Moving Least-Squares: A Numerical Differentiation Method for Irregularly Spaced Calculation Points

Transcription

1 SANDIA EPOT SAND Unliited elease Printed June 001 Moving Least-Squares: A Nuerical Differentiation Method for Irregularly Spaced Calculation Points Albert Gossler Prepared by Sandia National Laboratories Albuquerque, New Meico and Liverore, California Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, for the United States Departent of Energy under Contract DE-AC04-94AL Approved for public release; further disseination unliited.

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3 SAND Unliited elease Printed June 001 Moving Least-Squares: A Nuerical Differentiation Method For Irregularly Spaced Calculation Points Albert Gossler Theral/Fluid Coputational Engineering Sciences Departent Sandia National Laboratories P.O. Bo 5800 Albuquerque, New Meico ABSTACT Nuerical ethods ay require derivatives of functions whose values are known only on irregularly spaced calculation points. This docuent presents and quantifies the perforance of Moving Least-Squares (MLS), a ethod of derivative evaluation on irregularly spaced points that has a nuber of inherent advantages. The user selects both the spatial diension of the proble and order of the highest conserved oent. The accuracy of calculations is aintained on highly irregularly spaced points. Not required are creation of additional calculation points or interpolation of the calculation points onto a regular grid. Ipleentation of the ethod requires the use of only a relatively sall nuber of calculation points. The ethod is fast, robust and provides sooth results even as the order of the derivative increases. 3

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5 CONTENTS 1 INTODUCTION... 7 FOMULATION ESULTS Test Paraeters Application esults Nuber of Nearest Neighbors N esolution N s Scaled Isolation Distance D Gaussian adius Eecution Tie SUMMAY EFEENCES

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7 1 INTODUCTION The origin of least-squares ethods can be traced back to the beginning of statistical ethods perhaps to the first tie a ean was calculated on a set of data. The notion of oving least-squares (MLS), analogous to a oving average, in which a defined subset of data is used for each data reducing calculation, also has an etensive history. The use of MLS in one-diensional digital filters, for eaple, is well established, cf. [1][][3]. Shepard [4] first applied the ethod of MLS to the generation of two-diensional interpolants in a liited low-order case and Lancaster and Salkaukas [5] give a general higher-order approach of the ethod. Arentano and Duran [6] have recently given error estiates for one-diensional MLS approiations of functions and their 1 st and nd order derivatives. As an alternative to finite-eleent ethods, Nayroles et al. [7] and Belytschko et al. [8] adapted MLS approaches to eshless Galerkin approiations; a general overview of eshless ethods including MLS is given by Belytschko, et al. [9]. Variations of MLS within and across disciplines are referred to by different naes; in eshless ethods naes include the Diffuse Eleent Method (DEM) [7], the Eleent-Free Galerkin (EFG) ethod [8] and Partition of Unity Method (PUM) [10][11]; in data filter ethods, naes include Savitzky-Golay filters [1], and Digital Soothing Polynoials (DISPO) [3]. The present work is otivated by the developent of a three-diensional, vorticitybased, Lagrangian approach to fluid dynaics, a field to which MLS has been introduced relatively recently by Marshall and Grant [1]. Neither analytical error bounds for the MLS approiation of derivatives of ulti-diensional equations nor etensive analysis of the behavior of MLS estiations in such applications eist in the literature. This paper presents an MLS forulation for ulti-diensional applications and presents a detailed investigation of an MLS three-diensional application including errors in the approiation of 1 st, nd and 3 rd order derivatives. The governing equation for this fluid dynaics approach for a proble of incopressible, three-diensional fluid flow of variable density and constant viscosity occupying a region V is given by the vorticity transport equation dω dt ω 1 ( u ) ω ( ω ) u p t ω, (1) where the sybol refers to the gradient and the sybol refers to the nd order gradient, the Laplacian. In this approach, calculation points that need not be connected by a esh represent function values in V. These calculation points are advected at the local fluid velocity and, as such, are irregularly spaced. [The variables in equation (1) are not of priary concern in this discussion and coprise: vorticity (ω ), tie (t), velocity ( u ), density ( ), pressure (p), and kineatic viscosity ( ).] Nuerical evaluation of (1) clearly requires that the gradient and Laplacian be evaluated within V on the irregularly distributed calculation points. 7

