Quadrature Formulas for Multivariate Convex Functions - Science Direct

JOURNAL OF COMPLEXITY ARTICLE NO.

12, 5–16 (1996)

0003

Quadrature Formulas for Multivariate Convex Functions* CARSTEN KATSCHER AND ERICH NOVAK† Mathematisches Institut, Universita¨t Erlangen-Nu¨rnberg, Bismarckstrasse 1 sA , 91054 Erlangen, Germany AND

KNUT PETRAS‡ Institut fu¨r Angewandte Mathematik, Technische Universita¨t Braunschweig, Pockelsstrasse 14, 38106 Braunschweig, Germany Received June 21, 1995

We study optimal quadrature formulas for convex functions in several variables. In particular, we answer the following two questions: Are adaptive methods better than nonadaptive ones?, and are randomized (or Monte Carlo) methods better than deterministic methods?  1996 Academic Press, Inc.

1. INTRODUCTION AND RESULTS We study optimal rates of convergence of quadrature formulas for convex functions in several variables. We mainly consider the classes Fd 5 h f : [0, 1]d R [21, 1] u f convexj, but we will see that the lower bounds also hold for smaller classes (where we have a bound for the gradient of f ) and the upper bounds also hold for larger classes (where we replace ‘‘convex’’ by ‘‘convex in each xi-direc* Invited paper presented at the Schloss Dagstuhl Workshop on Continuous Algorithms and Complexity, October 17-21, 1994. † E-mail address: [email protected]. ‡ E-mail address: [email protected]. 5 0885-064X/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

6

KATSCHER, NOVAK, AND PETRAS

tion’’). We will prove that the error of optimal deterministic methods is of the order n22/d.

(1)

This order can be obtained by nonadaptive methods; adaptive methods do not give a better order of convergence. If we allow randomized (or Monte Carlo) methods then the order of the optimal methods is n22/d21/2,

(2)

where we have to use a slightly different notion of the error than in the deterministic case. The upper bound is proved by an adaptive method and we believe that nonadaptive methods are worse. We construct nonadaptive Monte Carlo methods with an error of the order n23/(2d)21/2

(3)

and conjecture that this is optimal. We could not prove the respective lower bound, however. The basic facts and, in particular, the statements (1)–(3) are known for the integration problem in one dimension; see Novak and Petras (1994). We give some comments and begin with deterministic quadrature formulas. For each quadrature formula, the error in the class of convex functions on, e.g., [0, 1] is not uniformly bounded. For the study of optimal quadrature formulas we therefore have to restrict the class of convex functions. The classes Fuv 5 h f : [0, 1] R R u f convex, f 91 (0) $ u, f 92 (1) # vj, where v . u, were studied by Glinkin (1984), Zwick (1988), and Novak (1993). The following is known for these classes. Let n [ N and ti 5 (2i 2 1)/(2n). Then the affine and nonadaptive formula Q*n ( f ) 5

O

1 1 n f (ti ) 2 ? (v 2 u) 1 16n n i51

(4)

is optimal even in the class of adaptive formulas of the form Qn ( f ) 5 f( f (t1 ), . . . , f (tn )), where the knot ti may depend on the ‘‘already known’’ function values, i.e., ti 5 ti ( f (t1 ), . . . , f (ti21 )), and f is an arbitrary mapping. Defining the

7

QUADRATURE FORMULAS FOR CONVEX FUNCTIONS

maximal error of Qn on a class F by Dmax (Qn , F ) 5 sup f[F

HUE

1

0

f (x) dx 2 Qn ( f )

UJ

,

we have Dmax (Q*n , Fuv ) 5

v2u . 16n2

We prefer, however, the more general classes Fd without the smoothness assumptions, because for each convex function on [0, 1]d, an appropriate nontrivial multiple is in Fd . The class F1 was studied by Braß (1982) who proved the following. Let cn be defined by

cn 5

H

2(n2 1 2n 1 1)21 if n is odd

2(n2 1 2n 1 2)21 if n is even.

Then the linear formula

Qn ( f ) 5

H

(n21)/2

cn ? (oi51

2if (i2cn ) 1 2if (1 2 i2cn ) 1 nf (1/2)) if n is odd

cn ? (oi51 2if (i2cn ) 1 2if (1 2 i2cn )) n/2

if n is even (5)

is optimal for F1 in the class of all nonadaptive quadrature formulas. The maximal error of the optimal formula is given by Dmax (Qn , F1 ) 5 cn .

