Generating dimension formulas for multivariate splines

Transcription

1 Geeatig dimesio fomulas fo multivaiate splies Simo Foucat Tatyaa Sookia Abstact Dimesios of spaces of multivaiate splies emai ukow i geeal. A computatioal method to obtai explicit fomulas fo the dimesio of splie spaces o simplicial patitios is descibed. The method is based o Hilbet seies ad Hilbet polyomials. It is applied to cojectue the dimesio fomulas fo splies o the Alfeld split of a simplex ad o seveal othe patitios. AMS classificatio: Pimay 41A15, Secoday 14Q99 Key wods ad phases: multivaiate splies, simplicial patitio, dimesio, Hilbet seies, Hilbet polyomial, Alfeld split. 1 Itoductio Let deote a simplicial patitio of a polyhedal domai Ω R, so that if ay two simplices i itesect, the thei itesectio is a facet of. The space of C splies of degee d i vaiables o is S d ( := { s C (Ω : s T P d, fo each simplex T }, whee P d, is the space of polyomials of degee d i vaiables. We ae iteested i the dimesio of the space Sd (. Fo fixed d ad, detemiig a closed fomula fo abitay patitios is still a majo ope poblem, eve i the bivaiate case, see [9]. I this case, it is kow [9, p. 40] that if is a shellable (egula with o holes tiagulatio, the ( ( d d 1 (1 dim Sd ( E I V I [( d ( ] v V I σ v, Dexel Uivesity, 141 Chestut Steet, Philadelphia, PA 19104, foucat@math.dexel.edu This autho is suppoted by the NSF gat DMS-1106 Towso Uivesity, 7800 Yok Road, Towso, MD 15, tsookia@towso.edu 1

2 whee E I is the umbe of iteio edges, V I is the umbe of iteio vetices, V I is the set of iteio vetices of, ad d σ v := max{ j 1 jm v, 0}, m v := umbe of diffeet edge slopes meetig at v. j=1 The ight-had side of (1 is the coect expessio fo the dimesio if d 1, see [9, p.47 ad p.7]. Not much is kow fo d, ad it is somewhat staggeig that the dimesios of S 1( ad of S 1( emai ucetai i geeal. Let us poit out that the ight-had side of iequality (1 ca be ewitte as a liea combiatio of biomial coefficiets a fom that is favoed i this pape: ( ( d d 1 dim Sd ( (E I V I V I ( d 1 µ V I ( d 1 ν ( V I ( µ 1 m v, v V I whee 1 1 µ :=, ν :=. I the case of a cell C a tiagulatio with oe iteio vetex v it is kow that the lowe boud is the coect dimesio, amely ( ( ( ( ( d d 1 d 1 µ d 1 ν µ 1 dim Sd (C = (E I ( m, whee m E I is the umbe of diffeet slopes of the E I iteio edges meetig at v. I fact, the fomula fo the cell is the basis of the agumet used to deive (1. This example demostates that the dimesio depeds ot oly o the combiatoics of umbe of vetices, edges, ad othe faces but also o its exact geomety. The poit of view adopted i this pape cosists i fixig the patitio ad lookig fo dimesio fomulas valid fo all d,, ad possibly. The mai expeimetal esult, amely Cojectue 1, coces the splie spaces o the Alfeld split of a sigle simplex. This split is a geealizatio of the Clough Toche split of a tiagle to highe spacial dimesios. The Clough Toche split of a tiagle has oe iteio vetex, thee iteio edges, ad thee subtiagles. The split of a tetahedo with oe iteio vetex, fou iteio edges, six iteio faces, ad fou subtetaheda was itoduced i []. We shall efe to the split of a simplex i R with ( 1 k iteio k-dimesioal faces, 0 k, as the Alfeld split A. The followig is ou cojectue o the dimesio. Cojectue 1. The dimesio of the space Sd (A of splies of degee d i vaiables ove the Alfeld split A of a simplex is give by ( d 1 ( ( 1, if is odd, d dim Sd (A = ( d 1 ( ( 1 d ( 1, if is eve.

