An intractability result for multiple integration. I.H.Sloan y and H.Wozniakowski yy. University of New South Wales. Australia.

Transcription

1 An intractability result for multiple integration I.H.Sloan y and H.Wozniakowski yy Columbia University Computer Science Technical Report CUCS y School of Mathematics University of ew South Wales Sydney 2052 Australia and yy Department of Computer Science Columbia University ew York, Y USA and Institute of Applied Mathematics University ofwarsaw ul. Banacha 2, Warsaw Poland December, 1995

2 1 Introduction In this note we consider the approximation of integrals over the d-dimensional unit cube, If = Z [0 1] d f(x 1 ::: x d ) dx 1 :::dx d = Z [0 1] d f(x)dx (1:1) under the assumption f 2 E d where, for arbitrary >1 E d is the set of complex-valued functions in L 1 ([0 1] d ), whose Fourier coecients satisfy j b f(h)j Here h =(h 1 h 2 ::: h d ) with integers h j and bf(h) = Z 1 (h 1 h d ) : [0 1] d f(x)e ;2ihx dx h x = P d j=1 h j x j and h j = max (1 jh j j): ote that a function f belonging to E d necessarily has a continuous 1-periodic extension, since the Fourier series for f 2 E d is absolutely convergent: h2zd j b f(h)e 2ihx j h2zd(h 1 h d ) ; < 1: Our aim is to show that in the worst-case setting the integration problem is intractable: that is, to achieve a given error ", "<1 for all f 2 E d the amount of information required is exponential in d. More precisely, we prove that the minimal error of any quadrature rule that uses <2 d points is one. The bound on is sharp, since the error of a quadrature rule that uses =2 d may be arbitrarily small for large. The space E d is a standard setting for the particular class of quadrature formulas known as lattice rules (for a survey see [4]). The implications of the intractability result for lattice methods are considered briey in Section 3. 2 The intractability result A quadrature rule approximating (1.1) is a linear functional of the form Qf = Q(d! t)f := j=1! j f(t j ) where the `weights'! := (! 1 :::! ) and `points' t := (t 1 ::: t ) satisfy! j 2 C t j 2 [0 1) d for j =1 ::: : Without loss of generality we can assume that t 1 ::: t are distinct points. The worst-case error for the quadrature rule Q = Q(d! t) for the class E d is P (Q) =P (d! t) := sup fjqf ; Ifj : f 2 E d g: 1

3 Since we are interested in a lower bound for P (Q) we dene It is an elementary fact that e( d ) := inf fp (d! t) :! 2 C t2 ([0 1) d ) g: since by taking! 1 =! 2 = :::=! =0we obtain e( d ) 1 (2:1) P (d 0 t) = sup fjifj : f 2 E d g = sup fj b f(0)j : f 2 E d g =1: The following theorem, which is our main result, states in eect that if <2 d then, in the worst-case setting and for the class E d the error is as bad as it can be, and the quadrature rule Q = 0 is a best possible rule. Theorem 1 If <2 d then e( d) =1: Proof. Let <2 d, and suppose that points t =(t 1 ::: t ) and weights! =(! 1 :::! ) are given. The theorem is proved by constructing a function g 2 E d, depending on t and!, such that Ig = 1 and Qg =0: From this it will follow thatp (Q) jqg ; Igj = jigj =1: Since this holds for any choice of points t and weights!, it follows that e( d) 1 which together with (2.1) proves e( d) =1. To accomplish the construction, let B d := f0 1g d and dene g to be a trigonometric polynomial of the form g(x) =(x) a h e 2ihx (2:2) where fa h 2 C : h 2 B d g is a non-trivial solution of the linear system 2ihtj a h e =0 j =1 ::: (2:3) and is a trigonometric polynomial which is yet to be specied. A crucial point in the proof is that, because the homogeneous linear system (2.3) has 2 d unknowns but fewer than 2 d equations, a non-trivial solution of (2.3) certainly exists. Let h 2 B d be such that ja h jja h j for h 2 B d.we scale our non-trivial solution of (2.3) so that It follows from (2.2) and (2.3) that g(t j )=(t j ) ja h j1 for h 2 B d and a h =1: a h e 2iht j =0 j =1 ::: from which it is clear that Q(d! t)g =0: ow wechoose (x) :=e ;2ih x 2

4 so that Clearly g(x) = a h e 2i(h;h )x : (2:4) Ig = bg(0) = a h =1: On the other hand g given by (2.4) is a trigonometric polynomial of degree 1ineach component ofx, since for h h 2 B d wehave This implies h j ; h j =0 1or ; 1forj =1 ::: d: (h 1 ; h 1) (h 2 ; h 2) (h d ; h d )=1forh h 2 B d : It therefore follows, since ja h j1 that g 2 E d, and so the theorem is proved. 2 Remark 1 Theorem 1 remains valid if we permit more general quadrature rules. amely, for xed points t j wemayapproximate the integral If by (f(t 1 ) ::: f(t n )), where is an arbitrary nonlinear mapping, : R n! R. Since the class E d is convex and symmetric (i.e., f 2 E d implies that ;f 2 E d )wemay apply Smolyak's theorem, see e.g., [6] p. 76. This theorem states that the mapping which minimizes the worst-case error is linear. For linear we have just proven that the error is at least one. We may also permit adaptive choice of points t j. That is, assuming that the points t 1 ::: t j;1 are already chosen and the function values f(t 1 ) ::: f(t j;1 ) are already computed, the next point t j may depend arbitrarily on f(t 1 ) ::: f(t j;1 ). By Bakhvalov's theorem, see e.g., [6], p. 59, the worst-case error of arbitrary quadrature rule that uses adaptive points t j cannot be smaller than the minimal worst-case error of linear quadrature rules that use nonadaptive (xed) points. Again the latter error is at least one. Hence, Theorem 1 also holds for adaptive choice of points t j. 3 Lattice rule results Our purpose in this section is merely to mention some known lattice rule results that cast on interesting light on the result in Theorem 1. In particular, we will see that the condition < 2 d in the theorem cannot be improved, and indeed that the behaviour of e( d) changes dramatically when reaches 2 d. A lattice rule is an equal-weight rule of the form where Qf = 1 j=1 f(t j ) ft 1 ::: t g = L \ [0 1) d and L is an `integration lattice' that is to say, L is a geometrical lattice containing Z d as a subset, where a geometrical lattice is a discrete subset of R d that is closed under addition 3

