Hierarchical cubature formulas

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Applied Mathematics

Hierarchical cubature formulas

V.L.Vaskevich

Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia e-mail:vask@math.nsc.ru

Received: March 10, 2001

Summary.

We study the properties of hierarchical bases in the space of continuous functions with bounded domain and construct the hierarchical cubature formulas. Hierarchical systems of functions are similar to the well-known Faber{Schauder basis. It is shown that arbitrary hierarchical basis generates a scale of Hilbert subspaces in the space of continuous functions. The scale in many respects is sim-ilar to the usual classication of functional spaces with respect to smoothness. By integration over initial domain the standard interpo-lation formula for the given continuous integrand, we construct the hierarchical cubature formulas and prove that each of these formulas is optimal simultaneously in all Hilbert subspaces associated with the initial hierarchical basis. Hence, we have constructed the universally optimal cubature formulas.

Key words:

calculation of integrals, guaranteed accuracy, cuba-ture formulas, bases in Sobolev spaces, hierarchical bases

Mathematics Subject Classication (1991): 65D32, 65D30, 41A55

1. Introduction

Let be a bounded domain in Rn_k _{is a non-negative integer and}

k = fx (k) j 2 j j = 12:::N(k)g is a nite subset of . We assume that0 1

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is dense in , i.e., 1 k=0 k = : The sequence = fkg 1 k=0 is said to be a multigrid in k is a

k{level of and vectorsx(k)

j 2 are nodes of .

Given a positive integerk, we introduce two real numbers hk and

hk by putting (1:1) hk = sup x2 inf y2k jx;yj h_k = inf_x 6=y jx;yjjxy2k :

By the denition, 0 < hk hk. If hk tends to zero as k !1 then

the union ofk is obviously dense in . We also assume that

(1:2) hk+1 < hk and

92(01) : hk hk hk k= 012::::

Together with k{levels of the multigrid we deal with their dier-ences dened as follows

knk ;1 = fx~ (k) j jj= 12:::(k)g k= 12:::: Ifk= 0 then ~x(0) j =x(0) j ,j= 12:::(0) =N(0). It is obvious that (0) +(1) ++(k) =N(k).

Let C( ) be the Banach space of functions which are continuous in . For u 2 C( ) andx 2 the value of u at x 2 is dened.

Hence, we can consider the vectors

Ak(ju) = ; u(~x(k) 1 )u(~x (k) 2 ):::u(~x (k) (k))

where k = 0, 1, :::. From the vectors Ak( j u) we compose the

following innite sequence

u = (A1( ju)A

2(

ju):::Ak(ju):::):

We will operate byu as an innite column-vector.

By the denition, we have

jju jl 1 jj= sup x2 ju(x)jsup x2 ju(x)j=jjujC( )jj:

Whence the linear operatorT :C( )!l

1 transformed the

func-tionsu(x) fromC( ) into the sequence u 2l

1is bounded. We call

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Let H be a countable subset ofC( ),

H =fh (k)

j (x)2C( )jj= 12:::(k)k= 01:::g:

In this paper, we considerHsuch that for members ofH the following equalities hold (1:3) h(k) j (~x(m) i ) = 08m < k k= 012::: h(k) j (~x(k) l ) =lj jl= 12:::(k):

Here _lj is the conventional Kronecker delta.

Denition 1.

If (1.3) holds thenH is called a hierarchical system.

From (1.3) it follows that all functionsh1,h2,:::,hM of a hierarchical

systemH are linearly independent.

By the denition, if H is a hierarchical system and h(k)

j (x) 2 H

then the trace h(k)

j

of h(k)

j on the multigrid has zeroes as the

entries in positions 12:::N(k;1)+j;1N(k;1)+j+1:::N(k)

and the entry of h(k)

j

in position N(k;1) +j equals 1.

Let H be a hierarchical system and let U be the matrix which has the traces h(k)

j as columns, i.e., U = (h(0) 1 h (0) 2 :::h (0) (0) :::h (k) 1 :::h (k) j :::h(k) (k) :::):

Then U is a subdiagonal matrix with 1 on the main diagonal. By the same way as in denition 1, we can dene a hierarchi-cal system in a Hilbert space. In 8], 9], and 3], it was shown that hierarchical systems in Sobolev{like spaces may be constructed as sequences of interpolatingDm_{{splines. It should be noted that}

hier-archical systems are frequently applied to the solution of boundary value problems by the method of nite elements (see, e.g., 2], 5], 6], and 11]). Hierarchical systems in Hilbert spaces such as Sobolev{like spaces are studied in 4].

Denition 2.

If a -hierarchical system H in C( ) is a basis of

C( ), then H is called a hierarchical basis.

Example 1.The well-known Faber|Schauder system is a hierarchi-cal basis ofC01], (see 7, p. 227]).

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Given a non-negative integer m and a hierarchical basis

H=fh (k)

j (x)2C( )jj = 12:::(k)k= 01:::g

inC( ), we dene the members of the following nite set

f! (i) lm(x)ji= 01:::m l= 12:::(i)g: by the equalities !(m) jm(x) =h(m) j (x) j= 12:::(m) !(i) lm(x) =h(i) l (x); m P k=i+1 (k) P j=1 h(i) l (~x(k) j )!(k) jm(x) i=m;1m;2:::0 l= 12:::(i):

By the denition ofH, the following equalities hold

!(i)

lm(~x(k)

j ) =kilj ik= 01:::m

l= 12:::(i) j = 12:::(k):

Given a continuous function '(x) with domain , we consider the following interpolation formula

'(x)= m X k=0 (k) X j=1 '(~x(k) j )!(k) jm(x) x2 :

By integration of the both sides of this approximate equality, we obtain the cubature formula

(1:4) Z '(x)dx =Xm k=0 (k) X j=1 c(k)o jm '(~x(k) j ) withc(k)o

jm the weights dened as follows

(1:5) c(k)o jm = Z ! (k) jm(x)dx k= 01:::m j= 12:::(k):

The formula (1.4) will be referred to asthe hierarchical cubature for-mula. Our goal in this paper is to study the properties of cubature formulas of the form (1.4).

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2. Hilbert spaces associated with a hierarchical basis

Let H =fh (k)

j (x) jj = 12:::(k)k= 01:::g be a hierarchical

basis in C( ). Given a function '(x) fromC( ), we can expand it in the series (2:1) '(x) = 1 X k=0 (k) X j=1 g(k) j h(k) j (x): The coecients g(k) j = g(k)

j (') of (2.1) are uniquely determined by

a function'. The partial sums of (2.1) converge to'(x) in the norm of C( ). If'(x) =h(m) l (x) then we have (2:2) g(k) j (h(m) l ) =ljkm j= 1:::(k) l= 1:::(m) km= 012::::

Together with the hierarchical basis H we introduce into considera-tion a numerical sequence fh(m)g

1 m=0, by putting h(m) = sup x2 fjh (m) j (x)jjj= 12:::(m)g: Letfamg 1

m=0 be a sequence of positive numbers such that

(2:3) LH(am) ( 1 X m=0 (m)h2(m) a2 m ) 1=2 <1: Given a sequence famg 1

m=0, we dene the linear subspace X

(am)( ) of C( ) as follows (2:4) X(am)( ) = 8 < : '2C( )jh'i 2= 1 X m=0 a2 m (m) X j=1 jg (m) j (')j 2< 1 9 = :

Since (2.2) holds, it follows thath(k)

l (x) belonging toHis also a

mem-ber of X(am)( ). Hence, X(am)( ) is an innite-dimensional linear

space.

We introduce a bilinear form in the spaceX(am)( ), by letting for

all functions'and

(2:5) h'i= 1 X k=0 a2 k (k) X j=1 g(k) j (')g(k) j ():

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Applying the Cauchy inequality for sums to the right side of (2.5), we obtain

jh'ijh'ihi<1:

Hence, for'andfromX(am)( ) the series in the right side of (2.5)

converges absolutely.

Theorem 1.

Functions from the hierarchical basis H are mutually orthogonal in the inner product hi dened by (2.5). If the

corre-sponding norm is denoted by hi then for a function ' belonging to

X(am)( ) (2.1) converges not only in the norm of C( ) but in the

norm hitoo.

Proof. To begin with, we consider the bilinear form (2.5). By the def-inition, it follows thath'iis a symmetric and linear form with

re-spect to'and. By (2.5), for a function'inX(am)( ) the inequality h''i 0 holds. If'= 0 theng

(k)

j (') = 0, andh''i= 0.

Converse-ly, let h''i= 0 for a function' inX

(am)( ). Then g (k)

j (') = 0 for

allk and j and the partial sums of (2.1) are equal to 0 everywhere in domain . Hence, the limit of these partial sums is also equal to 0, i.e.,'= 0.

Thus, we have proved that the bilinear form (2.5) is an inner prod-uct inX(am)( ). The corresponding norm is dened by the equality h'i=h''i

1=2.

Since (2.2) and (2.5) hold, it follows that

hh (m) l h(k) j i= 1 P i=0 a2 i (i) P n=1 g(i) n (h(m) l )g(i) n (h(k) j ) = 1 P i=0 a2 i (i) P n=1 _nlim_nj_ik=a2 mkmlj:

Hence, functions from the hierarchical basis H are mutually orthog-onal in the inner product hi, and for the norm ofh

(m)

l (x) in this

inner product we havehh (m)

l i=am.

