THE GIBBS PHENOMENON FOR RADIAL BASIS FUNCTIONS

Transcription

1 THE GIBBS PHENOMENON FOR RADIAL BASIS FUNCTIONS BENGT FORNBERG AND NATASHA FLYER Abstract What is now known as the Gibbs phenomenon was first observed in the contet of truncated Fourier epansions, but other versions of it arise also in situations such as truncated integral transforms and for different interpolation methods Radial basis functions(rbf) is a modern interpolation technique which includes both splines and trigonometric interpolations as special cases in -D, and it generalizes these methodologies to scattered node layouts in any number of dimensions We investigate here the Gibbs phenomenon for -D RBF interpolation, and find that it can differ also qualitatively from previously studied cases Key words Radial basis functions, RBF, Gibbs phenomenon, interpolation Introduction The best known version of the Gibbs phenomenon is the overshoot that arises when a discontinuous function is represented by a truncated set of Fourier epansion terms A similar situation arises if a truncated Fourier epansion is instead obtained by means of interpolation on an equispaced grid Figures a,b showmoredetailedpicturesnearaunitheightjumpinthesetwocases Inthelimits of increasingly many terms and of increasingly high node densities, respectively, eact formulas for the peak heights are + π π sin t dt 89 t and ma <ξ< { sin πξ π k= } ( ) k 4, ξ k ie the overshoots amount to approimately 89% and 4%, respectively, of the jumpheights Bothofthesecasesareanalyzedin[8],Section4;seealso[8],[9] The amplitudes of successive oscillations decay in inverse proportion with the distance from the jump AsFigure(c)shows,theGibbsphenomenonforsplinesdiffersfromtheprevious cases especially in terms of how fast the oscillations decay This is discussed further in Section We then, in Section 3, generalize splines to what has become known as radial basis functions (RBF) This is a modern and etremely fleible interpolation method, which includes both splines and trigonometric interpolants in different -D special cases, but which applies immediately also when the data is given at scattered node locations in any number of space dimensions The Gibbs phenomenon that arises for RBF interpolation has not been studied previously We find in Section 4 that it can take quite different character in different situations Section contains some concluding remarks University of Colorado, Department of Applied Mathematics, 6 UCB, Boulder, CO 839, USA (fornberg@coloradoedu) National Center for Atmospheric Research, Division of Scientific Computing, 8 Table Mesa Drive, Boulder, CO 83, USA (flyer@ucaredu)

2 a Truncated Fourier series 89 b Equispaced Fourier interpolation 4 c Cubic spline interpolation F The Gibbs phenomenon for (a) truncated Fourier series, (b) equispaced Fourier interpolation, and (c) cubic spline interpolation For (b) and (c), the nodes are located at the integers, with function value zero at positive integers, else one The Gibbs phenomenon for spline interpolations In the case of cubic splines, one can readily write down the cubic polynomials for each interval in closed form[9] InthecaseshowninFigurec,themaimumappearson[-,],withthe cubic over this interval given by +( )+(3 3 3 ) +(3 3) 3 The maimum value of 6 (8 3 6) 78 occurs at the location = 6 ( ) 384 The cubic B-spline takes, at successive nodes, the values {, 6, 3, 6,} Given thattheb-splineepansioncoefficientsb k for k inthecaseofthestepdatain Figure (c) will need to satisfy the linear recursion relation 6 b k + 3 b k+ 6 b k+= withcharacteristicequation 6 r + 3 r+ 6 =,itfollowsthattheb-splinecoefficients - and therefore the Gibbs oscillations- will decay for increasing k at the eponential rate b k =O(( 3) k ) O(e 37 k ) This fast rate may not be surprising given that the cubic spline, of all possible interpolants s() to the given data, minimizes [s ()] d Persistent oscillations