5 and subtraction. It is known [5] that if Q is a lattice rule that corresponds to an integration lattice L then Qf ; If = bf(h) for f 2 E d where L? is the `reciprocal lattice' of L, It follows in turn that h2l? h6=0 L? := fh 2 R d : x h 2 Z 8x 2 Lg Z d : P (Q) := sup fjqf ; Ifj : f 2 E d g = h2l? h6=0 One of the simplest of all lattice rules is the n d -point product-rectangle rule R n f := 1 n;1 n d n;1 k 1 =0 k 2 =0 ::: n;1 k d =0 f k 1 n k 2 n ::: k d n 1 (h 1 h d ) : (3:1) For this rule we can easily compute P (R n ) by using the fact that the corresponding integration lattice is L =(n ;1 Z) d from which it follows that L? =(nz) d. Specically, from (3.1) we nd 0 dy P (R n 1! j=1 h j 2nZ 1 d A 2() ; 1= 1+ ; 1 (3:2) h j n where (x) is the Riemann zeta function, (x) = In particular, on setting n =2wend 1 i=1 i ;x for x>1: P (R 2 )=(1+()=2 ;1 ) d ; 1: (3:3) In the product-rectangle rule with n =2wehave =2 d thus this example only just misses being covered by Theorem 1. On the other hand, we note from (3.3) that from which it follows that P (R 2 ) ;! 0 as ;! 1 e( 2 d d) ;! 0 as ;! 1: And since e( d) is clearly non-increasing in, it follows in turn that e( d) ;! 0 as ;! 1 for all 2 d : (3:4) The last result stands in striking contrast to the result from the theorem that e( d) =1 8 >1 if <2 d : 4! :

6 This result (3.4) shows that the condition <2 d in the theorem cannot be weakened, at least for large values of. The product-rectangle rule is not usually thought ofasaninteresting lattice rule, because lattice rules are traditionally designed to have good asymptotic convergence properties as ;! 1(for xed d.) By this test the n d -point product-rectangle rule performs poorly, since (3.2) gives the inferior result P (R n )=O(n ; )=O( ;=d ): Much greater interest attaches usually to the `method of good lattice points', a class of lattice rules of the form Q(z)f := ;1 j=0 f j z (3:5) where z 2 Z d is a well chosen integer vector, with no nontrivial factor in common with, and fxg for x 2 R d means that each component ofx is to be replaced by its fractional part in [0 1): The classical theorems of the method of good lattice points (see, for example, [3] or [4]) assert the existence of z = z() suchthat P (Q(z)) c( d) (log )( d) for some positive functions c and. Usually, ( d) is of order d. This result indicates on impressive rate of convergence for large, but asymptotic bounds of this kind either do not give explicit values of c( d) or do not provide useful bounds for smaller values of. Among the known explicit bounds, the authors of [2] assert that for prime values of up to approximately 10 d the bound in the following theorem is as good as any known bound: Theorem 2 For prime there exists a lattice ruleq(z) of the form (3.5) such that P (Q(z)) (1 + 2())d + ; 1 1 ; 2(1 ; 1; )() ; 1! d ; 1: (3:6) An analogous result for composite is given in [1]. For t 2 d it can easily be seen that the right-hand side of (3.6) is bounded below by (0:5+()) d ; 1whenever 2 d 2()+1: (3:7) Since () 1+1=( ; 1), it is easy to check that 1+ () 2 ; (): Thus, except possibly for the small values of and d that violate (3.7), the known theoretical result for the method of good lattice points is in fact worse than the result (3.3) for the humble 2 d -point product-rectangle rule. For still smaller values of it can be seen that the right-hand side of (3.6) always exceeds 1, so there is no violation of Theorem 1. 5

7 References [1] Disney, S.A.R. Error bounds for rank 1 lattice quadrature rules modulo composites. Monatshefte fur Mathematik, 110, (1990). [2] Disney, S.A.R. and Sloan, I.H. Error bounds for the method of good lattice points. Mathematics of Computation, 56, (1991). [3] iederreiter, H. Random number generation and quasi-monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia (1992). [4] Sloan, I.H. and Joe, S. Lattice methods for multiple integration. Oxford University Press, Oxford (1994). [5] Sloan, I.H. and Kachoyan, P.J. Lattice methods for multiple integration: theory, error analysis and examples. SIAM Journal on umerical Analysis, 24, (1987). [6] Traub, J.F., Wasilkowski, G. and Wozniakowski, H. Information-based complexity. Academic Press, Boston (1988). 6