Let 'm(x) be the partial sum of (2.1), i.e.,

'm(x) = m X k=0 (k) X j=1 g(k) j h(k) j (x):

The orthogonality of the basis functionsh(k)

j (x) implies h';'mi 2= 1 X k=m+1 a2 k (k) X j=1 jg (k) j (')j 2:

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By the denition ofX(am)( ), the sequence of the sums in the right

side of the last equality converges to 0 asmtends to innity. It means that (2.1) converges to'in the normhi. ut

LetHbe a hierarchical basis inC( ) and let'm(x) be the partial

sum of the series (2.1), i.e., (2:6) '0(x) = (0) P j=1 g(0) j (')h(0) j (x) 'm(x) ='m;1(x) + (m) P j=1 g(m) j (')h(m) j (x) for m 1:

Then the following equalities hold (2:7) 'm(x(m)

j ) ='(x(m)

j ) j = 12:::N(m):

The function 'm(x) is said to be a standard interpolant for '(x).

Lemma 1.

For an arbitrary'(x)2C( )coecientsg (k) j (')of (2.1) may be dened by (2:8) g(0) j (') ='(~x(0) j ) j = 12:::(0) g(m) j (') ='(~x(m) j );'m ;1(~x (m) j ) j= 12:::(m) m 1:

For a given positive integerN there is a positive real numberAN with

(2:9) f N X m=0 a2 m (m) X j=1 jg (m) j (')j 2 g 1=2 ANjj'jC( )jj:

Here AN does not depend on '. Hence, linear functionals g(k)

j () are

bounded on C( ).

Proof. By the denition ofH, (2.6), together with (2.7), yields (2.8). It is not hard to show by induction onm that the inequality holds

sup x2 j'm(x)jG(m) sup x2m j'(x)j whereG(0) =(0) andG(m) =(m)h(m)+G(m;1)(1+(m)h(m))

form 1. This, together with (2.8), yields jg

(m)

j (')j(1 +G(m;1))sup

x2

j'(x)j:

Hence, (2.9) holds withAN =f

N P m=0 (m)a2 mj1 +G(m;1)j 2 g 1=2. u t

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Theorem 2.

The space X(am)( ) with the norm

hiis complete and

so it is a Hilbert space. The embedding ofX(am)( )inC( )is

bound-ed and for an arbitrary function ' from X(am)( ) the following

in-equality holds

(2:10) jj'jC( )jjLH(am)h'i

with LH(am) the constant dened by (2.3).

Proof. Let '(x) be a member ofX(am)( ). Then the corresponding

series (2.1) converges to '(x) both in the norm of C( ) and in the norm ofX(am)( ). Whence and from the denition of

fh(m)g 1 m=0 it follows that j'(x)j 1 X k=0 (k) X j=1 jg (k) j (')jjh (k) j (x)j 1 X k=0 h(k)(k) X j=1 jg (k) j (')j:

Applying the Cauchy inequality for sums to the right side of this inequality, we obtain j'(x)j 1 X k=0 h(k)1=2(k) f (k) X j=1 jg (k) j (')j 2 g 1=2 LH(am)h'i:

Thus, we have proved (2.10). Letf'

(k)(x) g

1

k=1be a Cauchy sequence in the spaceX

(am)( ). By

(2.10), it also is a Cauchy sequence in the spaceC( ). Consequently, there exists a function '(x) inC( ) with

(2:11) _klim !1 sup x2 j'(x);' (k)(x) j= 0:

Moreover, there exists a positive real numberRsuch that sup

k1 h'

(k)

iR <1:

Let N be an integer and let 'N(x) be the partial sum of (2.1).

Then (2:12) h'Nif N X m=0 a2 m (m) X j=1 jg (m) j (';' (k)) j 2 g 1=2+ sup k1 h' (k) i:

To estimate the rst summand in the right side of (2.12) we use (2.9) and obtain (2:13) h'NiANjj';' (k) jC( )jj+ sup k1 h' (k) i

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whereAN is independent of'and '(k). Since (2.11) and (2.13) hold

it follows that for a given integerN there exists a numberK =K(N) such that fork K(N) the rst summand in the right side of (2.13)

is less then R. Hence,

h'Ni2R and h'i= lim

N!1

h'Ni2R <1

i.e.,'(x) is a member ofX(am)( ).