3 Spline Maimum value Oscillation decay rate degree O(e c k ); c= T Trends in the Gibbs phenomenon for spline interpolation of increasing orders cannot be present at increasing distances from the jump location because such would make this integral quantity large In the case of splines of higher orders, closed form epressions for the overshoot and for the decay rates become more involved, but the quantities are nevertheless easy to compute numerically Table summarizes some further cases In the limit of increasing spline orders, we can see how the trigonometric interpolation case is recovered (as also follows from the fact that spline interpolants of increasing odd orderstocardinal datas()= { if = otherwise, when Z approach sin π π uniformly in ; see for e[4]) The eponential coefficient decay is lost, and the slow algebraic rate has instead become dominant 3 Radial basis functions The RBF methodology was originated by Rolland Hardy around 97 in connection with a cartography application that required multivariate scattered-node interpolation [7] In a much noted 98 survey [6], this approach, using a certain type of basis functions known as multiquadrics(mq), was foundtobethepreferableoneofabout3thenknownmethods(scoringthebestin3 outof8tests,andsecondbestin3oftheremainingtests) Althoughunconditional non-singularity of the interpolation problem was known early in a very special case [], it was the breakthrough discovery in 986 of guaranteed nonsingularity also for MQs[]whichpropelledthedevelopmentofRBFintooneofthemostactiveareas in modern computational mathematics In the brief RBF introduction below (following a more etensive description in []), we first introduce RBF as a generalization of standard cubic splines to multiple dimensions The types of RBF we are most interested in depend on a scalar shape parameter ε The limit of ε (basis functions becoming increasingly flat) is of special interest, since the accuracy then can become particularly high, with both polynomial and trigonometric interpolants arising as special cases[6],[9],[3],[4] 3 Introduction to RBF via cubic splines Acubicsplineismadeupof a different cubic polynomial between each pair of adjacent node points(in -D), and itmayatthesenodepointsfeatureajumpinthethird derivative(ie thefunction, and its first two derivatives are continuous everywhere) The standard approach for computing the coefficients of the different cubics which form the spline requires only the solution of a tridiagonal linear system If the spacing between the sample points is 3

4 h,itiswellknownthatthesizeoftheerror,wheninterpolatingsmoothfunctions,will decreaselikeo(h 4 ) Itmakesalmostnodifferenceinthealgorithmifthenodesare not equally spaced However, simple generalizations to more space dimensions have inthepastbeenavailableonlyifthenodesarelinedupinthecoordinatedirections Another way to approach the problem of finding the -D cubic spline (for now omitting to address the issue of end conditions) is the following: At each data location i,i=,,n,placeatranslateofthefunctionφ()= 3,ie at i thefunction φ( i )= i 3 Wethenaskifitispossibletoformalinearcombinationofall these functions n s()= λ i φ( i ) (3) i= such that this takes the desired function values f i at the data locations i, i =,,n,ie suchthat s( i )=f i holds Thisamountstoaskingforthecoefficients λ i tosatisfyalinearsystemofequations φ( ) φ( ) φ( n ) φ( ) φ( ) φ( n ) φ( n ) φ( n ) φ( n n ) λ λ λ n = f f f n (3) Assuming that this system is non-singular, it can be solved for the coefficients λ i The interpolant s(), as given by (3), will then become a cubic function between the nodes and, at the nodes, have a jump in the third derivative We have thus found another way to create an interpolating cubic spline This time, we have arrived atafull linear system rather thanatridiagonal one However, as we will see net, this formulation opens up powerful opportunities for generalizing the form of the interpolant, and also for etending the methodology to scattered data in any number of space dimensions 3 Generalization to multiple dimensions Figure 3a illustrates the RBF ideain-dateachdatalocation i,wehavecenteredatranslateofoursymmetric functionφ() In-D,asillustratedinFigure3b,weinsteadusearotatedversionof the same radial function In d dimensions, we can write these rotated basis functions asφ( i ),where denotesthestandardeuclideannorm TheformoftheRBF interpolantandofthelinearsystemthatistobesolvedhashardlychangedfromthe -Dcase Insteadof(3)and(3),wenowuseasinterpolant n s()= λ i φ( i ) (33) with the collocation conditions φ( ) φ( ) φ( n ) φ( ) φ( ) φ( n ) φ( n ) φ( n ) φ( n n ) i= λ λ λ n = f f f n (34) In particular, we note that the algebraic compleity of the interpolation problem has notincreasedwiththenumberofdimensions-wewillalwaysendupwithasquare symmetricsystemofthesamesizeasthenumberofdatapoints Cubicsplineshave thus been generalized to apply also to scattered data in any number of dimensions 4