(6)

Though adaptive formulas might be slightly better than nonadaptive formulas for F1 , it follows from the results for Fuv that n22 is also the optimal order of convergence for adaptive formulas. The maximal error of the npoint Gaussian formula on the class F1 is less than f2 /3n2, see Petras (1993b). Therefore the Gaussian formulas are ‘‘almost optimal’’ for F1 . More of the known results concerning deterministic quadrature formulas in the onedimensional case can be found in the survey of Petras (1993a). Some results are even known for d . 1. See, in particular, Sonnevend (1983) and Gruber (1993). We want to point out, however, that we do not require smoothness, such as C 1 or even C 2, for our upper bounds. We also

8

KATSCHER, NOVAK, AND PETRAS

do not know any previous work concerning error bounds for stochastic methods. In the randomized setting we consider general (adaptive) stochastic rules of the form Q ng ( f ) 5 fg( f (t 1g ), . . . , f (t ng )), i.e., the knots t ig are random variables (which can be chosen nonadaptively or adaptively) and also the fg is randomly chosen. Then, as usual, the error of Q ng is defined in a ‘‘worst case stochastic sense’’, Dmax (Q ng , Fd ) 5 sup E f[F

SUE

[0,1]d

f (t) dt 2 Q ng ( f )

UD

.

It is known from Novak and Petras (1994) that, in the case d 5 1, nonadaptive stochastic methods are at most slightly better than (adaptive or nonadaptive) deterministic methods, the optimal order is n22. We also proved that adaptive Monte Carlo methods are better and here the optimal order is n25/2. These results are quite different from known results for unit balls of Ho¨lder or Sobolev spaces. It is known for all these classes that adaptive methods are not better than nonadaptive ones—for deterministic methods as well as for stochastic methods—while randomized methods are better than deterministic methods; see Novak (1988) and Traub et al. (1988). Classes of monotone functions were studied by Novak (1992) for d 5 1 and by Papageorgiou (1993) for d . 1. The results are similar to the results of the present paper. 2. PROOFS We have to prove five statements: (a) (b) (c) (d) (e)

the lower bounds (1) for adaptive deterministic methods; the upper bound (1) for nonadaptive deterministic methods; the lower bounds (2) for adaptive Monte Carlo methods; the upper bounds (3) for nonadaptive Monte Carlo methods. the upper bounds (2) for adaptive Monte Carlo methods;

(a) More precisely, we prove for deterministic methods Qn involving n 5 As kd nodes, where k/2 [ N, that Dmax (Qn , Fd ) $

bd ? (2n)22/d. 2 d(d 1 2) d

QUADRATURE FORMULAS FOR CONVEX FUNCTIONS

9

Here, bd $ (f(d 1 1))21/2 (2ef/d)d/2 denotes the volume of the d-dimensional unit ball. Let x 5 (x1 , . . . , xd ), and define f [ Fd by f (x) 5 21 1

OS

D

1 2 8 d xi 2 . d i51 2

We divide [0, 1]d into the d-cubes G1 , . . . , G2n , whose edges have the lengths k21. Assume that they are ordered in such a way that all nodes of Qn for the function f are in Gn11 < ? ? ? < G2n . Let yi be the center of Gi , let h˜ i : Rd R R represent the hyperplane which is tangent to f at the point yi , and let hi (x) 5 h˜ i (x) 1 2/(dk2). Note that hi (x) $ f (x) if and only if x is in the largest d-dimensional ball, which is contained in Gi . Define g 5 maxh f, h1 , . . . , hn j [ Fd . Using that hi (x) 2 f (x) 5 2/(dk2) 2 h if and only if ix 2 yi i 2 5 Ïhd/8, we obtain

dn :5

E

Gi

g(x) 2 f (x) dx 5

E

2/dk2 0

bd

SD hd 8

d/2

1 212dbd ? (2n)22/d. dh 5 ? n d(d 1 2)

From Qn (g) 5 Qn ( f ), we obtain the stated lower bound. (b) Let Qn be a given quadrature formula and let Q nd be the corresponding product formula on C ([0, 1]d) involving nd nodes (cf. Davis and Rabinowitz, 1984). Note that the function e[0,1]d21 f (x1 , . . . , xd21 , ?) dx1 ? ? ? dxd21 is convex on [0, 1] and that f (?, . . . , ?, xd ) is convex on [0, 1]d21. Therefore, we may apply the technique of Haber (1970) and obtain that Dmax (Q nd , Fd ) # Dmax (Qn , F1 ) ?