8 A weighted residual ethod for an r th r r1 r order proble y y, y,, y, be arranged in the for f can E y r r1 r f y, y,, y,. () where E is an error or residual vector of the original equation, is a vector of ξ independent variables and y th are vectors of the ξ derivative of the dependent variable, cf. [13]. This residual can be iniized using any of several ethods that cause a weighted average of the residual to vanish. Such approaches vary only in the anner in which the residual is weighted and include: (i) collocation, in which the actual values of the residual at selected points are ade equal to zero; (ii) Galerkin s ethod, in which the integrals of the residual weighted by selected shape functions are set to zero; (iii) a least-squares approach, in which the integral of the square of the residual is iniized. It cannot be said a-priori which approach yields the ost accurate solution for a given case. Hence, selection of the approach rests priarily on foreknowledge of the particular application or on other considerations. For eaple, the collocation approach does not require integration, providing for an easy ipleentation. However, there is no guarantee that this approach yields a solution that is sufficiently sooth for calculations of derivatives, particularly higher order derivatives. The shape functions in Galerkin s ethod can be used to create a syetric coefficient atri, often of advantage in finite eleent ethods. In the present application, the MLS approach is selected because it aintains good accuracy on irregularly spaced calculation points [1] and because of its inherent soothing properties. 8

9 FOMULATION The use of a Lagrangian nuerical approach to solve a transport equation, such as the vorticity transport equation (1), yields values at points n, n 1,..., N, which, in general, are irregularly spaced. An MLS approach is used here to evaluate the derivatives of a function, say f,t, at a calculation point located at. In this ethod, the values f n of f,t on these calculation points are interpolated locally by a polynoial in the coponents of the position difference. q k, t f C B i1, i, i. (3) In (3), the inde denotes the point about which the interpolation is perfored, C, i denotes a set of k undeterined coefficients of the polynoial, and B i, are the associated basis functions. The value of k is the total nuber of cobinations with repeat [14] possible for the higher order ters in diensions d and order h, given by k h h d, j) j1 j1 ( d 1)! j! d 1 j! (. (4) The prescribed order h polynoial fit is equal to the highest oent conserved by the basis functions B i,. The MLS approach using a polynoial fit of prescribed order h (calculated with a sufficient nuber of calculation points) represents an h th order function eactly and is referred to as an h th order MLS fit. The k basis functions are generated using the relationship a b y y z z abc c B, i (5) where i 1,..., k, and a, b, c are whole nubers with a b c j, j 1,..., h. For the three-diensional, second order case k 9 and the associated basis functions are d 3, h B, 1, (6a) y y B,, (6b) 9

10 10 z z B 3,, (6c) y y B 4,, (6d) z z B 5,, (6e) z z y y B 6,, (6f) 7, B, (6g) 8, y y B, (6h) 9, z z B. (6i) The paraeter is a length scale associated with the calculation point, and can be considered an effective point radius. The value of can be set for each point as a function of the local average calculation point spacing or can siply be based on typical spacing between calculation points. Either approach is used to ensure that for sall difference coponents the basis function values do not approach the coputer s floatingpoint precision (typically 6 10 for single precision or 1 10 for double precision). Ipleenting the MLS ethod in higher or lower diensions or other orders is straightforward. For eaple, to attain third-order polynoial fit ) ( 3 h in three diensions ) ( 3 d requires 19 higher order ters ) 19 ( 3 3, d h k. The corresponding basis functions are given by those in equations (6) with additional functions given by y y B 10,, (7a)

11 11 11, y y B, (7b) z z B 1,, (7c) 13, z z B, (7d) z z y y B 14,, (7e) 15, z z y y B, (7f) z z y y B 16,, (7g) 3 17, B, (7h) 3 18, y y B, (7i) 3 19, z z B, (7j) The coefficients i C, of the polynoial (3) are obtained by a localized least-squares procedure, in which the error E is epressed as 1 )], ( [ t q f L E n n n N n. (8) where N is the nuber of calculation points, the nearest neighbors, about point used in the MLS fit. The localization paraeter n L weights the contribution of different points to

12 the error E. The value of L n can be set equal to unity for the N nearest neighbors of and zero elsewhere or its value ay be set to decay with distance fro using any convenient function. Miniization of E with respect to each of the coefficients syste of linear equations of the for C, yields a i k k k G, ijc, j U, i j1, i 1,..., k, (9) where N G, ij LnBn, ibn, j n1, (10) N U, i, n1 B fn f LnBn i (11) B ( ). (1) n, i, i n Solution of the syste (9) yields k 1 C, i G, ij U, j j1, (13) where 1, ij G is the inverse of G, ij. Upon solving for coefficients C, i, the derivatives of f are approiated by differentiating the polynoial fit (3). As eaples, for the threediensional, second order polynoial case, the first order derivatives are given by C f n, 1 n, (14a) y f n, n C C f n, 3 z n, (14b), (14c) and the second order derivatives are given by 1

