High dimensional periodic sampling and cubature on Smolyak grids based on B spline quasi interpolation

We investigate linear algorithms of sampling recovery and cubature formulas on Smolyak grids of periodic dvariate functions having LipschitzH¨older mixed smoothness based on Bspline quasiinterpolation, and their optimality when the number d of variables and the number n of sampled function values may be very large. We establish upper and lower estimates of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in d and n.

High-dimensional periodic sampling and cubature on Smolyak grids

based on B-spline quasi-interpolation

Dinh D˜ung

Vietnam National University, Hanoi, Information Technology Institute

144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

n of sampled function values may be very large We establish upper and lower estimates of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit

in d and n.

Keywords and Phrases Linear sampling algorithm · Cubature formula · Smolyak grid · Lipschitz-H¨ older mixed smoothnesse space of mixed smoothness · Quasi-interpolation repre- sentations by B-spline series.

Mathematics Subject Classifications (2000) 41A15 · 41A05 · 41A25 · 41A58 · 41A63.

1 Introduction

The aim of this paper is to investigate linear algorithms of sampling recovery and cubature formulas

on Smolyak grids of periodic d-variate functions having Lipschitz-H¨older mixed smoothness based

on B-spline quasi-interpolation, and their optimality when the number d of variables and thenumber n of sampled function values may be very large We stress on explicitly estimating fromabove and below the error of the optimal sampling recovery and the optimal integration on Smolyakgrids as a function of two variables d and n

We consider only functions on Rd1-periodic at each variable It is convenient to consider them

as functions defined in the d-torus Td= [0, 1]d which is defined as the cross product of d copies ofthe interval [0, 1] with the identification of the end points To avoid confusion, we use the notation

Id to denote the standard unit d-cube [0, 1]d

For m ∈ N, the well known periodic Smolyak grid of sampled points Gd(m) ⊂ Td is defined as

d

s
, m < d

2m m−1d−1
, m > d

Notice the grid Gd(m) is full if and only if m ≥ d

Sparse grids Gd(m) for sampling recovery and numerical integration were first considered

by Smolyak [36] Temlyakov [37] – [39] and the author of the present paper [9] – [11] oped Smolyak’s construction for studying the sampling recovery for periodic Sobolev classes andNikol’skii classes having mixed smoothness Recently, Sickel and Ullrich [34] have investigated thesampling recovery for periodic Besov classes having mixed smoothness For non-periodic functions

devel-of mixed smoothness linear sampling algorithms on Smolyak grids have been recently studied byTriebel [40] (d = 2), Dinh D˜ung [14], Sickel and Ullrich [35], using the mixed tensor product hatfunctions and more general, B-splines In [16], we have constructed methods of approximation byarbitrary linear combinations of translates of the Korobov kernel κr,don Smolyak grids of functionsfrom the Korobov space K2r(Td) which is a reproducing kernel Hilbert space with the associatedkernel κr,d

In numerical applications for high-dimensional approximation problems, Smolyak grids was firstconsidered by Zenger [42] in parallel algorithms for numerical solving PDEs Numerical integration

on Smolyak grids was investigated in [21] For non-periodic functions of mixed smoothness ofinteger order, linear sampling algorithms on Smolyak grids have been investigated by Bungartz andGriebel [3] employing hierarchical Lagrangian polynomials multilevel basis There is a very largenumber of papers on Smolyak grids and their modifications in various problems of approximations,sampling recovery and integration with applications in data mining, mathematical finance, learningtheory, numerical solving of PDE and stochastic PDE, etc to mention all of them The readercan see the surveys in [3, 29, 22] and the references therein For recent further developments andresults see in [25, 24, 26, 19, 2]

For univariate functions f on R, the rth difference operator ∆rh is defined by

is defined by

∆r,uh := Y

i∈u

∆rhi, ∆r,∅h = I,

where the univariate operator ∆rh

i is applied to the univariate function f by considering f as afunction of variable xi with the other variables held fixed

Denote by C(Td) the normed space of all bounded continuous functions on Td with the maxnorm k · kC(Td ) For a function f on Td, let

If 0 < α ≤ r, we introduce the semi-norm |f |Hα

∞ (u) for functions f ∈ C(Td) by

The Lipschitz-H¨older space Hα

∞ of mixed smoothness α is defined as the set of functions

f ∈ C(Td) for which the norm

is finite Denote by U∞α the unit ball in H∞α

For a family Φ = {ϕξ}ξ∈Gd (m) of functions, we define the linear sampling algorithm Sm(Φ, ·)

∞ we will consider its subsets ˚Uα

∞ which consists of all functions f ∈ Uα

∞

such that f (x) = 0 if xj = 0 for some index j ∈ [d]