O iQ i .

d21

n

k

k50

If we choose Qn as in Eq. (5), we have Dmax (Q nd , Fd ) #

2d . (n 1 1)2

(c) Consider the equidistribution e on the following set of 22n functions, F d* :5 hmaxh f, hi1 , . . . , hik j u hi1 , . . . , ik j , h1, 2, . . . , 2njj , Fd ,

10

KATSCHER, NOVAK, AND PETRAS

where we adopt here and in the following the notation of part (a). We first want to estimate the expectation of the error of a deterministic (adaptive) method for the class F d* . It is obvious that an additional function value at ti [ Gj only gives additional information if f (ti ) , hj (ti ) and if no further node is in Gj . Therefore, we have 2n distinct combinations of function values depending on whether the function value at ti is f (ti ) or hj (ti ), where i 5 1, . . . , n. For each of these combinations, the sum of all occurring errors is at least

dn ?

O SnkDUk 2 n2U $ d ? 2 n

n

Ï2n

n22

k50

(for the last inequality, cf. Novak, 1992). We obtain Ee

SUE

d

[0,1]

f (x) dx 2 Qn ( f )

UD

$

2n ? dn 2n22Ï2n . 22n

Fubini’s theorem shows that this is also a lower bound for the worst-case expectation for the error of any adaptive stochastic formula. (d) Let k $ 3 be an odd number and

xi 5

5

2

(i 2 1)2 k11 if i # (k 2 1)2 2

1 2 xk112i

else.

We also put li 5 (xi11 2 xi ) and ti 5 As (xi 1 xi11 ), where i 5 1, . . . , k 2 1. We divide [0, 1]d into (k 2 1)d little boxes of the form Gi 5 [xi1 , xi111 ] 3 ? ? ? 3 [xid , xid11 ], where (i1 , . . . , id ) [ h1, . . . , k 2 1jd. The volume of Gi is d

uGu 5 p lir , r 51

and the midpoint is (ti1 , . . . , tid ). Let I iu ( f ) 5 uGi u22d

O f ( y˜ ), y˜

11

QUADRATURE FORMULAS FOR CONVEX FUNCTIONS

where y˜ runs through all 2d corners of Gi and let I il ( f ) 5 uGi u f (ti1 , . . . , tid ). We also put S kud,d ( f )

5

O

(k21)d i5 1

I iu ( f )

and

l S (k 21)d,d ( f )

5

O I ( f ).

(k21)d i51

l i

It is clear that l S (k 21)d,d ( f ) #

E

[0,1]d

f (x) dx # S kud,d ( f ).

By a computation that is similar to that in part (b) one obtains l S kud,d ( f ) 2 S (k 21)d,d ( f ) :5 F ( f ) #

8d . (k 2 1)2

(7)

For the following we need a bound for each single Fi ( f ) :5 I iu ( f ) 2 I il ( f ), where f [ Fd . We therefore estimate f ( y˜ ) 2 f (ti1 , . . . , tid ) for each corner y˜ 5 ( yi1 , . . . , yid ) of Gi . Choose without restriction Gi , [0, As]d and consider the representation f ( y˜ ) 2 f (ti1 , . . . , tid ) 5 f ( yi1 , . . . , yid ) 2 f (ti1 , yi2 , . . . , yid ) 1 f (ti1 , yi2 , . . . , yid ) 2 f (ti1 , ti2 , yi3 , . . . , yid ) 1? ? ? 1 f (ti1 , . . . , tid21 , yid ) 2 f (ti1 , . . . , tid ). Since g :5 f (ti1 , . . . , tir21 , ?, yir11 , . . . , yid ) is convex and igi # 1, we have for yir , tir that g( yir ) #

yir g(tir ) 1 (tir 2 yir )g(0) tir 2 0

,

12

KATSCHER, NOVAK, AND PETRAS

i.e., g( yir ) 2 g(tir ) # (tir 2 yir )

g(0) 2 g(tir )

#

tir

lir tir

5

4ir 2 2 8 . # 2i 2r 2 2ir 1 1 lir (k 2 1)2

We easily see that the same inequality holds for yir . tir . Finally, we obtain

S D OO O S d

Fi ( f ) #

p lir 22d

r 51

#

32d

2 (k 2 1)2

8 232d 25 (k 2 1)2 r51 lir (k 2 1) d

y˜

D

2k 2 4 (k 2 1)2

d?

y˜

d21

# 2d

OOpl d

d

y˜

S D 2 k21

r51 j51 j?r

ij

(8)

d11

.