5 F 3 Illustration of the RBF concept in -D and in -D 33 Differenttypesofradialfunctions TheerrorO(h 4 )forcubicsplinesin -DwillbecomeO(h 6 )inthecaseofquinticsplines AnditfallstoO(h )forlinear splines (corresponding to φ() = ) In general, if we take the RBF approach as outlinedabove,anduseφ()= m+,theerrorwillbecomeo(h m+ )(evenpowers in φ() will not work; for e if φ() =, the interpolant (3) will reduce to a single quadratic polynomial, no matter the value of n, and attempting to interpolate morethanthreepointswillhavetogiverisetoasingularsystem) Thesizesofthese errorscorresponddirectlytowhichderivativeofφ()isitthatfeaturesajumpatthe origin This leads to the obvious question: why not choose a φ() which is infinitely differentiable everywhere, suchas φ()= +? This idea isanecellent one - andcanbeappliedtogoodadvantageinanynumberofdimensions Ifwestillignore boundary issues(possibly leading to some counterpart of the Runge phenomenon for polynomials), the accuracy will become spectral: better than any polynomial order, andgenerallyoftheformo(e cn ),wherec>andnisthenumberofpoints[6] Table 3 lists a number of possible choices of radial functions, with illustrations infigure3 Inthepiecewisesmoothcategory(wherewesofarhavediscussedonly φ(r)= r m+ ),weincludenowalsothinplatesplinesφ(r)= r m ln r Theseare commonlyusedin-d,especiallythenwithm= Justlikethenaturalcubicspline (ie withendconditionss (a)=s (b)=)minimizes b a [s ()] doverallpossible -D interpolants, the RBF interpolant using φ(r) = r logr achieves the equivalent minimization for -D scattered data[7] The spectral accuracy noted above for smooth radial functions holds in any num-

6 MN φ(r)=r 3 Piecewise smooth 3 TPS φ(r)=r log r - 3 Infinitely smooth MQ φ(r)=(+r ) / IMQ φ(r)=(+r ) -/ IQ φ(r)=(+ r ) - GA φ(r)=e r F 3 Illustrations of the si different radial functions in Table 3 ber of spatial dimensions For the non-smooth RBF, the order accuracy actually improves with the number of dimensions, d For eample, for φ(r) = r m+ and for φ(r) = r m ln r, the order of accuracy becomes O(h m++d ) and O(h m+d ) respectively[3] For the infinitely smooth RBF in Table 3, we have also introduced a shape parameter,whichwedenotebyε Forsmallvaluesofε,thebasisfunctionsbecome very flat, andfor large values, they become sharply spiked(eg for IQ, IMQ, GA) or, inthecaseofmq,itapproachesthepiecewiselinearcase Table3alsoshows the Fourier transform of the radial functions(or generalized Fourier transform if the integral that defines the regular version would be divergent, cf [], [], []; we û(ξ)eiξ dξ, û(ξ)= u()e iξ d) The use here the conventionu()= π following are some key theorems regarding RBF interpolation For proofs and further discussions,seefore[3],[6],[],[]: AllthesmoothRBFchoiceslistedinTable3willgivecoefficientmatrices Ain(34)whicharenonsingular,ie thereisauniqueinterpolantoftheform (33) no matter how the (distinct) data points are scattered in any number ofspacedimensions InthecasesofIQ,IMQandGA, thematriispositive definite and, for MQ, it has one positive eigenvalue and the remaining ones all negative Interpolation using MN and TPS can become singular in multi-dimensions However, low degree polynomials can be added to the RBF interpolant to guarantee that the interpolation matri is positive definite(a stronger condition than non-singularity) For eample, for cubic RBF and TPS in d dimensions,thisbecomesthecaseifweuseasinterpolants()= d+ k= γ kp k ()+ 6