For sampling recovery of functions from ˚Uα

∞, we use the linear sampling algorithm ˚Sm(Φ, ·) onthe grids ˚Gd(m) by

C0(d, α, p) 2−αmmd−1 ≤ ˚sm(˚U∞α)p ≤ sm(U∞α)p ≤ C(d, α, p) 2−αmmd−1 (1.9)

In high-dimensional approximation problems using function values information, the number m

of sampled values is the main parameter in the study of convergence rates of the approximationerror with respect to m going to infinity However, the parameter d may hardly affect this ratewhen d is large In the present paper, inspired by the relation (1.9), we establish upper andlower bounds for the quantities sm(U∞α)p and ˚sm(˚U∞α)p explicitly in n and d as a function of twovariables m, d, in particular, upper bounds for the constant C(d, α, p) and C0(d, α, p) in (1.9) Toobtain these upper and lower bounds we construct linear sampling algorithms on Smolyak grids

of the form (1.5) based on a B-spline quasi-interpolation series specially on Faber-Schauder seriesrelated to the well-known hat functions As consequences we obtain upper and lower bounds forthe quantities Intm(U∞α)p and ˚Intm(˚U∞α)p of optimal cubature formula on Smolyak grids explicit

in n and d Related to the problems investigated in the present paper, is problems of hyperboliccross approximation of functions having mixed smoothness in high-dimensional setting in terms ofn-widths and ε-dimensions which have been invesigated in [4, 17]

The paper is organized as follows In Section 2, we establish upper and lower bounds weconstruct linear sampling algorithms on Smolyak grids based on Faber-Schauder series for ˚sm(˚U∞α)p

and sm(Uα

∞)p for 0 < α ≤ 2 As consequence, we derive upper and lower bounds for the error

of cubature formulas on Smolyak grids and of optimal integration on Smolyak grids ˚Intsm(˚U∞α)p

and Intsm(U∞α)p In Section 3, we extend the results for sm(U∞α)p and Intsm(U∞α)p in Section 2 tothe case of large mixed smoothness α > 2, based on B-spine quasi-interpolation representationsfor function from H∞α In Section 4, we give some example of polynomials inducing generatingquasi-interpolation operators

2 Sampling recovery based on Faber-Schauder series

2.1 Faber-Schauder series

Let M2(x) = (1 − |x − 1|)+, x ∈ I, be the piece-wise linear B-spine with knot at 0, 1, 2 Since thesupport of functions M2(2k+1·) for k ∈ Z+is the interval [0, 2−k] we can extend these functions to

an 1-periodic function on the whole R Denote this periodic extension by ϕk

The univariate periodic Faber-Schauder system of the hat functions is defined by

Lemma 2.1 The d-variate periodic Faber-Schauder system Fd is a basis in C(Td) Moreover,function f ∈ C(Td) can be represented by the series

converging in the norm of C(Td)

Proof For the univariate case (d = 1), this lemma can be deduced from its well-known counterpartfor non-periodic functions on I (see, e.g., [30, Theorem 1, Chapter VI]) For the multivariate case(d > 1), it can be proven by the tensor product argument

Put Zd+(u) := {k ∈ Zd+: supp k = u}, where supp k denotes the support of k, i.e., the subset ofall j ∈ [d] such that kj 6= 0

Theorem 2.1 Let 0 < p ≤ ∞ and 0 < α ≤ 2 Then a function f ∈ H∞α can be represented by theseries (2.1) converging in the norm of C(Td) Moreover, we have for every k ∈ Zd+(u),

≤ 2−|u|ωur(f, 2−k)

≤ 2−|u|2−α|k|1|f |Hα

∞ (u)

The inequality (3.42) is proven Let us prove (2.3) We first consider the case 1 ≤ p < ∞ Wehave for every k ∈ Zd+(u),

kqk(f )kpp =

Z

Td

X

s∈Z d (k)

λk,s(f )ϕk,s(x)

s∈Z d (k)

ϕk,s(x)

 t

1 − t

n−s

For nonnegative integer n, we define the function

For m ∈ Z+, we define the operator ˚Rm for f ∈ ˚U∞α by

(2α− 1)s, β(α, 0, m) := 1

Theorem 2.2 Let 0 < p ≤ ∞ and 0 < α ≤ 2 Then we have for every f ∈ ˚U∞α and and every

m ≥ d − 1,

kf − ˚Rm(f )kp ≤ [2(p + 1)1/p]−d2−αmβ(α, d, m) ≤ exp(2α− 1) [b(α, p)]−d2−αmmd−1 (2.13)