Let us check thath';' (k)

i!0 ask!1. It is well-known that

there exists a completionX( ) of a space X(am)( ) with the inner

product hi. In addition, X

(am)( ) is dense in X( ). Since X( )

is complete it follows that there exists an element 2 X( ) such

that h;' (k)

i ! 0 as k ! 1. By (2.10), we may consider as

a continuous function with domain . Moreover, the initial sequence

f' (k)(x)

g 1

k=1 converges to in the norm of C( ). By uniqueness of

the limit,we conclude that'=. In other words,X(am)( ) coincides

withX( ), i.e.,X(am)( ) is a Hilbert space. u t

Corollary 1.

Let '(x) be a member of X(am)( ) and let '

m(x) be the partial sum of (2.1). Then

(2:14) '(x) ='0(x)+ 1 X k=1 ('k(x);'k ;1(x)) = 1 X k=0 (k) X j=1 g(k) j (')h(k) j (x) The series in the right side of (2.14) converges to' in the normhi.

By (2.14), we can split the identical operator into the direct sum of projections of X(am)( ) to the nite-dimensional subspaces of

X(am)( ). Hence, (2.14) is similar to the multi-level splitting of nite

element spaces (see 11, p. 383]).

3. The optimality of the hierarchical cubature formulas

In this section, we consider cubature formulas of the form

(3:1) Z '(x)dx =Xm k=0 (k) X j=1 c(k) jm'(~x(k) j ):

Integrable functions are assumed to be members of some Hilbert space

X(ak)( ) embedded into C( ) the nodes of formula (3.1) are the

members of m-level m of the multigrid and the number of the

nodes equals N(m).

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(3:2) (lm') = Z '(x)dx ; m X k=0 (k) X j=1 c(k) jm'(~x(k) j ):

The error is a linear functional, therefore also referred to as error functional, since we require that the rules for choosing the nodes and the weights of (3.1) be independent of specifying an integrable function. We consider the following problem.

Problem 1.

Given a Hilbert space X(ak)( ) and a positive integer

m, nd the error (3.2) with N(m) nodes and the minimal norm in the space duel to X(ak)( ).

The cubature formula corresponding to the solution of Problem1 is said to be an X(ak)( )-optimal formula.

Theorem 3.

For a positive integer m there is a unique X(ak)( )

-optimal cubature formula of the form (3.1). The weights ofX(ak)( )

-optimal cubature formula are dened by (1.5), i.e., this formula is hierarchical cubature formula (1.4).

Proof. By the Riesz Theorem, the error functionallmdened by (3.2)

may be written as inner product

(3:3) (lm') =hum'i 8'2X

(ak)( ):

Hereumis a uniquely determined member ofX(ak)( ) called the

ex-tremal functionoflmor, more verbosely,X(ak)( )-extremal function.

Moreover, the following equalities hold (3:4) jjlmjX (ak)( ) jj 2= humi 2= 1 X k=0 a2 k (k) X j=1 jg (k) j (um)j 2:

The extremal functionum expands into a series inh(k)

j (x). Moreover, the coecientg(k) j (um) of h(k) j (x) is dened by (3:5) g(k) j (um) =humh (k) j i=a 2 k = (lmh(k) j )=a2 k:

This, together with (3.4), yields (3:6) jjlmjX (ak)( ) jj 2 = 1 X k=0 (k) X j=1 j(lmh (k) j )j 2 a2 k :

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If k > m then the values ofh(k)

j at the points ofm are equal to 0.

Hence, we get (lmh(k) j ) = Z h (k) j (x)dxb (k) j for k > m:

Inserting these equalities in (3.6), we obtain (3:7) jjlmjX (ak)( ) jj 2 1 X k=m+1 (k) X j=1 jb (k) j j 2 a2 k : Letl0

m be the error corresponding to hierarchical cubature

formu-la (1.4) and letu0

m be the extremal function ofl0

m. By the denition,

we have (l0

mh(k)

j ) = 0 for k= 01:::m j= 12:::(k):

Whence and from (3.6) it follows that (3:8) jjl 0 mjX (ak)( ) jj 2= 1 X k=m+1 (k) X j=1 jb (k) j j 2 a2 k :

Since (3.7) and (3.8) hold, the X(ak)( )-optimality of hierarchical

cubature formula (1.4) is immediate.

By the parallelogram law, anX(ak)( )-optimal cubature formula

is unique. ut

It is well-known that the same cubature formula may be optimal simultaneously in many normed spaces not necessarily equivalent to one another. The formulas with such properties is called to be univer-sally optimalcubature formulas(see, e.g., 1] and 10]).By Theorem 3, hierarchical cubature formula (1.4) is the universally optimal cuba-ture formula on the family of Hilbert spaces introduced in section 2.

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9. Rozhenko, A.I. (1999):The Abstract Theory of Splines in Russian], Novosi-birsk University Press, NovosiNovosi-birsk.

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Hierarchical cubature formulas

Applied Mathematics