3 This fomula was obtaied usig the computatioal method that we itoduce i Sectio. I Sectio, we descibe the steps leadig to Cojectue 1, ad epot without details othe fomulas obtaied via this method fo seveal tetahedal patitios. I Sectio 4, we discuss the potetial of the method. The computatioal method I this sectio, we show how to deive a explicit fomula fo the dimesio of S d (, i the fom of a liea combiatio of biomial coefficiets, usig computed values of this dimesio fo a fiite umbe of paametes ad d. We fist show why the sequece {dim S d ( } depeds oly o a fiite umbe of its values. Let us fo ow fix the umbe of vaiables, the simplicial patitio, ad the smoothess paamete. It is well-kow that the dimesio of Sd ( agees with a polyomial of degee i vaiable d whe d is sufficietly lage. This polyomial is called the Hilbet polyomial, ad it is deoted by H := H, thoughout this pape. We deote by d := d, the smallest itege such that dim Sd ( = H(d fo all d d. The sequece {dim S d ( } is detemied by its fist d 1 values. Ideed, the tems {dim S d (, d d d } defie {dim S d ( } d d by itepolatio of the Hilbet polyomial, while the values {dim S d (, 0 d d 1} complete the fist d tems of the sequece. The estimatio of d emais a key questio. Ou method icopoates the widely accepted assumptio that ( d, 1. This is suggested by the techique of patitioig the miimal detemiig set ito oovelappig subsets associated with each face, see [4]. Moeove, fo the subspace of Sd ( imposig additioal (o supe smoothess j 1 acoss evey j-dimesioal face of, it was show i [5] that the dimesio is ideed a polyomial i d fo d 1. The boud ( is likely to be a oveestimatio, though. The examples of Sectio ad the impoved boud d, obtaied i [8] fo shellable tiagulatios suppots this belief. Reducig the boud would educe the umbe of dimesio values to be computed. Sice splies with degees ot exceedig smoothess ae simply polyomials, we have ( d dim Sd ( = fo d.

4 Thus, assumig (, oly the ( 1 1 values {dim S d (, 1 d 1} ae left to be computed. A additioal savig ca be made by usig the values fo smalle degees, sice we have [ dim S d ( = dim P d, = ( ] d = [ dim S k ( = dim P k, = ( k ] fo k d. Assumig that computig dim S d ( is possible fo ay d 0, the above descibed method gives us access to the whole sequece {dim S d ( }. To obtai a explicit fomula, we ely o the cocept of Hilbet seies, i.e., the geeatig fuctio of the sequece {dim S d ( }. Accodig to [6, Theoem.8], it satisfies ( dim Sd ( z d = P (z (1 z 1, fo some polyomial P := P, with itege coefficiets. Deotig these coefficiets by a k = a k,,, ad deotig the degee of P by k = k,, that is, P (z = a k z k, a k 0, k two futhe paticulas ae established i [6, Theoem 4.5]: (4 P (1 = a k = N, P (1 = ka k = ( 1 F it, k whee N ad F it epeset the umbe of simplices ad iteio facets of, espectively. I the paticula case whe is a sigle simplex, the space Sd ( is just the space P d, of polyomials of degee d i vaiables. The it ca be see that P = 1 fom the idetity (5 ( d z d = k 1 (1 z 1. This idetity is clea fo = 0 ad is iductively obtaied by successive diffeetiatios with espect to z fo 1. While the deivatio of the polyomial P fom the dimesios dim S d ( was staightfowad, idetity (5 covesely povides a explicit fomula fo the dimesios dim S d ( i tems of the coefficiets of P. Ideed, the fomula (6 dim Sd ( k ( d k = a k was isolated i [6] ad it also follows fom dim Sd ( z d = k a k z k (1 z 1 = k 4 ( d a k z dk = k ( d k a k z d

