Uν , H γ ) is bounded polynomially in d and ε −1 from above As a consequence, according to [26, (2.3)], we find that the problem is polynomially tractable In addition, the case γ = β, which contains the classical situation with Uνα1 instead of U α1 in (1.2), gives as well the polynomial tractability, see Theorems 4.10 and 4.11 Let us mention the relation to the results of Novak and Wo´zniakowski on weighted tensor product problems with finite order weights [26, 5.3] Their approach also limits the number ν of active variables in a function via a finite order weight sequence (of order ν) However, since in this paper in most cases neither the spaces H α,β of the functions to be approximated, nor the space H γ , where the approximation error is measured, are tensor products of univariate spaces [35], our results are not included in [26, Theorem 5.8] Apart from that, totally different approaches for the approximation of functions depending on just a few variables in high dimensions are given in [9, 44] The paper is organized as follows In Sect 2, we describe a dyadic harmonic decomposition of periodic functions from H α,β used for norming these classes suitably for high-dimensional approximations In Sect 3, we prove upper bounds for α,β α,β hyperbolic cross approximations of functions from U = U α,β , U∗ and Uν by linear methods, and for the dimensions of the corresponding approximation spaces By means of these results, we are able to estimate dn (U, H γ ) and nε (U, H γ ) from above In Sect 4, we prove the optimality of these approximations by establishing lower bounds for dn (U, H γ ) In Sect 5, we discuss the extension of our results to biorthogonal wavelets and more general decompositions Dyadic Decompositions Let N denote the natural numbers, Z the integers, Z+ = N ∪ {0} the natural numbers including zero, R the real numbers, and C the complex numbers The number d is always reserved for the number of variables of the functions under consideration Indeed, we will consider functions on Rd which are 2π -periodic in each variable, as functions defined on the d-dimensional torus Td := [−π, π]d Denote by L2 := L2 (Td ) the Hilbert space of functions on Td equipped with the inner product (f, g) := (2π)−d Td f (x)g(x) dx As usual, the norm in L2 is f := (f, f )1/2 For s ∈ Zd , let fˆ(s) := (f, es ) be the sth Fourier coefficient of f , where es (x) := ei(s,x) Found Comput Math Let S(Td ) be the space of functions on Td whose Fourier coefficients form a rapidly decreasing sequence, and S (Td ) the space of distributions which are continuous linear functionals on S(Td ) It is well-known that, if f ∈ S (Td ), then the Fourier coefficients fˆ(s), s ∈ Zd , of f form a tempered sequence (see, e.g., [33, 40]) A function in L2 can be considered as an element of S (Td ) For f ∈ S (Td ), we use the identity fˆ(s)es f= s∈Zd holding in the topology of S (Td ) Denote by [d] the set of natural numbers from to d, and by σ (x) := {i ∈ [d] : xi = 0} the support of the vector x ∈ Rd For r ∈ Rd , the rth derivative f (r) of a distribution f is defined as the distribution in S (Td ) given by the identification (is)r fˆ(s)es , f (r) := (2.1) s∈Zd0 (r) where (is)r := dj =1 (isj )rj , (ia)b := |a|b e(iπb sign a)/2 for a, b ∈ R, and Zd0 (r) := {s ∈ Zd : sj = 0, j ∈ σ (r)} Let us recall the definition of some well known function spaces with isotropic and anisotropic smoothness The isotropic Sobolev space H γ , γ ∈ R For γ ≥ 0, H γ is the subspace of functions in L2 , equipped with the norm d f := f Hγ + f (γ j) , j =0 where j := (0, , 0, 1, 0, , 0) is the j th unit vector in Rd For γ < 0, we define H γ as the L2 -dual space of H −γ The space H r of mixed smoothness r ∈ Rd is defined as the tensor product of the spaces H rj , j ∈ [d]: d H r := H rj , j =1 where H rj is the univariate Sobolev space in variable xj For a finite set A ⊂ Rd , denote by H A the normed space of all distributions f for which the following norm is finite: f HA := f Hr r∈A For α, β ∈ R, let us define the space H α,β as follows If β ≥ 0, we put H α,β := H A , where A= α1 + β j : j ∈ [d] (2.2) Found Comput Math and := (1, 1, , 1) ∈ Rd If β < 0, we define H α,β as the L2 -dual space of H −α,−β The space H α,β has been introduced in [22] Notice that H α,0 = H α1 and H 0,β = H β We will need a dyadic harmonic decomposition of distributions We define for k ∈ Z+ , Pk := s ∈ Z : 2k−1 ≤ |s| < 2k , k > 0, P0 := {0}, and for k ∈ Zd+ , d Pk := Pkj j =0 For distributions f and k ∈ Zd+ , let us introduce the following operator: fˆ(s)es δk (f ) := s∈Pk If f ∈ L2 , we have by Parseval’s identity f 2 = (2.3) δk (f ) k∈Zd+ Moreover, the space L2 can be decomposed into pairwise orthogonal subspaces Wk , k ∈ Zd+ , by L2 = Wk , k∈Zd+ with dim Wk = |Pk | = 2|k|1 , where Wk is the space of trigonometric polynomials g of the form g= cs es s∈Pk and |Q| denotes the cardinality of the set Q Put |k|1 := dj =0 kj and |k|∞ := max1≤j ≤d kj for k ∈ Zd+ Lemma 2.1 For any α, β ∈ R, we have the following norm equivalence: f 22(α|k|1 +β|k|∞ ) δk (f ) 2 H α,β k∈Zd+ Proof We need the following preliminary norms equivalence for r ∈ Rd : f 2 Hr 22(r,k) δk (f ) k∈Zd+ (2.4) Found Comput Math Indeed, for the univariate case (d = 1), by the definition f H r is the norm of the isotropic Sobolev space H γ for γ = r Consequently, by (2.3) f Hr δk (f ) Hγ k∈Z+ Observe that δk (f ) 2H γ 22γ |k|1 δk (f ) This inequality is implied from the definition (2.1) for γ ≥ 0, and from the L2 -duality of H γ for γ < Hence, we prove (2.4) for the univariate case Since in the multivariate case, H r is the tensor product of isotropic Sobolev spaces it is easy derive (2.4) from the univariate case Let us prove the lemma We first consider the case β ≥ Taking A for the definition of H α,β as in (2.2), by (2.4) we get f HA max f r∈A Hr k∈Zd+ 22(r,k) δk (f ) max r∈A k∈Zd+ ≤ 22 maxr∈A (r,k) δk (f ) (2.5) k∈Zd+ Let us decompose Zd+ into the subsets Zd+ (r), r ∈ A, such that Zd+ = Zd+ (r), Zd+ (r) ∩ Zd+ (r ) = ∅, r = r, r∈A and max r , k = (r, k), r ∈A k ∈ Zd+ (r) (Obviously, such a decomposition is easily constructed and some of the Zd+ (r) may be empty sets.) Then we have 22(r,k) δk (f ) max r∈A = max r∈A k∈Zd+ 22(r,k) δk (f ) r ∈A k∈Zd+ (r ) ≥ 22(r ,k) δk (f ) r ∈A k∈Zd+ (r ) = 22 maxr∈A (r,k) δk (f ) k∈Zd+ This and (2.5) show that f 2 HA 22 maxr∈A (r,k) δk (f ) k∈Zd+ Found Comput Math By a direct computation one can verify that maxr∈A (r, k) = α|k|1 + β|k|∞ This proves the lemma for the case β ≥ If β < 0, by the definition, the L2 -duality and (2.3) f 2−2(−α|k|1 −β|k|∞ ) δk (f ) H α,β k∈Zd+ 22(α|k|1 +β|k|∞ ) δk (f ) = k∈Zd+ On the basis of Lemma 2.1, let us redefine the space H α,β , α, β ∈ R as the space of distributions f on Td for which the following norm is finite: f H α,β 22(α|k|1 +β|k|∞ ) δk (f ) := (2.6) k∈Zd+ With this definition we have H 0,0 = L2 We put H 0,β = H β and H α,0 = H α1 as in the traditional definitions Denote by U α,β the unit ball in H α,β Regarding (2.6) it is worth mentioning the following important thing In traditional approximation problems where the parameter d is small and fixed, the convergence rates with respect to different equivalent norms only differ by moderate constants The picture completely changes for high-dimensional approximation problems where we stress the importance of finding an accurate dependence of the convergence rate on the number d of variables and the dimension n of the approximation space In fact, it essentially depends on the choice of the norm of a function class (i.e., its unit ball) and a norm measuring the approximation error In some high-dimensional problems it is more convenient to define the function spaces based on a mixed dyadic decomposition in terms of (2.6) The problem itself changes if we use another characterization for H α,β and H γ instead of (2.6) For instance, one can define the following equivalent norm of H α,β in terms of the Fourier coefficients by using an ANOVA-type decomposition: f H˜ α,β fˆ(s) + := s∈Zd |sj |2(α+β) e⊂[d],e=∅ s∈Zd,e j ∈e |s |2α fˆ(s) , ∈e, =j where Zd,e := {s ∈ Zd : σ (s) = e} Note that H α,β and H˜ α,β coincide as function spaces However, if d is large the unit balls with respect to the norms of these spaces differ significantly α,β α,β α,β We define the subsets U∗ and Uν , ≤ ν ≤ d − 1, in U α,β as follows U∗ is the subset in U α,β of all f such that d δk (f ) = if kj = j =0 α,β The subset Uν is the set of all f ∈ U α,β such that δk (f ) = if σ (k) > ν Found Comput Math 4.2 The Case α > γ − β > Lemma 4.3 Let < t ≤ 1/2 and k, n be integers such that ≤ k ≤ n/2 Then k t s=0 Proof Since (1 − t)/t ≥ 1, n s 1−t t s n s s 1−t t n s n = n n−s ≤ n n−s ≤ and ≤ k ≤ n/2, we have s 1−t t , s = 0, , k Hence, k s=0 n s k s 1−t t n n−s ≤ s=0 s 1−t t , and consequently, k n s tn s=0 1−t t s ≤ tn n s=0 n s 1−t t s = Lemma 4.4 Let < θ ≤ Then for any natural numbers d and n satisfying the condition θ −1 n+ , (4.1) d≤ 2θ − θ we have the inequality 2|k|1 ≥ 2−1/(θ−1) d2d−2 − 2−1/(θ−1) −d n k∈Ind Proof Consider the subsets Ind (j ), j ∈ [d], in Ind defined by Ind (j ) := k ∈ Ind : |k|∞ = kj , |k|1 ≥ 2(θ − 1) n + d − + 2/θ 2θ − We prove that Ind (j ) ∩ Ind (j ) = ∅ for j = j Fix j ∈ [d] and let k be an arbitrary element in Ind (j ) Then by the definitions we have kj ≥ θ |k|1 − (θ − 1)n − θ (d − 1) 2θ (θ − 1) n + θ (d − 1) + − (θ − 1)n − θ (d − 1) 2θ − θ −1 = n + 2θ − ≥ (4.2) Found Comput Math On the other hand, θ |k|1 − kj + (θ − 1)kj = θ |k|1 − |k|∞ ≤ (θ − 1)n + θ (d − 1) Hence, θ −1 n+2 2θ − |k|1 − kj ≤ θ −1 (θ − 1)n + θ (d − 1) − θ −1 (θ − 1) = θ −1 n + d − + 2/θ 2θ − Take an arbitrary j ∈ [d] such that j = j Then, since ki ≥ for any i ∈ [d], and θ > 1, from the last inequality and (4.2) we get kj ≤ |k|1 − kj − (d − 2) θ −1 n + d − + 2/θ − (d − 2) 2θ − θ −1 = n + 2/θ − 2θ − ≤ < kj This proves that Ind (j ) ∩ Ind (j ) = ∅ for j = j Therefore, we have the inequality 2|k|1 ≥ k∈Ind d 2|k|1 j =1 k∈Ind (j ) Due to the symmetry, all the sums k∈Ind (j ) 2|k|1 , j ∈ [d] are equal Thus, in order to prove the lemma it is enough to show, for instance, that 2|k|1 ≥ 2−1/(θ−1) 2d−2 − 2−1/(θ−1) −d n (4.3) k∈Ind (d) Observe that for k ∈ Ind (d), |k|1 can take the values (2(θ − 1)/(2θ − 1))n + d − + 2/θ , , n + d − Put |k|1 = n + d − − m for m = 0, 1, , M, where M := n + d − − (2(θ − 1)/(2θ − 1))n + d − + 2/θ Fix a nonnegative integer m with ≤ m ≤ M Assume that |k|1 = n + d − − m Then clearly, k ∈ Ind (d) if and only if kd ≥ n − θ m It is easy to see that the number of all such k ∈ Ind (d) is not smaller than (n + d − − m) − n − θ m d −1 We have = d − + (θ − 1)m d −1 Found Comput Math M 2|k|1 ≥ k∈Ind (d) 2n+d−1−m m=0 M = 2n+d−1 d − + (θ − 1)m d −1 d − + (θ − 1)m d −1 2−m m=0 =: 2n+d−1 A(n) (4.4) Put ε := (θ − 1)−1 and N := ((θ − 1)M) Replacing m by τ := m/ε in A(n), we obtain d −1+ τ d −1 2−ετ A(n) = τ ∈ε −1 {0,1, ,M} 2−ε( ≥ τ +1) τ ∈ε −1 {0,1, ,M} d −1+ τ d −1 Since < θ ≤ 2, the step length of τ is 1/ε ≤ Therefore, we have A(n) ≥ 2−ε N 2−εs s=0 d −1+s d −1 (4.5) By (3.4) we have N B(n) := s=0 2−εs d −1+s d −1 = (1 − t)−d − t d+N +1 d−1 s=0 N +d s 1−t t s t=2−ε By the assumptions of the lemma < 2−ε ≤ 1/2 and d − ≤ (d + N + 1)/2 Applying Lemma 4.3 gives B(n) ≥ (1 − t)−d t=2−ε = 1 − 2−1/(θ−1) −d (4.6) Combining (4.4)–(4.6) proves (4.3) Remark 4.5 By using weaker assumptions we can prove the following slightly worse lower bound compared to Lemma 4.4 If < θ ≤ 2, then for any natural numbers d and n, we have the inequality 2|k|1 ≥ 2−1/(θ−1) 2d−2 − 2−1/(θ−1) −d n k∈Ind Lemma 4.6 Let α, β, γ ∈ R satisfy the conditions 2(γ − β) ≥ α > γ − β > and ≤ ν ≤ d − Then we have Found Comput Math (i) for any ξ ≥ (2α + δ)(d − 1), d 1 dim V d (ξ ) ≥ 2−ρ/δ d + ρ/δ −1 2ξ/δ , (ii) for any ξ ≥ (2α + δ)(d − 1), dim V∗d (ξ ) ≥ d 2ρ/δ − −d ξ/δ , (iii) for any ξ ≥ (2α + δ)(ν − 1), ν d dim Vνd (ξ ) ≥ 2−ρ/δ ν + ν(2ρ/δ − 1) 2ξ/δ Proof We first prove the second inequality in the lemma Put θ := α/ρ and n := (ξ − α(d − 1))/δ We have J∗d (ξ ) ⊃ Ind Since ξ ≥ (2α + δ)(d − 1), Condition (4.1) is satisfied Hence, by Lemma 4.4, 2|k|1 ≥ dim V∗d (ξ ) = k∈J∗d (ξ ) 2|k|1 k∈Ind ≥ 2−1/(θ−1) d2d−2 − 2−1/(θ−1) −d n ≥ 2−1/(θ−1) d2d−2 − 2−1/(θ−1) −d (ξ −α(d−1))/δ−1 ≥ d 2ρ/δ − −d ξ/δ 2 For proving (i) we start in a way similar to the proof of Lemma 3.5 (see the first three equations in (3.6)) and conclude by using the previous relation d dim V d (ξ ) ≥ k=1 d k 2ρ/δ − k = 2ξ/δ d k=1 −k ξ/δ d! 2ρ/δ − (d − k)!(k − 1)! = 2ξ/δ 2ρ/δ − −1 d−1 d k=0 d −1 k ρ/δ −1 − d + 1/ 2ρ/δ − d = 2−ρ/δ d + 1/ 2ρ/δ − 2ξ/δ = −k 2ρ/δ − −k d−1 ξ/δ (4.7) Found Comput Math Finally, we prove (iii) with a similar computation as done in (4.7) Indeed, we obtain ν dim Vνd (ξ ) ≥ k=1 d k 2ρ/δ − k = 2ξ/δ ≥ = ν ν d!