7 Type of radial function Piecewise smooth -D Fourier transform φ(ξ) MN monomial r m+ ( ) m+ (m+)! 8π TPS thinplatespline r m ln r Infinitely smooth MQ multiquadric IQ inverse quadratic IMQ inverse MQ +(εr) +(εr) +(εr) ξ m+ ( ) m+ (m)! π 3 ξ m+ ( ) K ξ ε ξ π ε ε K ξ e ε ( ξ ε π GA Gaussian e (εr) ( ) π πξ SH Sech sech(εr) ε sech ε ) ε e ξ /(4ε ) T 3 Definition and Fourier transforms for some cases of radial functions n k= λ k φ( k )togetherwiththeconstraints n j= λ jp k ( j )=,k=,,d+ Here, p k () denotes a basis for polynomials of degree one in d dimensions, ie in the case of d = 3 (with = (,, 3 )), we have p =, p =, p 3 =, p 4 = 3 In -D, the RBF interpolant in the limit of ε converges to Lagrange interpolation polynomial[6](no matter how the distinct nodes are scattered) For -D periodic data, this same limit reproduces standard trigonometric interpolants (again, for all node distributions) For scattered nodes on a sphere, it reproduces a spherical harmonics interpolant[] 4 TheGibbsphenomenonforRBF AlthoughthemainpointinusingRBF is their fleibility in allowing smooth mesh-free interpolation of scattered data in any number of dimensions, we limit ourselves in this first study of the Gibbs phenomenon for RBF interpolants to -D unit-spaced data For each of the radial functions in Table 3, we can introduce a π-periodic function Ξ(ξ)= k= φ(k)e ikξ = j= φ(ξ+πj) (4) The second sum above, following from Poisson s summation formula, will converge alsointhecaseswherethefirstonediverges As a key step towards analyzing the Gibbs phenomenon, we consider cardinal data, defined at the integer lattice points as f = and f k = at = k non-zero integer It has previously been shown[4],[] that the corresponding RBF epansion 7

8 coefficients then become λ k = π π and the RBF cardinal interpolant becomes coskξ Ξ(ξ) dξ (4) s C ()= π φ(ξ) cosξ Ξ(ξ) dξ (43) Derivations of these formulas are given in Appendi A If we, in place of cardinal data,considerstep(gibbs-)data(f k =at=knon-positiveintegerandf k =at = k positive integer), adding translates of(43) gives the Gibbs interpolant as s G ()= Substituting(43) into(44) leads to s G ()= 4π s C (+j) (44) j= φ(ξ) sin( )ξ Ξ(ξ) sin ξ dξ (4) Incontrastto(43)and(44),thissimplifiedepressionisonlyvalidincasesforwhich Ξ(ξ)sin ξ is everywhere non-zero, (eg MN, TPS and MQ) The main task of the presentstudyistoanalyzeanddisplays G () Ourstartingpointwillbetoinvestigate howλ k varieswithk,asdescribedby(4) Thesimilaritybetweentheintegrals(4) and(43)willthenpermittheessentialobservationsforλ k tobecarriedover,firstto s C ()andthenby(44)tos G () Thenetsubsectionsdescribethesestepsinmore detail 4 Analysis of cardinal epansion coefficients λ k Since this task was carriedoutinsomedetailin[],welimitourselvesheretoeplainingthemainideas andresults fromthatstudy Forsome types of RBF, itturnsoutto be possible to etensively simplify the epressions that are obtained from combining the Fourier transforms from Table 3 with(4) and(4) For eample: CubicRBF(MNwithj=;sameascubicsplines): λ = 4+3 3, λ ± = and λ ±k = ( )k 3 3 (+ 3) k, k IQ: λ k = ( )k ε sinh( π ε ) π cos kω π cosh(ω/ε) dω, GA: λ k = e(εk) j=k ( )j e ε (j+ ) j= ( )j (j+ (j+ ), )e ε SH: λ k = j= ( )j sech (εj) ( )k sech(εk) Instead of pursuing more simplifications of this type, it turns out that we obtain more insights by evaluating the integral in (4) by means of Cauchy s theorem and the calculus of residues The case of MQ is used below to illustrate this approach 8