13 fn n C,7, (15a) fn y n C,8, (15b) fn z n C,9. (15c) For the three-diensional, third order polynoial case, the first and second order derivatives are again given by equations (14) and (15) and third order derivatives are given by 3 fn 6 3 n 3 fn 6 3 y n 3 fn 6 3 z n C,17 3 C,18 3 C,19 3, (16a), (16b). (16c) The coefficient atri in equation (9) can be solved using a variety of linear equation solvers including Gauss-Jordan eliination, Gaussian eliination, LU decoposition and Cholesky decoposition. However, under typical conditions the condition nuber of the coefficient atri ay becoe very large. Any atri is singular if its condition nuber is infinite and can be defined as ill-conditioned if the reciprocal of its condition nuber approaches the coputer s floating-point precision. In such cases, the cited solvers ay fail or ay return a highly inaccurate solution. To avoid such nuerical probles, a singular value decoposition (SVD) linear equation solver is soeties recoended for use in conjunction with the MLS ethod. The SVD solver identifies equations in the atri that are, within a specified tolerance, redundant (linear cobinations of the reaining equations) and eliinates the, thereby iproving the condition nuber of the atri. The reader is referred to reference [15], Chapter 15 for a helpful discussion of SVD pertinent to linear least-squares probles. To eaine their relative erits, two linear equation solvers are used in this report: Gauss-Jordan eliination (GJE) with full pivoting, a direct solver that is robust and relatively siple; SVD, a solver that reains robust even for initially illconditioned sets of linear equations. 13

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15 3 ESULTS This section etensively eplores a Moving Least Squares application in order to gain working knowledge of the approach and so develop helpful coents, sall hints and general advice to ake MLS a practical tool. 3.1 Test Paraeters The MLS approach is tested by using it to evaluate the 1 st, nd and 3 rd derivatives of a three-diensional Gaussian test function f given by 0.5 y 0.5 z 0.5 f, y, z ep. (17) δ The paraeter δ is the Gaussian radius and deterines the width of the single wave in this soothly varying function. As δ decreases, the function narrows. Before introducing a graph of this function, a thin center slice will be defined as a nearly one-diensional function space along the -ais within the region ( 0.5 ) y (0.5 ) and ( 0.5 ) z (0.5 ), where is the average spacing between calculation points. Figure 1 presents a graph of the Gaussian function given by equation (17) on the test function space used throughout this section, a 1 11 cube etending fro the origin to 1 on each Cartesian ais. The radius used for this graph is given by δ This radius is used for all tests within this section ecept for tests on the effect of varying the radius. While this radius yields the widest Gaussian function considered, this function itself is not of priary interest; its derivative functions are. The waves of higher order derivatives of this function becoe increasingly narrow with the ratio of wave aplitude to wave width becoing increasingly severe. The test function gradient f is shown in Figure, the Laplacian f in Figure 3, and 3 rd 3 order gradient f in Figure 4. The wave aplitude to width ratio can be seen to increase rapidly fro approiately 10 for f to a value of approiately 35 for f and then to a value of approiately 400 for 3 f. Because MLS is an approach applied to solution techniques in which function values are known generally on irregularly spaced calculation points, a topic of interest is the effect of the randoness of the point arrangeent on the accuracy and stability of the result. Toward this end, functions are tested using both evenly spaced calculation points with unifor spacing and with randoized calculation points with average local spacing. The randoized point locations are generated by perturbing the uniforly spaced points in each Cartesian direction by α where α is a rando nuber in the range 0 α ε. 15

16 f Figure 1. A thin center slice of the Gaussian test function f with δ f Figure. A thin center slice showing the gradient δ f of the Gaussian test function f with 16

17 f Figure 3. A thin center slice showing the Laplacian δ f of the Gaussian test function f with f Figure 4. A thin center slice showing the third order gradient function f with δ f of the Gaussian test 17