(ν − k)! ρ/δ −1 k k (d − k)!ν! k=1 ξ/δ −k ξ/δ ν k=1 k ν d k k ν 1d ν 2ρ/δ − 4ν 2ρ/δ − −1 ξ/δ ν−1 k=0 = ρ/δ −1 = ρ/δ + d/ν − ≥ −ρ/δ d ν 1+ ν(2ρ/δ − 1) −1 d 1+ −1 −k ν −1 k k d ν 2ρ/δ − −k ν−1 d 2ξ/δ ν(2ρ/δ − 1) d 1+ −k ν d 2ξ/δ ν(2ρ/δ − 1) ν 2ξ/δ Theorem 4.7 Let α, β, γ ∈ R satisfy the conditions 2(γ − β) ≥ α > γ − β > Then, we have (i) for any integer n ≥ αδ d2d(2α/δ+1) (1 + 1/(2ρ/δ − 1))d , dδ + ρ+3δ 2ρ/δ − δd n−δ ≤ dn U α,β , H γ ≤ δ α δ 22ρ+δ d δ + 2ρ/δ − δd n−δ , (ii) for any integer n ≥ αδ d2d(2α/δ+1) (2ρ/δ − 1)−d , δ d δ 2ρ/δ −1 −δd −δ n α,β ≤ d n U∗ , H γ ≤ α δ δ 22ρ+δ d δ 2ρ/δ −1 −δd −δ n (iii) for any integer n ≥ αδ ν2ν(2α/δ+1) (1 + d/(2ρ/δ − 1))ν , νδ + ρ+3δ d ν(2ρ/δ − 1) δν n−δ ≤ dn Uνα,β , H γ ≤ α δ δ 22ρ+δ ν δ + d 2ρ/δ − δν n−δ , Found Comput Math Proof Due to Theorem 3.6, we have to prove the lower bounds in this theorem Let us prove the lower bound for dn (U α,β , H γ ) The other lower bounds can be proved in a similar way It has been shown in the proof of Theorem 3.6 that the function ϕ(ξ ) := dim V d (ξ ) in variable ξ satisfies (3.7) and (3.8) For a given n satisfying the condition in (i) of the theorem, let ξm be the number such that dim V d (ξm ) ≥ n + > dim V d (ξm−1 ) (4.8) Hence, by the corresponding restriction on n in the theorem it follows that ξ ≥ (2α + δ)(d − 1) By Lemma 4.6 and (3.8) we obtain n ≥ 2−ρ/δ d + 1/ 2ρ/δ − −ρ/δ ≥ d + 1/ 2ρ/δ − d ξm−1 /δ d ξm /δ , or, equivalently, 2−ξm ≥ 1/2ρ/δ+3 d δ + 1/ 2ρ/δ − δ δd −δ (4.9) n Consider the set B(m) := {f ∈ V d (ξ ) : f H γ ≤ 2−ξm } in H γ By Lemma 4.2 B(m) ⊂ U α,β and consequently, by Lemma 4.1 and (4.8) dn U α,β , H γ ≥ dn B(m), H γ ≥ 2−ξm The last inequalities combining with (4.9) prove the desired inequality Theorem 4.8 Let α, β, γ ∈ R satisfy the conditions 2(γ − β) ≥ α > γ − β > Then we have (i) for any ε ≤ 2−(2α+δ)(d−1) , d 1+ ρ/δ+2 2ρ/δ − d ε −1/δ ≤ nε U α,β , H γ ≤ α 2ρ/δ d + ρ/δ δ −1 d ε −1/δ , (ii) for any ε ≤ 2−(2α+δ)(d−1) , d 2ρ/δ − −d −1/δ ε α,β ≤ n ε U∗ , H γ ≤ α 2ρ/δ d 2ρ/δ − δ −d −1/δ ε , (iii) for any ε ≤ 2−(2α+δ)(ν−1) , ν 1+ ρ/δ+2 d ν(2ρ/δ − 1) ν ε −1/δ ≤ nε Uνα,β , H γ ≤ α 2ρ/δ d ν + ρ/δ δ −1 ν ε −1/δ Found Comput Math Proof Due to Theorem 3.7, we have to prove the lower bounds in this theorem Let us prove the lower bound for nε (U α,β , H γ ) The other lower bounds can be proved in a similar way For a given ε ≤ 2−(2α+δ)(d−1) , put ξ = | log ε| Consider the set B∗ (ξ ) := {f ∈ V∗d (ξ ) : f H γ ≤ 2−ξ } in the subspace V∗d (ξ ) of H γ By Lemma 4.2 α,β B∗ (ξ ) ⊂ U∗ Hence, by (4.12) and Lemma 4.1 we have dn U∗ , H γ ≥ dn B∗ (ξ ), H γ ≥ 2−ξ = ε, α,β where n := dim V∗d (ξ ) − Therefore, by the definition and Lemma 4.6(ii), α,β nε U∗ , H γ ≥ dim V∗d (ξ ) − 1 ≥ d 2ρ/δ − ≥ d 2ρ/δ − −d ξ/δ −d −1/δ ε The last inequalities combined with (4.11) concludes the proof Remark 4.9 (a) In Theorems 4.7 and 4.8 the upper and lower bounds (almost) match Therefore, we have sharp bounds for every d, large enough n and small enough ε (depending on d) (b) We have exponentially growing d-dependence in the constants in Theorems 4.7(i)–(ii) and 4.8(i)–(ii) However, this does not imply the curse of dimensionality here For instance, it is easy