Approximation by Piecewise Constants on Convex Partitions

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1 Aroxiation by Piecewise Constants on Convex Partitions Oleg Davydov Noveber 4, 2011 Abstract We show that the saturation order of iecewise constant aroxiation in L nor on convex artitions with N cells is N 2/(d+1), where d is the nuber of variables. This order is achieved for any f W 2 (Ω) on a artition obtained by a sile algorith involving an anisotroic subdivision of a unifor artition. This iroves considerably the aroxiation order N 1/d achievable on isotroic artitions. In addition we show that the saturation order of iecewise linear aroxiation on convex artitions is N 2/d, the sae as on isotroic artitions. 1 Introduction Let Ω be a bounded doain in R d, d 2. Suose that is a artition of Ω into a finite nuber of subdoains ω Ω called cells, such that ω ω = if ω ω, and ω ω = Ω, where ω denotes the Lebesgue easure (d-diensional volue) of ω. A artition is said to be convex if each cell ω is a convex doain. We assue throughout the aer that Ω adits a convex artition. With a slight abuse of notation, we denote by D the cardinality of a finite set D, so that stands for the nuber of cells ω in. Given a function f : Ω R, we are interested in the error bounds for its aroxiation by iecewise olynoials in the sace { { } S n ( ) = q ω χ ω : q ω Π d 1, if x ω, n, χ ω (x) := 0, otherwise, ω Deartent of Matheatics and Statistics, University of Strathclyde, 26 Richond Street, Glasgow G1 1XH, Scotland, UK, oleg.davydov@strath.ac.uk 1

2 where Π d n, n 1, is the sace of olynoials of total degree < n in d variables. The best aroxiation error is easured in the L -nor := L(Ω), Clearly, where E n (f, ) := inf f s, 1. s S n( ) ( 1/ E n (f, ) = ω E n(f) L (ω)) if <, ax ω E n (f) L (ω) if =, E n (f) L(ω) := inf f q L(ω) q Π d n is the error of the best olynoial aroxiation of f on ω. If ω is a bounded convex doain and f ω belongs to the Sobolev sace W n (ω), then the error E n(f) L(ω) is estiated as (1) E n (f) L(ω) C d,n dia n (ω) f W n (ω), (2) where C d,n denotes a ositive constant deending only on d and n [3], and Note that f W n (ω) := n f, α := α x L(ω) α α d for α Z d +. α =n f f ω L(ω) 2E 1 (f) L(ω), f ω := ω 1 ω f(x) dx, see for exale [2], and hence (2) ilies that the Poincaré inequality f f ω L(ω) ρ d dia(ω) f L(ω), f W 1 (ω), (3) holds with a constant ρ d deending only on d when ω is bounded and convex, where ( d f L(ω) := f 2) 1/2 L(ω). x k k=1 Indeed, it is easy to check that f L(ω) is equivalent to the Sobolev seinor f W 1 (ω), as f L(ω) f W 1 (ω) d ax{1 2,1 1 } f L(ω), 1. We refer to use f L(ω) in (3) because this seinor is invariant under orthogonal transforations of the coordinate syste, which silifies soe 2

3 calculations below. It is iortant for the roof of Theore 1 that ρ d does not deend on the geoetry of the doain. It follows fro (2) that for any convex artition, E n (f, ) C d,n dia n ( ) f W n (Ω), dia( ) := ax ω dia(ω). Obviously, dia( ) C 1/d, where C deends only on Ω and d. Hence, in ters of, the aroxiation order that can be obtained fro the last estiate is not better than E n (f, ) = O( n/d ). (4) This order is achieved for exale for Ω = (0, 1) d on convex artitions, = 1, 2,..., defined by slitting the cube (0, 1) d uniforly into = d equal subcubes of edge length 1/. Asytotically otial bounds for the L -error e n (f, ) of the interolation by iecewise olynoials of degree < n on anisotroic triangulations of a olygonal doain in R 2 have been studied in [1, 5]. There, for n 2, an exact constant C n is found such that li inf N N n/2 e n (f, N ) C n as soon as the sequence of triangulations { N } satisfies dia( N ) = O( N 1/2 ). Moreover, a sequence { N } with this roerty exists such that li su N N n/2 e n (f, N ) C n. In [2, Theore 2] we have shown that assuing higher soothness of f does not hel to irove the order E 1 (f, N ) = O( N 1/d ) if the sequence of artitions { N } is isotroic, that is there is a constant c > 0 such that dia(ω) cρ(ω) for all ω N N, where ρ(ω) is the axiu diaeter of d-diensional balls contained in ω. More recisely, if E 1 (f, N ) = o( N 1/d ), N, for a function f C 1 (Ω) and soe isotroic sequence of artitions { N } with li dia( N) = 0, then f is N a constant. Thus, 1/d is the saturation order of the iecewise constant aroxiation on isotroic artitions. In this aer we show that the order of aroxiation by iecewise constants can be iroved to E 1 (f, ) = O( 2/(d+1) ) on suitable anisotroic convex artitions obtained by a sile algorith if f W 2 (Ω), Ω = (0, 1) d (Algorith 1 and Theore 1). Moreover, according to Theore 2, 2/(d+1) is the saturation order of iecewise constant aroxiation in L -nor on convex artitions as it cannot be further iroved for any f C 2 (Ω) whose Hessian is ositive definite at soe oint. Finally, Theore 3 shows that the saturation order of iecewise linear aroxiations on convex artitions is 2/d, that is the sae as on isotroic artitions. In the bivariate case the saturation order N 2/3 has been shown by a different ethod in [4] for suitable sequences of artitions N of (0, 1) 2 into 3