9 F 4 Magnitude of h(ξ), as given by (46) over the domain Reξ π, 8 Imξ 8 4 Cardinal data epansion coefficients by contour integration We considermqrbfandchooseforsimplicityε=thetaskbecomestoevaluate where λ k = 4π h(ξ)= π h(ξ)e ikξ dξ j= K ( πj+ξ ) πj+ξ Thefunctionh(ξ)isπ-periodic,andcanoverξ [,π]bewritten,withouttaking magnitudes, as h(ξ)= K (πj+ξ) j= πj+ξ + (46) K (πj ξ) j= πj ξ In this latter form, h(ξ) can be etended as a single-valued analytic function throughout the strip Reξ π, < Imξ < Figure 4 illustrates the magnitude of this function, and Figure 4 shows its schematic character We change the integration path as is indicated in Figure 4, and note that the two leading contributions to the integral, when k increases, will come from (i) the first pole and (ii) from the (non-cancelling) contributions from the vicinities of the branch points at ξ = and ξ = π Along the line ξ = π+i t, t real, the function h(ξ) is purely real and /h(ξ) features decaying oscillations The first pole of h(ξ) appears near π+69iandhasaresidueofapproimately-34866,thuscontributingaterm of 7433( ) k e 6k toλ k Thesingularityofh(ξ)aroundtheorigin(repeated at ξ =π) comes fromonly one term in the denominator of (46), taking the form 9

10 F 4 Character of the function h(ξ) in the comple plane The original and the modified integration paths are shown ξ K (ξ) =ξ +( 4 γ + ln lnξ )ξ4 +Thebranchsingularityistoleadingorder oftheform ξ4 lnξ= ξ4 (ln ξ +i argξ)(andsimilarlyaroundξ=π) What doesnotcancelbetweenthetwosidesofthecontourbutinsteadaddsup(hencethe factorbelow)amountsto( 4π ) i {some δ>} ( )ξ4 i π e ikξ dξ Lettingξ=it andnotingthat,ask,wecanchangetheupperintegrationlimittoinfinity,this simplifiesto 8 t4 e kt dt= 3 k Forincreasingk,wethusobtain λ k 7433( ) k e 6k + }{{} eponential part 3 k + }{{} algebraic part (47) Figures 43a,b compare, using log-linear and log-log scales respectively, the true values for λ k againstthe-termapproimationin(47) Theagreementisseentobenearly perfect Thesameprocedureasabovecanbecarriedthroughforanyvalueoftheshape parameterεandalsoforalloftherbftypeslistedintable3 Therewillinevery case be an eponential decay process, featuring oscillations in sign It will depend onthe regularity of φ(ξ)atξ=(branchpointor not whencontinuedto comple ξ) whether there will also be an algebraic non-oscillatory decay present If this is present,itwilldominatewhenkissufficientlylarge Giventheepressionsfor φ(ξ) intable3,wecanseethatλ k willdecayeponentiallyforallkinthecasesofmn (confirming what we obtained earlier when we considered MN splines in Section ), GAandSH,whereastherewillbeatransitionfromeponentialtoalgebraicdecayin thecasesoftps,mq,iqandimq 4 Cardinaldatainterpolants s C () Theformulasforλ k (4)ands C () (43)differonlyinafewrespects: A trivial multiplicative factor, Thefreeparameteriscalledinsteadofk(andwewill consideritalsofor non-integer values),

11 λ k λ k k - k F 43 Comparison between correct values of λ k for MQ in -D, ε = (dots) and the -term asymptotic formula (47) (solid line) The subplot to the left is log-linear and the one to the right of type log-log F 44 Original and modified integration contours for evaluating (a) the integral (4) for λ k and (b) the integral (43) for s C () Thereisanetrafactor φ(ξ)inthenumeratorof(43), Theintegrationintervalis[, ]insteadof[,π] Inthesamewayaswedeformedthecontourfortheintegral(4),aswasshownin Figure 4, and is again shown more schematically still in Figure 44a, we now deform thecontourfor(43)asisshowninfigure44b SinceΞ(ξ)isπ-periodic,thepoleswill in the two cases have the same imaginary parts, and therefore the eponential decay rateswillbethesamefors C ()aswasfoundforλ k Thesingularityattheoriginwill becancelledbythefactor φ(ξ)inthenumeratorof(43),butthecontributionsfrom othermultiplesofπwillnotbecancelled,soalgebraicdecayrates(ifatallpresent) willalsobethesame(toleadingorder)forλ k ands C () WeillustratethegeneralobservationsabovewiththecaseofIQInthiscase,it transpires that we can simplify(43) to sinh π ε s C ()= sinπ πε(cosh π ε cosπ) π cosξ cosh ( ξ ε ) dξ