Now we are ready to define a nonadaptive Monte Carlo method with an order n23/(2d)21/2. For i 5 1, . . . , (k 2 1)d we independently and randomly choose a d

g [ p [2li j /2, li j /2] j51

according to the normed Lebesgue measure and put Fi (g)( f ) 5 uGi u22d

O

(a1,...,ad)[h21,1jd

f (ti1 1 a1g1, . . . , tid 1 ad gd).

It is easy to see that E(Fi ( f )) 5

E

Gi

f (x) dx

and hence the Monte Carlo method Q( f ) 5

O F (f)

(k21)d

i

i51

has expectation E(Q( f )) 5 e[0,1]d f (x) dx. By induction one can prove I il ( f ) #

E

Gi

f(x) dx # I iu( f )

13

QUADRATURE FORMULAS FOR CONVEX FUNCTIONS

and I il ( f ) # Fi (g)( f ) # I iu( f ) and so obtains

UE

U

f (x) dx 2 Fi (g)( f ) # I iu( f ) 2 I il ( f ) 5 Fi ( f )

Gi

for each g and f [ Fd . For the stochastic error of Q( f ) we obtain by (7) and (8)

E

SUE

d

[0,1]

f (x) dx 2 Q( f )

UD S O D O S S S D S D #

(k21)d

1/2

Fi ( f )2

i5 1

# max Fi ( f ) · i

, 2d 5 2d

2 k 21

2 k 21

(k21)d i51

d11

·

D D

1/2

Fi ( f )

8d (k 2 1)2

1/2

(d13)/2

Since we need 2d(k 2 1)d function values to compute Q( f ) we obtain the claimed order of convergence. (e) The basic idea is the same as in Novak and Petras (1994) for the case d 5 1. We define an adaptive Monte Carlo method in three steps. (i) The first step is deterministic and nonadpative. For a given l k $ 2 we compute S kud,d ( f ) and S (k 21)d,d ( f ), as defined in part (d). We need d d k 1 (k 2 1) function values to do this. (ii) The second step is still deterministic, but depends on f. We do not compute new function values but define an adaptive partition of [0, 1]d into boxes Gi, j . For i 5 1, . . . , (k 2 1)d we define

li 5

LFF((ff)) (k 2 1) J. i

d

14

KATSCHER, NOVAK, AND PETRAS

We divide each box Gi into li subsets Gi, j . For the total number l of sets Gi, j we obtain by li , (Fi ( f )/F( f ))(k 2 1)d 1 1 that l , 2(k 2 1)2. The definition of Gi, j is as follows. We set

F

Hj :5 ti1 2 j

li 1 2li

, ti1 1 j

li1 2li

G

d

3 p [xir , xir11],

j 5 1, . . . , li

r 52

and Gi,1 5 H1 ,

Gi, j 5 Hj \ Hj21 , j 5 2, . . . , li .

The Gi, j are sets with symmetry center in (ti1 , . . . , tid ). We define I i,u j ( f ) 5 uGi, j u22d

O f( y˜), y˜

where y˜ runs through all corners of Hj , and

I i,l j ( f ) 5 uGi, j u f (ti1 , . . . , tid ) 5

uGi u f (ti1 , . . . , tid ). li

Again, we obtain I i,l j ( f ) #

E

Gi, j

f (x) dx # I i,u j ( f ).

Furthermore, the numbers I i,u j ( f ) 2 I i,l j ( f ) are increasing for increasing j and I i,uli ( f ) 2 I i,l li ( f ) 5 (1/ li )(I iu( f ) 2 I il ( f )), such that

I i,u j ( f ) 2 I i,l j ( f ) #

Fi ( f ) F( f ) # . li (k 2 1)d

(iii) The stochastic step is as in part (d) with the nonadaptive boxes Gi replaced by the adaptive sets Gi, j . The only difference in the definition

15

QUADRATURE FORMULAS FOR CONVEX FUNCTIONS

of the quadrature formula for Gi, j is that g is randomly chosen in

SF

ti1 2 j

li1 2li

d

F

, ti1 2 ( j 2 1)

3p 2 r52

li1

G F

2li

G

< ti1 1 ( j 2 1)

li1 2li

, ti1 1 j

li1

GD

2li

lir lir . , 2 2

For the stochastic error of the resulting method we obtain

E

SUE

[0,1]d

f (x) dx 2 Q( f )

UD SO #

l

D

F( f )2 2d i51 (k 2 1)

1/2

, 8Ï2 d(k 21)222d/2.