4 olygons with cell boundaries consisting of totally O(N) straight line segents. 2 Otial iecewise constant aroxiation on convex artitions In this section we rovide a sile algorith that generates iecewise constant aroxiations with the aroxiation order 2/(d+1) on convex olyhedral artitions with totally O( ) facets. For the sake of silicity we only consider Ω = (0, 1) d. Algorith 1. Assue f W1 1 (Ω), Ω = (0, 1) d. Slit Ω into N 1 = d cubes ω 1,..., ω N1 of edge length h = 1/. Then slit each ω i into N 2 slices ω ij, j = 1,...,N 2, by equidistant hyerlanes orthogonal to the average gradient g i := ω i 1 ω i f(x) dx on ω i. Set = {ω ij : i = 1,...,N 1, j = 1,..., N 2 }, and define the iecewise constant aroxiation s (f) by s (f) := f ω χ ω, f ω := ω 1 f(x) dx. (5) ω ω Clearly, = N 1 N 2 and each ω ij is a convex olyhedron with at ost 2(d+1) facets. This algorith is illustrated in Fig. 1. Theore 1. Assue that f W 2 (Ω), Ω = (0, 1) d, for soe 1. For any = 1, 2,..., generate the artition by using Algorith 1 with N 1 = d and N 2 =. Then f s (f) C d 2/(d+1) ( f W 1 (Ω) + f W 2 (Ω)), (6) where C d is a constant deending only on d. Proof. We only consider the case < as = can be derived by obvious odifications of the arguents given here. Note that a different roof in the case = can be found in [2]. By construction, N 1 f s (f) = N 2 f f ωij L (ω ij ). i=1 For a fixed i, let {σ 1,..., σ d } be an orthonoral basis of R d such that σ d = g i 1 g i if g i 0, and let ϕ : R d R d be the linear aing defined by 4 j=1

5 Figure 1: Algorith 1 (d = 2, N 2 = = 4). Average gradients g i on the squares ω i are deicted as arrows. The cells ω ij, j = 1,...,4, are shown only for one square. the atrix diag(1,..., 1, N 2 ) with resect to the basis {σ 1,...,σ d }. We set ω ij = ϕ(ω ij ), f = f ϕ 1. Then ω ij = N 2 ω ij, dia( ω ij ) d/, and f f ωij L (ω ij ) = N 1 2 f f ωij L ( ω ij ). Since f ωij = f ωij and ω ij is bounded and convex, (3) shows that f f ωij L( ω ij ) ρ d dia( ω ij ) f L( ω ij ), where ρ d deends only on d. We have f L ( ω ij ) = ( d k=1 ) 1/2 2 D σk f L ( ω ij ) ( = N 2 N2 2 D d 1 ) 1/2 σ d f 2 + D σk f 2 k=1 k=1 L (ω ij ) N 1 2 D σd f L (ω ij ) + N d 1 2 D σk f L (ω ij ), where D σk f = f T σ k denote the directional derivatives of f. Since ω i D σk f(x) dx = 0, k = 1,..., d 1, 5