12 F 4 Cardinal interpolant s C () in the case of IQ, ε =, shown over the intervals (a) [,4] and (b) [4,] (now a finite integration interval and no Poisson sum) Figure 4 illustrates how this cardinal data interpolant at first decays in an oscillatory manner at an eponential rate, followed by algebraic decay without changes of sign, entirely as predicted by the general argument above 43 Gibbs data interpolant s G () Considering the fast decay of cardinal RBF interpolants, it is clear that superposing translates of these according to(44) willgiveresultsfors G ()whicharequalitativelythesameasthoseforthecardinal interpolants C () Withhelpoftheformulas(43),(44),or(4),onecanreadilycomputetheGibbs interpolants for any radial function Figures 46a-c show the Gibbs oscillations in a number of cases In accordance with the analysis, we can note that the oscillations decayeponentiallyforalldistancesincaseswhen φ(ξ) isanalyticaroundtheorigin (here shown only in the case of GA), but otherwise there will at some distance be a transition to one-sided oscillations which decay at a slower algebraic rate For the infinitelysmoothrbf,itisalsoofinteresttoseehowthegibbsphenomenonvaries with the shape parameter ε As ε, the oscillations seen in Figure 47 c (MQ, ε = ) increasingly resemble the trigonometric interpolation case shown in Figure b The transition point between eponential and algebraic decay, visible around =4inthecaseofε=(Figure47a)andaround=6forε=(Figure47b) hasintheε=casemovedtoofarouttobevisibleincomputationscarriedoutin standard 6-digit numerical precision In this limit, the eponential decay has itself slowed up, and turned into the slow algebraic one of trigonometric interpolation 44 Eamples of Gibbs and Runge phenomena on a finite interval In practical applications, RBF are almost always used on finite domains and with irregularly placed nodes In contrast, analysis is most easily carried out- as we have done above- on equispaced infinite(or periodic) lattices A key question that always must be asked following such analysis is to what etent the main observations will carry over to more practical situations We will here focus this discussion on the test caseofinterpolatingf()=arctanover [,] When using for eample splines or finite elements, one can increase local resolution in areas of steep gradients by local insertion of additional nodes Given that RBF also allow for arbitrary node locations, one might then attempt a similar strategy As Figure 48 shows, this can trigger oscillations away from the gradient area, that

13 a TPS b IQ, ε = c GA, ε = F 46 The Gibbs oscillations around a jump, and further out to the right, in the cases of (a) TPS, (b) IQ, ε=, and (c) GA, ε= even might grow(rather than decay) with increasing distance The issue is discussed insomedetailin[] Itisfoundtherethattheeffectisbetterseenasananalogto the polynomial Runge phenomenon, and that care in selecting node locations or use ofspatiallyvariableshapeparameterε(usingε k atnode k )cannotonlyovercome the oscillations, but in fact offer spectacular accuracies This is illustrated in Figure 49 Bringing the nodes more smoothly towards the center(with the two center ones very close), together with a larger value of ε, reduces the error by two orders of magnitude The new error pattern would appear to be related to the (surprising) one-sided Gibbs oscillation pattern that emerged from our analysis, and which was notable inthe toptworows of subplots infigure 47 Nearly two further orders of magnitude can be gained when allowing spatially variable ε It can easily be verified that Chebyshev interpolation(polynomial interpolation at the locations of Chebyshev polynomial etrema; a standard procedure for non-periodic pseudospectral methods) will require n=7 nodes to achieve the same 3 accuracy as MQ RBF here didwithn=nodes The optimizations used to generate the data for Figure 49 were carried out using Matlab sga (geneticalgorithm)toolbo Whilethisisnotcertaintogive thetrue optima, nor is practical to use on a routine basis, it suffices very well for finding simple principles that characterize particularly accurate RBF approimations (such aschoosingε k largewherethenodesareclosetogether) TheeamplesshowninFigure49weredesignedtoillustratehowRBFcanprovide very high accuracy also in steep gradient situations Several aspects of the 3