5 by idetifyig the coefficiets i fot of each z d. Takig ito accout that (d k (d k 1 (d k 1, if d k, ( d k! (d k (d k 1 (d k 1 = 0 =, if k d k 1,! 0, if d k 1, we obseve that, fo d k, the dimesio of S d ( agees with the Hilbet polyomial (d k (d k 1 (d k 1 H(d := a k.! k Moeove, fo d = k 1, we have H(k 1 dim S k 1 ( = a k ( ( 1( (! 0 = ( 1 a k 0. The defiitio of d theefoe yields d = k, ad cosequetly, we see that k = d. This was ituitively aticipated because the detemiatio of the sequece {dim Sd ( } equies d 1 pieces of ifomatio while the equivalet detemiatio of the polyomial P equies the k 1 pieces of ifomatio coespodig to its coefficiets. Now we descibe a pactical way to detemie these coefficiets fom the computed values {dim Sd ( } d d=0. It is simply based o the obsevatio that (7 a k = 1 d k P (z k! dz k z=0 = 1 k! = 1 k! = 1 k! = l=0 l=0 d k ( dz k (1 z 1 dim S d ( z d z=0 k ( k d k l ( l dz k l (1 z 1 d l ( z=0 dz l dim Sd ( z d z=0 k l=0 ( k ( 1 k l ( 1! l ( 1 k l! l! dim S l ( k ( 1 ( 1 k l dim Sl k l (. I paticula, the value dim S0 ( = 1 yields a 0 = 1, the the value of dim S1 ( yields a 1, the values of dim S1 ( ad of dim S ( yield a ad so o. This shows that the computatio of the coefficiets a k ca be pefomed sequetially, alog with the computatio of the dimesios dim Sk (. As log as dim Sk ( equals ( k, idetity (5 esues that the coefficiets a k agee with the coefficiets of the costat polyomial P = 1: a 0 = 1, a 1 = 0, a = 0,, a d = 0, 5

6 whee d deotes the lagest itege such that dim Sd ( = ( d. As a matte of fact, applyig (7 to a patitio cosistig of a sigle simplex, we obtai 0 = k l=0 ( 1 k l ( 1 k l We may theefoe also expess the coefficiet a k as (8 a k = l=0 ( l, k 1. k ( 1 ( 1 k l δl k l (, k 1, whee δl ( is the codimesio of the polyomial space P l, i the splie space Sl (, i.e., ( l δl ( := dim Sl (, which is less costly to compute tha the dimesio of S l (. We fially ote that at most mi{, k d } ozeo tems ete the sum i (8, sice the summad is ozeo oly whe l k 1 ad l d 1. The computatioal method descibed above exploits the specific fom of the Hilbet seies. As a coclusio to this sectio, we make the side obsevatio that ( ca be deived by simple meas. It suffices to set u d = dim S d ( i the followig lemma. Lemma 1. Let {u d } be a sequece fo which thee is a polyomial Q of degee m such that u d = Q(d wheeve d d fo some d. The thee exists a polyomial R such that u d z d = Poof. We wite the polyomial Q as Q(d =: of the sequece {u d }, we have u d z d = Q(d z d m 1 = q k (1 z k1 R(z (1 z m1. m (u d Q(d z d = d d=0 ( d k q k. The, fo the geeatig fuctio k m ( d k q k z d k (u d Q(d z d. The latte ideed takes the fom R(z/(1 z 1 fo some polyomial R. d d=0 (u d Q(d z d 6

7 The pevious lemma also eables to epove (4. Ideed, we otice that the lowe ad uppe bouds deived i [] yield ( ( d d 1 dim Sd ( = N ( 1F it O(d. 1 Theefoe, fo d lage eough, the quatity ( ( d d 1 (9 u d := dim Sd ( N ( 1F it 1 educes to a polyomial of degee. The claim implies that, fo some polyomial R, P,(z (1 z 1 N ( 1F it (1 z 1 (1 z = u d z d = Reaagig the latte, we obtai P,(z = N ( 1(1 zf it (1 z R(z, which i tu shows that P,(1 = N ad P, (1 = ( 1F it. R(z (1 z 1. Applicatio of the method to specific patitios I this sectio, we demostate the usefuless of ou computatioal method o seveal specific patitios. We ecall that ou method elies o the computatio of dim S d ( fo fixed d,, ad. This step was pefomed usig the iteactive applet [1] fo =, ad othe codes i Java ad Fota fo >, all witte by Pete Alfeld. Alfeld split of a simplex. We ecall that the split of a simplex A i R with ( 1 k iteio k-dimesioal faces, 0 k, is the Alfeld split of A. Fo =, Theoem 9. i [9] yields ( d dim Sd (A = ( d 1 µ ( d 1 ν, µ := 1, ν := 1 Fo = ad = 0, 1,,, we wee able to compute eough values of the dimesios to deive the sequece a ( := (a ( 0, a( 1, a(,... of coefficiets of the polyomial P A, with cetaity. We obtaied a (0 = (1, 1, 1, 1, 0,... a (1 = (1, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0,... a ( = (1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... a ( = (1, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