18 esults of applying aiu randoization perturbations of ε and ε on approiately 1000 point positions are shown in Figure 5. The randoization with ε 0.05 is barely detectable by eye whereas the randoization with ε is plainly evident. andoization with ε represents the aiu randoization possible such that the average spacing local to any calculation point reains equal to the unifor spacing. The effective point radius in equation (5) is set equal to. While the effect of changing is not eplicitly eplored, inforal observation shows varying this paraeter within an order of agnitude greater or less than has virtually no effect on the MLS results presented. The value of L n is equation (8) is set equal to unity for the N nearest neighbors of and zero elsewhere. The effect of this paraeter is a cople consideration. In general, one-diensional MLS fits, as are used in digital signal processing, generate soother function approiations f n (by reoving high frequency variations in the saple function) for L n functions that decay with distance fro. The interested reader is referred to reference [3], Chapter 5, for an introductory discussion of this topic. The paraeters N s and N are used in investigating resolution effects later in this section. The nuber of points per side of each face of the 1 11 test space is given by N s. 3 The total nuber of points N N s resolving the three-diensional functions is set by varying N s within the range 1 N s 51 such that the range of N spans nearly 1.5 agnitudes, 961 N A scaled isolation distance D used in later tests is introduced here. The distance between an isolated calculation point and its nearest neighbor is easured as the scaled isolation distance D ties the average calculation point spacing. Two paraeters are calculated for coparison of test results. One paraeter is C a, the aiu condition nuber of the coefficient atri G, of equation (9) prior to eliination of redundant and nearly redundant equations using SVD. The other paraeter is an nor error, a scaled rs (root-ean-square) calculation given by ij E rs 1 a 1 N g i gi. (18) g N i1 where g and g represent the analytic and calculated values, respectively, of 3 f. f, f, or 18

19 For all tests, floating-point calculations are perfored using double precision. The perforance of MLS using either the GJE or SVD linear equation solver is quantified by evaluating graphs of calculation point values and by evaluating the error and the condition nuber, suarized in Table 1, as a function of the independent paraeters suarized in Table y 0.5 (a) ε y 0.5 (b) ε Figure 5. Approiately 1000 point positions randoized using different aiu perturbation aplitudes ε. 19

20 Description Ca Maiu condition nuber. Ers Scaled rs calculation given by equation (18). Table 1. Values evaluated in the MLS tests. Description ange N Nuber of nearest neighbors used in the MLS fit. 1 N 100 N s Nuber of calculation points per side of each face of the test volue. The test volue is resolved by the total 3 nuber of calculation points N N s. 1 N 51 s ε Maiu randoization perturbation. Calculation points of unifor spacing are perturbed in each Cartesian direction by α where α is a rando nuber in the range 0 α ε. 0 ε 0.50 D Scaled isolation distance: the distance between an isolated calculation point and its nearest neighbor is easured as D ties the average calculation point spacing. 1 D 10 δ The radius of the Gaussian test function f given by equation (17) δ Table. Suary of paraeters varied in MLS tests. 0

21 3. Application esults The titles of the subsections within this section indicate the paraeter that is the focus of tests within that subsection. This priary paraeter is varied and results presented and discussed. Also, generally the two different linear equation solvers are used and suppleentary paraeters such as the aiu randoization perturbation ε and the MLS order of fit h are varied to help investigate the effects of the priary paraeter of a subsection. Use of the GJE solver is eplicitly stated; otherwise, use of the SVD solver is iplied. A short suary of ain results appears at the end of each subsection Nuber of Nearest Neighbors N. The first (and last) consideration in this application is choosing the nuber of nearest neighbors N of equation (8) to use in the MLS fit. For this series of tests, the resolution is set to N s 51. Beginning with a 1 st order MLS fit and using either the GJE solver or the SVD solver, the condition nuber C a and the error E rs are calculated as a function of the nuber of nearest neighbors N for uniforly spaced calculation points ( ε 0) and for the randoized points with ε Figure 6 presents the results for these tests using the uniforly spaced calculation point and randoized points, respectively. andoized points are seen to sooth out the coefficient condition nubers C a ; with decreasing N, Ca increases abruptly for uniforly spaced points and increases gradually for randoized points. The nuber of nearest neighbors N required to achieve the lowest possible error E rs is slightly greater for the randoized points. The results using the GJE solver are indistinguishable fro those using the SVD solver presented in Figure 6(a) and 6(b) in the 1 region where the condition nuber C a is bounded to values C a 10. For uniforly spaced calculation points, the GJE solver abruptly fails (for N 5 ) due to the atri becoing effectively and abruptly singular. For the randoized points, as the condition nuber C a quickly increases, the GJE solver generates large E rs errors (for 4 N 5) and then fails (for N 3) as C a becoes unbounded. Selected results using the GJE solver are included in Figure 6(b) to highlight the difference in behavior between the two solvers. For uniforly spaced and randoized calculation points, the SVD solver does not fail as the condition nuber C a increases. ather, the SVD solver provides solutions with the sallest possible least-square coefficient values (as opposed to values approaching infinity!). The errors E rs of these solutions increase substantially and gradually with decreasing N. Lines drawn through the error values E rs are broken into two parts corresponding to regions in which the condition nuber is sharply increasing or nearly constant. In the region of sharply increasing condition nuber C a, error E rs increases with decreasing N at slightly greater than nd order (as labeled) for the uniforly spaced calculation points and with nd order (not labeled) for the randoized points. 1