to see that lim d→∞ log nε(d) =0 ε(d)−1 + d where ε(d) := 2−(2α+δ)(d−1) in Theorem 4.8(i) According to the notions in [26, 29] this, surprisingly, indicates weak tractability here So far this is not a proof since Theorem 4.8 does not give the behavior of nε for “larger” ε and therefore the quantity lim ε −1 +d→∞ log nε ε −1 + d is not yet proven to be zero This issue will be completely clarified in a forthcoming paper of Ullrich and coauthors, see also [28] (c) In Theorem 3.6(ii), depending on ρ and δ, the constant (2ρ/δ − 1)−δd might decay exponentially in d However, the statement is given for n > Cα/ρ 2ρ/δ d2αd/δ (2ρ/δ − 1)−d where α > ρ Hence, if the constant decays exponentially one might have to wait exponentially long (with respect to d) Therefore, the above result so far does not imply a break of the curse of dimensionality in Theorems 4.7(ii) and 4.8(ii) In fact, this refers to the “footnote” in [22] on page 2224 where the opposite is stated Found Comput Math (d) In contrast to d we assume that ν is a fixed parameter Due to the upper bound in Theorem 3.6(iii) we can break the curse of dimensionality here (e) Based on Theorems 3.7 and 4.8, we have similar statements on the curse of dimensionality in terms of nε We mention here the related paper [27] discussing the intractability of L∞ -approximation of infinitely differentiable functions on Id 4.3 The Case α > γ − β = Theorem 4.10 Let α, β, γ ∈ R satisfy the conditions α > γ − β = Then the following relations hold true (i) For any d ≥ and n ≥ 2d 4−α (1 + log e)(d − 1) −α(d−1) −α n (log n)α(d−1) ≤ dn U α,β , H γ d −1 2e ≤ 4α −α(d−1) n−α (log n)α(d−1) (ii) For any integer n ≥ 2d+1 and d ≥ 4, 4−α (1 + log e)(d − 1) −α(d−1) −α n (log n)α(d−1) α,β ≤ d n U∗ , H γ d −1 e ≤ 4α −α(d−1) n−α (log n)α(d−1) (iii) If in addition ≤ ν ≤ d/2, then for any n ≥ 4−α (1 + log e)(ν − 1) √ −α(ν−1) −αν αν −α ν 2ν−1 ν−1 5+3 d ν d n 22ν+1 , (log n)α(ν−1) ≤ dn Uνα,β , H γ ≤ √ 5+3 α ν−1 e −α(ν−1) ν e −αν d αν n−α (log n)α(ν−1) Proof Due to Theorem 3.13, we have to prove the lower bounds in this theorem Let us prove the lower bound for dn (U α,β , H γ ) The other lower bounds can be proved in a similar way For a given n ≥ 2d+1 , there is an unique m ≥ d + such that (d − 1)−(d−1) 2m (m − 1)d−1 ≥ n + > (d − 1)−(d−1) 2m−1 (m − 2)d−1 Hence, by the inequality a log e > log(1 + a), a ≥ 0, we obtain m + (d − 1) m−1 m−1 − log e ≥ m + log d −1 d −1 d−1 ≥ log(n + 1), (4.10) Found Comput Math and consequently, m≥ log(n + 1) + d log e , + log e From the last inequality and (4.10) we derive 2−αm ≥ 2−α(m−1) (d − 1)α(d−1) (m − 2)−α(d−1) 2−α (d − 1)−α(d−1) (m − 2)α(d−1) ≥ 2−α (d − 1)−α(d−1) log(n + 1) + d log e −2 + log e α(d−1) (n + 1)−α We have (n + 1)−α ≥ 2−α n−α Moreover, by the inequality d ≥ + 2/ log e one can verify that log(n + 1) + d log e log n −2≥ , + log e + log e and consequently, 2−αm ≥ 4−α (1 + log e)(d − 1) −α(d−1) −α n (log n)α(d−1) (4.11) From Lemma 3.8 and (3.11), we obtain 2|k|1 ≥ (d − 1)−(d−1) 2m (m − 1)d−1 ≥ n + dim V∗d (αm) = (4.12) k∈K∗d (m) Consider the set B∗ (m) := {f ∈ V∗d (ξ ) : f H γ ≤ 2−αm } in the subspace V∗d (αm) α,β of H γ By Lemma 4.2 we have B∗ (m) ⊂ U∗ Hence, by (4.12) and Lemma 4.1 we obtain dn U∗ , H γ ≥ dn B∗ (m), H γ ≥ 2−αm α,β The last inequality combined with (4.11) finishes the proof of the desired lower bound Theorem 4.11 Let α, β, γ ∈ R satisfy the conditions α > γ − β = Then the following relations hold true (i) For any < ε ≤ 2−αd , α(d − 1) −(d−1) −1/α ε | log ε|d−1 ≤ nε U α,β , H γ ≤4 α(d − 1) 2e −(d−1) ε −1/α | log ε|d−1 (ii) For any < ε ≤ 2−αd and d ≥ 4, 2α(d − 1) −(d−1) −1/α ε α,β | log ε|d−1 ≤ nε U∗ , H γ ≤4 α(d − 1) e −(d−1) ε −1/α | log ε|d−1 Found Comput Math (iii) If in addition ≤ ν ≤ d/2 then for any < ε ≤ 2−2αν , 2α(ν − 1) −(ν−1) −ν ν −1/α ν d ε | log ε|ν−1 ≤ nε Uνα,β , H γ ≤2 √ 5+3 α(ν − 1) e −(ν−1) (ν/e)−ν d ν ε −1/α | log ε|ν−1 Proof Due to Theorem 3.14, we have to prove the lower bounds in this theorem Let α,β us prove the lower bound for nε (U∗ , H γ ) The other lower bounds can be proved −αd we take m ≥ d ≥ such that in a similar way For a given ε ≤ 2−αm ≥ ε > 2−α(m+1) The right-hand inequality gives 2m ≥ ε −1/α and m ≥ α −1 | log ε| − Consider the set B∗ (m) := {f ∈ V∗d (ξ ) : f H γ ≤ 2−αm } in the subspace V∗d (αm) α,β of H γ By Lemma 4.2 we have B∗ (m) ⊂ U∗ Hence, by (4.12) and Lemma 4.1 we have dn U∗ , H γ ≥ dn B∗ (m), H γ ≥ 2−αm ≥ ε, α,β where n := dim V∗d (αm) − Therefore, by Lemma 3.8, (3.11) and the inequality | log ε| ≥ 4α we get α,β nε U∗ , H γ ≥ dim V∗d (αm) − m−1 ≥ 2m d −1 ≥ (d − 1)−(d−1) (m − 1)d−1 ε −1/α d−1 −1/α ≥ (d − 1)−(d−1) α −1 | log ε| − ε −(d−1) −1/α ≥ 2α(d − 1) ε | log ε|d−1 Remark 4.12 Note that in [22] the authors did not prove any lower bounds for the dimensions of the optimized sparse grid spaces and the approximation error for the α,β linear approximation in H γ of functions from the class U∗ (which is defined via a biorthogonal wavelet decomposition, see next section) Found Comput Math Biorthogonal Wavelet Decompositions In this section, we discuss how to extend the results for the dyadic harmonic decomposition in this paper to periodic biorthogonal wavelet decompositions and other periodic and non-periodic decompositions Let Φ := {ϕk,s }k∈Z+ ,s∈Qk and Φ˜ := {ϕ˜k,s }k∈Z+ ,s∈Qk be biorthogonal systems in L2 , where Qk := {s ∈ Z : ≤ s < 2k } We will assume that {ϕk,s }k∈Z+ ,s∈Qk forms a Riesz basis for L2 , that is, |ck,s |2 ck,s ϕk,s k∈Z+ s∈Qk k∈Z+ Therefore, every f ∈ L2 has a unique representation f= (f, ϕ˜ k,s )ϕk,s , k∈Z+ s∈Qk and we have the dyadic biorthogonal wavelet decomposition f= qk (f ), k∈Z+ with the norm equivalence f 2 qk (f ) , k∈Z+ where qk (f ) := (f, ϕ˜ k,s )ϕk,s s∈Qk One of the most important cases of biorthogonal systems in L2 which has wide applications are wavelet biorthogonal systems Univariate periodic wavelet biorthogonal systems Φ := {ϕk,s }k∈Z+ ,s∈Qk and Φ˜ := {ϕ˜ k,s }k∈Z+ ,s∈Qk are of the form ϕk,s (x) = ϕ k x − 2π2−k s , ϕ˜ k,s (x) = ϕ˜ k x − 2π2−k s , where {ϕ k }k∈Z+ and {ϕ˜ k }k∈Z+ are the sequences of mother wavelets which in particular, can be received from the mother wavelets ψ and ψ˜ of univariate non-periodic wavelet biorthogonal systems by the periodization formula ϕ k (x) = ψ 2k (x + 2πs) , ψ˜ 2k (x + 2πs) ϕ˜ k (x) = s∈Z s∈Z We assume the following conditions on Φ There hold the