8 Fo 4, the dimesios we could compute yielded the stat of the sequece a ( with cetaity, but we caot be totally sue that all ozeo coefficiets have bee foud. We obtaied a (4 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... a (5 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... a (6 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0,... a (7 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0,... a (8 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,.... Ispectio of the sequeces a (1,..., a (8 stogly suggests the patte of ozeo coefficiets a ( 1 = a( = a( = 1 fo eve, a( = fo odd. Fo = 4, 5, 6, we wee also able to compute some values of the dimesios fo the space of C -splies of degee d ove the Alfeld split A. These ivestigatios lead us to the followig cojectue: ( d 1 ( ( 1 d (10 dim Sd (A = ( d 1 ( 1 ( d, if is odd, ( 1, if is eve. Let us ote that fo eve values of the fomula ca be expessed diffeetly sice ( d ( 1 j = j=1 ( d ( 1 1 ( d ( 1. 1 The esult ca be equivaletly fomulated via the polyomial P A, of ( as 1 z 1, odd, P A,(z= 1 z (1j, eve. j=1 We epot below futhe cojectues poduced fom ou method. Desciptios ad illustatios of the coespodig tetahedal patitios ae available i Alfeld s applet meu, see [1]. Type-I split of a cube (B I. This patitio of a cube cosists of six tetaheda, all shaig oe mai diagoal of the cube. This diagoal is the oly iteio edge of the patitio. Thee ae o iteio split poits. Type-I split has 6 iteio tiagula faces, ad 18 bouday edges 8

9 compised of 1 edges of the cube ad six diagoals of its faces. Based o computatios fo 8, we cojectue that ( d ( (, odd, d d ( 1 dim Sd (B I= ( d ( d 4, eve. Wosey Fai split of a tetahedo (W F. This patitio is a efiemet of the Alfeld split A of a tetahedo obtaied by applyig the Clough Toche split A to each face of the tetahedo. The Wosey Fai split cosists of 1 subtetaheda meetig at oe iteio poit. This patitio has 18 iteio tiagula faces ad 8 iteio edges. Based o computatios fo 8, we cojectue that ( d ( d dim Sd (W F = 8 ( d 4 4 ( d 4 ( d ( ( ( d ( 1 d (, odd, ( d (, eve. Geeic octahedo (OCT. This patitio of a octahedo cosists of eight tetaheda meetig at oe iteio split poit. This split poit caot be colliea with ay two vetices of the octahedo. Thee ae 1 iteio tiagula faces ad 6 iteio edges i this patitio. Based o computatios fo 8, we cojectue that dim S d (OCT = ( d ( d ( 1 ( 1 ( d ( 1 ( 7 ( d ( ( d ( ( 1 ( 1 ( d ( ( d (, = mod,, othewise. Geeic 8-cell (C 8. The easiest way to visualize this patitio is to stat with a efiemet of the Alfeld split A of a tetahedo obtaied by applyig the Clough Toche split A to 9