22 C a E rs 10-1 Slope = -.3 (a) uniforly spaced points N 10 1 C a E rs (b) randoized points ε N Figure 6. esults for the 1 st order MLS fit. The condition nuber error E rs for the gradient (SVD: ; GJE: ). C a ( ) and the

23 If the basis functions for the N points were unique, the linear equations in the coefficient atri would be unabiguous and the condition nuber C a and the error Ers would reain roughly constant. Hence, the basis functions are effectively redundant when evaluated nuerically. The randoization of calculation point positions helps to aintain unique contribution of the associated basis functions. This result, while perhaps not surprising, nonetheless counicates an uncoon knowledge: for the MLS approach, and possibly other iniization approaches, calculating function values using randoized points can be ore accurate and ore stable than using uniforly spaced points. The nuber of nearest neighbors to use in the 1 st order MLS fit can now be chosen based on the presented results. Assuing for the oent that there ust be soe calculation penalty (to be evaluated later) in using ore nearest neighbor calculation points for the fit, a iniu nuber of points is selected that, in this case, (i) aintains an error of approiately E rs 10 and (ii) avoids solving an ill-conditioned atri. That nuber is N 7. Using 7 points will provide a sufficient nuber of unique basis equations to deterine the unknown MLS coefficient atri of equation (9) for this application under all conditions. For siilar conditions, 1 st and nd order gradients are calculated using a nd order MLS fit. Coparing the condition nubers C a of results for uniforly spaced calculation points with those for randoized points, shown in Figure 7, the tendency of randoization is again to sooth out (lessen the agnitude and ake ore gradual the change in) the condition nubers C a. In the region where the condition nuber C a is bounded to 1 values C a 10, the results using the GJE solver are indistinguishable fro those using the SVD solver presented in Figure 7(a) and Figure 7(b). For uniforly spaced calculation points, the GJE solver fails (for N ) as the atri abruptly becoes singular, consistent with the 1 st order MLS results. The condition nuber C a when using randoized points quickly increases and the GJE solver generates large E rs errors (for N 10 and N 11) and then fails (for N 9 ) as C a increases without bound. In contrast, the SVD solver again provides solutions with increased E errors for poorly conditioned (relatively large 1 rs condition nubers but C a 10 ) and even for ill-conditioned atrices ( C a 10 ). To deonstrate this change in behavior between results generated using the two solvers, selected results using the GJE solver are included in Figure 7(b). 1 3

24 C a E rs Slope = 0.0 Slope = -1.0 (a) uniforly spaced points N C a E rs (b) randoized points ε N Figure 7. esults for the nd order MLS fit. The condition nuber C a ( ) and the error E rs for the gradient (SVD: ; GJE: ) and the Laplacian (SVD: ; GJE: ). 4

25 C a E rs Slope = Slope = N Figure 8. esults using the sae paraeters as those of Figure 7 ecept for randoization, which is reduced to ε The condition nuber C a ( ) and the error E rs for the gradient (SVD: ; GJE: ) and the Laplacian (SVD: ; GJE: ) for the nd order MLS fit. For the nd order fit and randoized calculation points, if the criterion is siply that error E rs reains below 10 for unifor or randoized calculation points, then N 14 is adequate. Let the acceptable iniu nuber of nearest neighbors for a nd order MLS fit using randoized points be N in 14 for future reference. If the criterion is added to avoid solving an ill-conditioned atri, a value of N 3 ight be chosen. Let the acceptable aiu nuber of nearest neighbors for a nd order MLS fit using randoized points be N a 3. Selecting the actual value of N to be used will be deferred until after review of test results presented in succeeding subsections. Forally applying the iniu and aiu nearest neighbors criteria to the 1 st order MLS fit discussed above gives N N 7 for that case. in a andoization with ε shown in Figure 7(b) is reduced to ε and shown in Figure 8 to gage the effect of the degree of randoization. The trends are siilar in these figures: even inial randoization soothes the condition nuber C a ; error E rs values are in close agreeent between the figures using either the GJE or SVD solvers. Considering the region of greatest interest where the lowest nuber of nearest neighbors N provides the lowest error E rs, a severe reduction in condition nuber C a is achieved by randoizing calculations points even slightly while there is only a sall and gradual change in the E rs errors as randoization is increased. As with the greater randoization, Ers errors when using the GJE solver are indistinguishable fro those when using the SVD 5