In this step we compute l · 2d , 2(k 2 1)d2d function values, together certainly less than 2d12kd. Hence we obtain the claimed order of convergence. Remarks. (a) Again we stress that we cannot prove that the upper bound (3) is optimal. This open problem seems to be related to difficult discrepancy problems that are studied in the book of Beck and Chen (1987). A related open problem concerns the integration of multivariate monotone functions. Papageorgiou (1993) proved optimal rates for deterministic and for randomized methods but it is unclear whether (for d . 1) adaptive Monte Carlo methods are better than nonadaptive ones. (b) In all proofs of upper bounds we did not really use that the functions are convex. It is enough to demand that f is convex in each xidirection. So the upper bounds even hold for larger classes. (c) Similarly, the lower bounds even hold (with different constants) for smaller classes where we demand, in addition, that the functions are C 1 with a Lipschitz constant 1. (d) The ratios of our upper and lower bounds are increasing exponentially with increasing dimension. It might be interesting to study the exact behavior of the distinct constants with respect to the dimension in the different settings.

REFERENCES 1. BECK, J., AND CHEN, W. W. L. (1987), ‘‘Irregularities of Distributions,’’ Cambridge Tracts in Mathematics, Vol. 89, Cambridge Univ. Press, Cambridge. 2. BRAß H. (1982), Zur Quadraturtheorie konvexer Funktionen, in ‘‘Numerical Integration,’’

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3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14.

15. 16.

KATSCHER, NOVAK, AND PETRAS

(G. Ha¨mmerlin, Ed.), International Series of Numerical Mathematics, Vol. 57, pp. 34–47, Birkha¨user, Basel/Boston. DAVIS, P. J. AND RABINOWITZ, P. (1984), ‘‘Methods of Numerical Integration,’’ 2nd ed., Academic Press, New York. GLINKIN, I. A. (1984), Best quadrature formula in the class of convex functions, Math. Notes 35, 368–374; Mat. Zametki 35, 697–707 [in Russian]. GRUBER, P. M. (1993). Aspects of approximation of convex bodies, in ‘‘Handbook of Convex Geometry’’ (P. M. Gruber and J. M. Wills, Eds.), pp. 319–345, Elsevier, New York. HABER, S. (1970), Numerical evaluation of multiple integrals, SIAM Rev. 12, 481–526. NOVAK E. (1988), ‘‘Deterministic and Stochastic Error Bounds in Numerical Analysis,’’ Lecture Notes in Mathematics, Vol. 1349, Springer-Verlag, New York. NOVAK, E. (1992), Quadrature formulas for monotone functions, Proc. Amer. Math. Soc. 115, 59–68. NOVAK E. (1993), Quadrature formulas for convex classes of functions, in ‘‘Numerical Integration IV,’’ (H. Braß and G. Ha¨mmerlin, Eds.), International Series in Numerical Math., Vol. 112, pp. 283–296, Birkha¨user, Basel. NOVAK, E., AND PETRAS, K. (1994), Optimal stochastic quadrature formulas for convex functions, BIT 34, 288–294. PAPAGEORGIOU, A., (1993), Integration of monotone functions of several variables, J. Complexity 9, 252–268. PETRAS, K. (1993a), Quadrature theory of convex functions, in ‘‘Numerical Integration IV,’’ International Series in Numerical Math., Vol. 112, pp. 315–329, Birkha¨user, Basel. PETRAS, K. (1993b), Gaussian quadrature formulae—Second Peano kernels, nodes weights, and Bessel functions, Calcolo 30, 1–27. SONNEVEND, G. (1993), Optimal passive and sequential algorithms for the approximation of convex functions in Lp([0, 1]s ), p 5 1, y, in ‘‘Constructive Function Theory, Proc. Int. Conference, Varna, Bulgaria, 1981,’’ pp. 535–542. TRAUB, J. F., WASILKOWSKI, G. W., AND WOZ´ NIAKOWSKI, H. (1988), ‘‘Information-Based Complexity,’’ Academic Press, New York. ZWICK, D. (1988), Optimal quadrature for convex functions and generalizations, in ‘‘Numerical Integration III,’’ (H. Braß and G. Ha¨mmerlin, Eds.), International Series in Numerical Math., Vol. 85, pp. 310–315, Birkha¨user, Basel.