10 two faces of the tetahedo. Let us deote the ew split poits o the face u ad v. This patitio cosists of 8 subtetaheda meetig at oe iteio poit. Note that the vetices u ad v ca be moved to the exteio of the oigial tetahedo without chagig the topology of the patitio. This pocess esults i a patitio that has the same umbe of iteio ad bouday faces, edges, ad vetices as the octahedal patitio descibed above. Howeve, coectivities of the faces ae diffeet. Fo example, each iteio edge of the octahedal split is shaed by exactly fou tetaheda. I the 8-cell, two iteio edges ae shaed by five tetaheda, aothe two ae shaed by fou tetaheda, ad the emaiig two edges ae shaed by thee tetaheda. Based o computatios fo 8, we cojectue that, fo, ( d dim Sd (C 8= ( d ( 1 ( d ( 1 ( 1 ( 9 ( 7 ( d ( ( d ( ( d ( ( d ( We ote that the cases = 0 ad = 1 do ot follow the geeal patte., odd,, eve. 4 Discussio Towads theoetical impovemets. The mai shotcomigs of ou method ae its high complexity ad limited eliability. Complexity. At peset, we eed to compute a expoetial i umbe of values of splie dimesios. As iceases, the cost of computig each dimesio goes up. This quickly becomes pohibitive. Oe way to esolve this issue is to lowe the boud o d. Ideally, it would be a dop fom 1 dow to a quatity that is liea i. Such estimate o the lowe boud o d is suppoted by seveal obsevatios. If =, fo shellable tiagulatios, we have d. Whe =, easoably low values of d ca be ifeed fo the examples of Sectio. We also obseved liea behavio i of d fo the Alfled splits. Oe ca also evisio that futhe theoetical ifomatio will help to educe the umbe of computatios. Fo istace, if a specific patitio is kow to yield oegative coefficiets a k, the we ca stop computig dim Sd ( as soo as the coditios d a k = F ad d ka k = (1F 1 it ae satisfied. Reliability. Eve if all ecessay values of dim Sd ( ae available fo a fixed, the fomula we deduce is oly valid fo this fixed. At peset, the fomula we ife fo all values of elies o a plausible guess. Some theoetical ifomatio o the type of depedece of dim Sd ( o would be decisive i this espect. The esults of Sectio suggest depedece o the paity 10

11 of, sometimes depedece o divisibility of by, ad occasioally the pedicted depedece is ot valid fo smalle values of. Towads computatioal impovemets. To compute the dimesio of S d (, Alfeld s codes taslate the set of smoothess coditios ito a liea system fo the Bestei Bézie coefficiets, the the matix of the system is educed by Gaussia elimiatio, ad its ak is detemied. It may be possible to fid faste alteatives. The discussio i [7] hits at a pactical method usig Göbe bases. Additioally, whe computig dim S d (, it should be possible to use the kowledge of the dimesios of the spaces with lowe degee ad smoothess, sice the values {dim S d (, 0 d d } ae detemied sequetially. Fially, to deduce the coefficiets a k, it may be sesible to compute oly the quatities δ d ( appeaig i (8, o some suitable liea combiatios of {dim S d (, 0 d d }. This latte appoach could take advatage of the fact that the sequece {a k } appeas to have oly few ozeo tems. A optimistic fial pespective. Should the theoetical ad computatioal impovemets mateialize, a stad-aloe pogam fo the explicit detemiatio of the dimesios ought to be implemeted. With mode (o futue computatioal powe, the dimesio fomulas could be obtaied fo a wide vaiety of patitios. It is ot uealistic that some expessios fo the coefficiets a k could the be ifeed i tems of the smoothess, the combiatoial paametes, ad othe topological paametes especially i the geeic case whee the geomety does ot play a ole. Ackowledgemets. We thak Pete Alfeld fo the essetial computatioal tools he ceated ad fo the elighteig thoughts he shaed with us. Refeeces [1] P. Alfeld, pa/dmds/ [] P. Alfeld, A tivaiate Clough Toche scheme fo tetahedal data, Comput. Aided Geom. Desig 1 (1984, [] P. Alfeld, Uppe ad lowe bouds o the dimesio of multivaiate splie spaces, SIAM J. Nume. Aal. (1996, [4] P. Alfeld, L. L. Schumake, ad M. Sivet, O dimesio ad existece of local bases fo multivaiate splie spaces, J. Appox. Theoy 70 (199, [5] P. Alfeld ad M. Sivet, A ecusio fomula fo the dimesio of supesplie spaces of smoothess ad degee d > k, Appoximatio Theoy V, Poceedigs of the Obewolfach Meetig (1989, W. Schempp ad K. Zelle (eds, Bikhäuse Velag,

12 [6] L. J. Billea ad L. L. Rose, A dimesio seies fo multivaiate splies, Discete ad Computatioal Geomety 6 (1991, [7] L. J. Billea ad L. L. Rose, Göbe basis methods fo multivaiate splies, i: Mathematical methods i compute aided geometic desig, T. Lyche ad L. L. Schumake (eds., 1989, [8] D. Hog, Spaces of bivaiate splie fuctios ove tiagulatio, Appox. Theoy Appl. 7 (1991, [9] M.-J. Lai ad L. L. Schumake, Splie fuctios o tiagulatios, Cambidge Uivesity Pess, Cambidge,