Moreover, if in addition, m ≥ 2(d − 1),

kf − ˚Rm(f )kp ≤ a◦(α, p, d) 2−αm

m

The inequality (2.14) can be derived from (2.17) by applying Lemma 2.3 for t = 2−α

Let us prove (2.16) where m = d From (2.17), and (2.5) we have

 t

1 − t

d−1−s

= 2−d(α+1)(p + 1)−d/p

1

1 − t

d

= [2(2α− 1)(p + 1)1/p]−d

(2.18)

For m ∈ Z+, we define the operator Rm by

k∈Z d + : |k| 1 ≤m

k∈Z d + : |k| 1 ≤m

2−l(p + 1)−l/p 1

1 − t

t



2−l(p + 1)−l/p

t

Next, let us verify (2.20) Applying Lemma 2.3 to Fm,l−1(t) in (2.23) we have for m ≥ 2(d − 1),

 t



2−l(p + 1)−l/pbl−1(2−α)

= a(α, p, d) 2−αm

m

d − 1



gk(x) := X

s∈I 1 (k)

From the identity M4(2)(x) = M2(x) − 2M2(x − 1) + M2(x − 2), x ∈ R, it follows that

|M4(2)(x)| ≤ 2M2(2−1x), x ∈ R,

and consequently,

One can also verify that

supp gk,s = Ik,s =: [2−ks, 2−k(s + 1)], int Ik,s∩ int Ik,s0 = ∅, s 6= s0, (2.32)and by the equationR01M4(x) dx = 1,

Let us prove this inequality for u = [d] and h ∈ Rd+, the general case of u can be proven in

a similar way with a slight modification For h ∈ R and k ∈ N, let h(k) ∈ [0, 2−k) is the numberdefined by h = h(k)+ s2−k for some s ∈ Z For h ∈ Rd and k ∈ Nd, put

The inequality (2.39) is proven for u = [d] and h ∈ Rd

+.From (2.38), (2.32), (2.33) we can also verify that if m is given then for arbitrary n ≥ m,

d − 1

(2α− 1)d−1

= (2α− 1)−12−7d2−αm

m

d−1

2−αm

m

Proof We take the univariate functions gk,s, gk as in (2.29), (2.30), and for every u ⊂ [d], definethe functions

By (2.37) we have Sm(Φ, φm,n) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary

(2α− 1)l

(2α− 1)l

(2α− 1)l

l

X

s=0

d

d − 1

(2α− 1)d−1

l

X

s=0

d − 1s



2−7(s+1)(2α− 1)−(s+1)

= [2−7(2α− 1)]−12−αm

m

d − 1

(2α− 1)d−1

d−1

X

s=0

d − 1s

[2−7(2α− 1)]−s

= [2−7(2α− 1)]−1(2α− 1)d−1[1 + 2−7(2α− 1)−1]d−12−αm

m

d−1

2−αm

m

d − 1



(2.51)

which proves (2.44)

From the fifth inequality in (2.52) we have

(2α− 1)l

l

X

s=0

d − 1s

(2α− 1)l

l

X

s=0

 ls

[2−7(2α− 1)]−s

(2α− 1)l[1 + 2−7(2α− 1)]l

Hence, it is easy to see that

|I(f ) − Λsm(f )| ≤ kf − Sm(Φ, f )k1,

and, as a consequence of (1.8) and (2.54),

For functions f ∈ ˚U∞α, based on the grids ˚Gd(m), we can define the cubature formula ˚Λsm(f ) :=

˚Λ( ˚Gd(m), f ) on grids ˚Gd(m), and the quantity of optimal cubature ˚Intsm(Fd) on Smolyak grids

d − 1



≤ ˚Intsm(˚U∞α) ≤ a◦(α, d) 2−αm

m

Proof The upper bounds in (2.61)–(2.63) follow from (2.58) and Theorem 2.2

To prove the lower bounds, we take the function fm,n ∈ ˚U∞α as in (2.35) with the property as

Comparing with (2.41), we can see that ˚Intsm(˚Uα

∞) can be estimated from below as in the proof ofTheorem (2.4) for ˚ssm(˚U∞α) This proves the lower bounds in in (2.61)–(2.63)



4−lβ(α, l, m)

In a similar way we can prove

Theorem 2.7 Let 0 < p ≤ ∞ and 0 < α ≤ 2 Then we have for every every m ≥ 2d−1,

d−1

2−αm

m

d − 1



≤ Intsm(U∞α) ≤ a(α, d) 2−αm

m

d − 1

, (2.67)where

repre-3.1 Periodic B-spline quasi-interpolation representations

We introduce quasi-interpolation operators for functions on Rd For a given natural number `,denote by M` the cardinal B-spline of order ` with support [0, `] and knots at the points 0, 1, , `