Jackson type inequality inf g∈Σk f −g L2 ≤ C2−mk f Hr , Found Comput Math for some m ∈ N, and the Bernstein type inequality f ≤ C2lk f Hl L2 , f ∈ Vk , for some l ≤ r with < r ≤ m, where Vk := span{ϕk,s : s ∈ Qk } We also assume that similar inequalities hold for the dual system Φ˜ with parameters m ˜ and r˜ For distributions f and k ∈ Zd+ , let us introduce the following operator: d qk (f ) := qkj (f ) j =0 where the univariate operator qkj is applied to f as a univariate function in variable xj while the other variables are held fixed If f ∈ L2 , we have f 2 qk (f ) k∈Zd+ α,β Let us use the notation H∗ to denote the subspace in H α,β of all functions f having the biorthogonal wavelet decomposition f 2 qk (f ) k∈Nd The following lemma has been proved in [22] Lemma 5.1 Let α, β ∈ R satisfy the restrictions ≤ α < r and ≤ α + β < r, where r is the parameter in the Jackson and Bernstein type inequalities Then we have the following norm equivalence: f α,β H∗ 22(α|k|1 +β|k|∞ ) qk (f ) L2 k∈Nd α,β Lemmas 2.1 and 5.1 show that functions f ∈ H∗ have similar dyadic harmonic and biorthogonal wavelet decompositions with the same equivalent norms There are α,β other analogous decompositions of H∗ not only for periodic functions but for nonα,β periodic functions defined on a d-dimensional cube Indeed, we can treat spaces H∗ α,β α,β α,β as well H α,β , Hν and classes U α,β , U∗ , Uν in a more general form which are suitable for different applications Let H be a separable Hilbert space and H have the following dyadic decomposition Namely, H is decomposed into pairwise orthogonal subspaces Wk , k ∈ Zd+ , H= Wk , k∈Zd+ with dim Wk = 2|k|1 Found Comput Math Then every f ∈ H can be decomposed into a series f= pk (f ), pk (f ) ∈ Wk , = pk (f ) k∈Zd+ with f k∈Zd+ We define H α,β as the Hilbert space of formal series f= pk (f ), pk (f ) ∈ Wk , k∈Zd+ for which the following norm is finite: f H α,β 22(α|k|1 +β|k|∞ ) pk (f ) = k∈Zd+ With this definition we have H 0,0 = H For α = 0, we put H 0,β = H γ for β = γ α,β α,β We define the subspaces H∗ and Hν , ≤ ν ≤ d − 1, in H α,β as follows The α,β subspace H∗ is the set of all f ∈ H α,β such that such that d pk (f ) = if kj = j =0 α,β The subspace Hν is the set of all f ∈ H α,β such that pk (f ) = α,β α,β if σ (s) > ν α,β α,β Denote by U α,β , U∗ and Uν the unit ball in H α,β , H∗ and Hν , respecα,β α,β tively (For convenience, here we use the same notations H α,β , U α,β , U∗ , Uν as in Sect for the harmonic dyadic decomposition.) 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(Springer, Berlin, 2010) ... at most n It represents a characterization of the best linear approximation error There is a vast amount of literature on optimal linear approximations and the related Kolmogorov and linear n-widths. .. used for norming these classes suitably for high-dimensional approximations In Sect 3, we prove upper bounds for α,β α,β hyperbolic cross approximations of functions from U = U α,β , U∗ and Uν... spaces and H -norm approximation of functions with dominating mixed second derivative Then the total cost for the solution of Poisson’s equation is of the order nε ≤ C(d)ε −1 , see also [41] for