14 a MQ, ε = b MQ, ε = c MQ, ε = F 47 The Gibbs oscillations for MQ in the case of different values of the spape parameter (a) ε=, (b) ε=, and (c) ε= (a) 4 nodes -4 - (b) 6 nodes -4 - (c) 8 nodes F 48 MQ ε = interpolants to f() = arctan over [,]: (a) 4 equispaced nodes, (b) with two etra nodes inserted near the center, and (c) with still two more nodes inserted equispaced infinite lattice analysis for the Gibbs phenomenon can be observed, such asoscillationsinsomecasesemanatingfromthe jump,andinsomecasestheonesided oscillation pattern This simple eample illustrates that opportunities eist - mostly still uneplored - to overcome both Gibbs phenomena and related Runge phenomena Conclusions There is much in common between the Gibbs phenomenon for RBF interpolation and for other interpolation types(such as splines and trigonometric 4

15 Equi-spaced k ; Optimize ε Interpolant Error ε values - - Optimize k and ε Optimize k and ε k F 49 Ten-node MQ interpolations of f() = arctan Top row: Equispaced nodes, ε (same at all nodes) optimized Middle row: Node locations k and ε(same at all nodes) optimized Bottom row: Both k and ε k optimized functions), but there are also significant differences, especially in terms of how the oscillations decay away from the jump In RBF cases, both eponential and algebraic decays can be present and, if so, they will dominate at differentdistances from the jump discontinuity Furthermore, oscillations without sign changes have not been seen previously in connection with the Gibbs phenomenon The possibility of using a spatially variable shape parameter in RBF interpolation, as eplored in[], offers ecellent opportunities for eliminating the adverse effects of the Gibbs phenomenon 6 Acknowledgements The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation Natasha Flyer was supported by NSF Grant ATM-668 The work of Bengt Fornberg was supported by the NSF Grants DMS-3983, DMS-668 and ATM-668 The present study builds on results for cardinal epansion coefficient decay rates, described in[] We acknowledge gratefully many discussions with Susan(Tyger) Hovde, Cécile Piret and Julia Zuev Appendi A This appendi consists of three parts: Derivation of(4) Derivation of(43)

16 3 Verifyingthatthecardinalinterpolants C (),asgivenby(43),indeedsatisfies { n= s C (n)= n=±,±, () Derivation of (4) In terms of the π-periodic function Λ(ξ)= k= and Ξ(ξ), as defined in(4), the cardinal coefficient relationship k= λ k φ(n k)= λ k e ikξ, () { n= n, n Z can be epressedas Λ(ξ) Ξ(ξ)= Therefore, Λ(ξ)=/Ξ(ξ) Accordingto(), thefouriercoefficientsofthisfunctionaretheepansioncoefficientsλ k Therefore, we obtain(4) Derivation of (43) The RBF cardinal interpolant becomes s C () = λ k φ( k) k= = π e ikξ (π) Ξ(ξ ) dξ φ(ξ )e iξ ikξ dξ k= π = π ( ) (π) e ik(ξ ξ) dξ Ξ(ξ ) φ(ξ )e iξ dξ π k= }{{} πδ(ξ ξ ) = δ(ξ ξ ) φ(ξ ) π Ξ(ξ ) eiξ dξ dξ = φ(ξ) π Ξ(ξ) eiξ dξ 3 Directverificationthats C ()satisfies () Withn Z,weget s C (n) = π = π = π π π π π φ(ξ)e inξ j= φ(ξ+πj) dξ= π π k= π ( ) k= φ(ξ+πk) e inξ dξ j= φ(ξ+πj) }{{} = { n= e inξ dξ= n=±,±, 6 φ(ξ+πk) j= φ(ξ+πj) einξ dξ

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