6 Poincaré inequality (3) also ilies Hence D σk f L(ω i ) ρ d dia(ω i ) D σk f L(ω i ), k = 1,..., d 1. N 2 d 1 j=1 k=1 ( D σk f L (ω ij ) d dρ ) d f W (ω 2 i ). By cobining the above estiates we obtain f s (f) ( dρd ) N 1 2 i=1 N 1 N 2 i=1 ( dρd ) N 1 [ d ( dρd ) d ( dρ d f L ( ω ij ) j=1 ( dρ d ) f W(ω 2 i ) + N 2 N 2 j=1 ) f W 2 (Ω) + ( dρd N 2 ) f W 1 (Ω). D σd f L (ω ij ) Since N 1 = d, N 2 =, we have = d+1, and (6) follows with C d = d 5/2 ρ 2 d. 3 Saturation Orders The ain result of this section is the following theore which, together with Theore 1 shows that the saturation order of iecewise constant aroxiation on convex artitions is 2/(d+1). Theore 2. Assue that f C 2 (Ω) and the Hessian of f is ositive definite at a oint ˆx Ω. Then there is a constant C f,d deending only on f and d such that for any convex artition of Ω, E 1 (f, ) C f,d 2/(d+1). The roof of Theore 2 will be given at the end of the section. It turns out that iecewise linear aroxiations on convex artitions have the saturation order 2/d. Thus, in contrast to iecewise constants, there is no iroveent of the order in coarison to isotroic artitions. Theore 3. Assue that f C 2 (Ω) and the Hessian of f is ositive definite at a oint ˆx Ω. Then there is a constant C f,d deending only on f and d such that for any convex artition of Ω, E 2 (f, ) C f,d 2/d. ] 6

7 Proof. Since f C 2 (Ω), there is δ > 0 and a cube Q Ω such that the sallest eigenvalue of the Hessian of f is at least δ everywhere in Q. Assue that ω has nonety intersection with Q, and let x 1, x 2 ω Q be such that x 1 x dia(ω Q). Since the univariate function 2 g := f [x1,x 2 ] is convex with second derivative at least δ everywhere in [x 1, x 2 ], the error of its best L -aroxiation by (univariate) linear olynoials is δ greater or equal x 16 1 x Indeed, by araetrising g with t [0, 1] and assuing without loss of generality that g(0) = g(1) = 0, we have g (t) δ x 1 x 2 2 t(t 1) 1 2 and g(t) = 2 0 g (τ)m t (τ) dτ t(t 1) δ x 2 1 x 2 2 2, where M t is the Peano kernel of the second divided difference [0, 1, t]. Since g( 1) 2 δ x 8 1 x 2 2 2, Chebyshev theore ilies the clai. Hence, It follows that E 2 (f, ) E 2 (f) L (ω Q) δ 64 dia2 (ω Q). Q µ d 2 d ω Q ( 16 ) d/2e2 dia d (ω Q) µ d (f, ) d/2 δ, where µ d denotes the volue of the d-diensional ball of radius 1. Thus, E 2 (f, ) δ Q 2/d 2/d. 16µ 2/d d Proof of Theore 2. We first choose δ > 0 and a cube Q Ω such that the sallest eigenvalue of the Hessian of f is at least δ everywhere in Q. Clearly, f( x) 0 for soe x Q. Since the gradient of f is continuous, there is a constant γ > 0 and a cube Q Q containing x such that D σ f(x) γ for all x Q, where σ = f( x)/ f( x) 2. We assue without loss of generality that Q = Q. The arguents in the roof of Theore 3 lead to the estiate E 1 (f, ) E 2 (f, ) δ 64 dia2 (ω Q) for any ω with nonety intersection with Q. Moreover, if [x 1, x 2 ] is an interval in ω Q arallel to σ, then f(x 2 ) f(x 1 ) γ x 2 x 1 2, which ilies that E 1 (f, ) γ 2 x 2 x

8 Hence ω Q is contained between two hyerlanes orthogonal to σ, with distance between the not exceeding 2E γ 1(f, ). The enultiate dislay shows that the intersection of ω Q with any interediate hyerlane is contained in a (d 1)-diensional ball of radius ( 64 E ) 1/2 δ 1(f, ). Therefore, we ay estiate the volue of ω Q as which ilies and Theore 2 follows. ω Q 2 γ E 1(f, ) µ d 1 ( 64 δ E 1(f, ) ) (d 1)/2, Q 2µ d 1 γ ( 64 ) (d 1)/2E1 (f, ) (d+1)/2, δ References [1] V. Babenko, Y. Babenko, A. Ligun and A. Shueiko, On asytotical behavior of the otial linear sline interolation error of C 2 functions, East J. Arox, 12(1), 2006, [2] O. Davydov, Algoriths and error bounds for ultivariate iecewise constant aroxiation, in Aroxiation Algoriths for Colex Systes, (E. H. Georgoulis, A. Iske and J. Levesley, Eds.), Sringer Proceedings in Matheatics, Vol. 3, Sringer-Verlag, 2011, [3] S. Dekel and D. Leviatan: The Brable-Hilbert lea for convex doains, SIAM J. Math. Anal. 35, 2004, [4] A. S. Kochurov, Aroxiation by iecewise constant functions on the square, East J. Arox. 1, 1995, [5] J.-M. Mirebeau, Otial eshes for finite eleents of arbitrary order, Contr. Arox. 23, 2010,