We introduce quasi-interpolation operators for functions on Rd In what follows we fixed r ∈ Nand consider the cardinal B-spline M := M2r of even order 2r Let Λ = {λ(s)}|j|≤µ be a given

finite even sequence, i.e., λ(−j) = λ(j) for some µ ≥ r − 1 We define the linear operator Q forfunctions f on R by

If Q is a quasi-interpolation operator of the form (3.1)–(3.2), for h > 0 and a function f on R,

we define the operator Q(·; h) by

From (3.4) and (3.5) we get for a function f on R,

and the univariate coefficient functional aki,si is applied to the univariate function f by considering

f as a function of variable xi with the other variables held fixed

Since M (2r2kx) = 0 for every k ∈ Z+ and x /∈ (0, 1), we can extend the univariate B-spline

M (2r.2k·) to an 1-periodic function on the whole R Denote this periodic extension by Nk anddefine

where Id(k) := di=1I(ki) Then we have for functions f on Td,

with the convergence in the norm of C(Td)

From the definition of (3.16) and the refinement equation for the B-spline M , in the univariatecase, we can represent the component functions qk(f ) as

qk(f ) = X

s∈I d (k)

where ck,s are certain coefficient functionals of f In the multivariate case, the representation(3.18) holds true with the ck,s which are defined in the manner of the definition (3.10) by

See [14] for details Thus, we have proven the following

Lemma 3.2 Every continuous function f on Td is represented as B-spline series

X

s∈I d (k)

converging in the norm of C(Td), where the coefficient functionals ck,s(f ) are explicitly constructed

as linear combinations of at most m0 function values of f for some m0 ∈ N which is independent

of k, s and f

The formula (3.12) with the coefficients valued functionals ak,s(f ) given as in (3.3) and (3.10),defines a multivariate periodic quasi-interpolation operator for functions on Td While the formula(3.20) with the coefficients valued functionals ak,s(f ) given as in (3.16), defines a multivariateperiodic B-spline quasi-interpolation representation for functions on Td They are completely gen-erated from the initial quasi-interpolation operator Q given as in (3.1) and (3.2) For this reason

we will call Q the generating quasi-interpolation for Qk and for the representation (3.20)

3.2 A formula for the coefficients in quasi-interpolation representations

If h ∈ Rd, we define the shift operator Ths for functions f on Td by

where sh := (s1h1, , sdhd) Sometimes we also write Th[P ]= Th[P (z)]

Notice that any operation over polynomials generates a corresponding operation over operators

Th[P ] Thus, in particular, we have

where Pj(zj) are univariate polynomial in variable zj.

Lemma 3.3 Let P be a tensor product Laurent polynomial, h ∈ Rd with hj 6= 0, and l ∈ N.Assume that Th[P ](g) = 0 for every polynomial g ∈ Pl Then P has a factor Dl and consequently,

Th[P ] = Th[P∗]◦ ∆lh, P (z) = DlP∗(z),

where P∗ is a tensor product Laurent polynomial

Proof By the tensor product argument it is enough to prove the lemma for the case d = 1 Weprove this case by induction on l Let P (z) =Pn

s=−mcszs for some m, n ∈ Z+ Consider first thecase l = 1 Assume that Th[P ](g) = 0 for every constant functions g Then replacing by g0 = 1 in(3.22) we get

By B´ezout’s theorem P has a factor (z − 1) This proves the lemma for l = 1 Assume it is true for

l − 1 and Th[P ](g) = 0 for every polynomial g of degree at most l − 1 By the induction assumption

we have

Th[P ] = T[P1 ]

We take a proper polynomial gl of degree l − 1 (with the nonzero eldest coefficient) Hence

ψl = ∆l−1h (gl) = a where a is a nonzero constant Similarly to the case l = 1, from the equations

0 = Th[P ](gl) = T[P1 ]

h (ψl) we conclude that P1 has a factor (z − 1) Hence, by (3.24) we can seethat P has a factor (z − 1)l The lemma is proved

Let us return to the definition of quasi-interpolation operator Q of the form (3.1) induced

by the sequence Λ as in (3.2) which can be uniquely characterized by the univariate symmetricLaurent polynomial

PΛ(z) := zr X

|s|≤µ

˚Λ( ˚Gd(m), f ) on grids ˚Gd(m), and the quantity of optimal cubature ˚Intsm(Fd) on Smolyak grids

d... class="page_container" data-page="24">

From (3.4) and (3.5) we get for a function f on R,

and the univariate coefficient functional aki,si is applied... (1.8) and (2.54),

For functions f ∈ ˚U∞α, based on the grids ˚Gd(m), we can define the cubature formula ˚Λsm(f ) :=

˚Λ(