26 solver in the region where the condition nuber is bounded to values C a 10. Included in Figure 8 are selected GJE results to show the divergent behavior of the two solvers. The 1 st, nd and 3 rd order gradients are net calculated using a 3 rd order MLS fit. These results are presented in Figure 9. These results, along with the 1 st and nd order MLS fit results, indicate that for randoized calculation points the reducing and soothing of condition nubers C a increases with the order of the MLS fit. Furtherore, these results support the stateent that the higher order basis functions are ore likely to be nearly redundant. In contrast to the 1 st and nd MLS order fit results, the iniu nuber of nearest neighbors N required to achieve the lowest possible error E rs is less for randoized calculation points. The GJE solver results are indistinguishable fro those presented for the SVD solver in Figure 9(a) and Figure 9(b) in the region where the condition nuber is bounded to values 1 10 C a, siilar to the results for lower order MLS fits. However, in contrast to the previous results, the condition nuber does not abruptly becoe unbounded for uniforly 17 spaced calculation points (rather, it quickly grows to approiately 10 ). As a result, the GJE solver does not abruptly fail; the error E rs increases quickly over the range 40 N 75 and then fails at N 40. Using randoized points, the condition nuber C a quickly increases and the GJE solver generates large E rs errors (for 14 N 0) and eventually fails as C a increases without bound. In contrast, the SVD solver again provides solutions with increased E rs errors for poorly conditioned and ill-conditioned atrices. The difference in behavior between results generated using the two solvers is shown graphically by the selected GJE solver results included in Figures 9(a) and 9(b). As with lower order MLS fits, the SVD solver provides accurate solutions for a range of poorly conditioned and ill-conditioned atrices for the 3 rd order fit. If the criterion in setting N for the 3 rd order fit is siply that the error E rs be below 10 for uniforly spaced or randoly spaced calculation points, then N 40 is satisfactory. Let the acceptable iniu nuber of nearest neighbors for a 3 rd order MLS fit be N in 40. If the criterion is again added to avoid solving an ill-conditioned atri, a value of N 76 ight be chosen. Let the acceptable aiu nuber of nearest neighbors for a 3 rd order MLS fit be N a 76. As with the nd order MLS fit, selecting the actual value to be used will be deferred until after review of test results presented in succeeding subsections. 1 6

27 C a E rs Slope=-1.4 Slope=-1.4 (a) uniforly spaced points Slope= N C a E rs (b) randoized points ε N Figure 9. esults for the 3 rd order MLS fit. The condition nuber C a ( ) and the error E rs for the gradient (SVD: ; GJE: ), the Laplacian (SVD: ; GJE: ) and the 3 rd order gradient (SVD: ; GJE: ). 7

28 In Figure 9(a), the error in the Laplacian using very few nearest neighbors is seen to be arkedly better than that of the 1 st and 3 rd order gradients for the case of uniforly spaced calculation points. This eperience of an approiation yielding an eceptionally higher order of approiation of a given function occurs in coputational approaches eploying grids with uniforly spaced calculation points. Such a gift is eplained by the canceling out of ters of higher order than those calculated in the approiation, producing an effectively higher order approiation. Further details of such occurrences are not within the scope of this MLS investigation and this result will siply be accepted with equaniity. The value of N fies the volue of a sphere, the MLS window, circuscribing the N nearest neighbors. An effective radius r of this MLS window can be quickly deterined 3 3 fro the relation 4 N r, where the average calculation point spacing is given by 3 1. Hence, 1 N s 1/ 3 N r (19) 1 N s Therefore, the size of MLS window changes by nearly a factor of 4 over the range tested for the 1 st and nd order MLS fits and by approiately a factor of for the 3 rd order fits. The insensitivity of E rs to the MLS window size is observed in the presented results; with increasing N N in, E rs errors either reain fairly constant (for the 1 st and 3 rd order MLS fits) or increase only slightly (for the nd order fit). As the MLS fits of this section are applied to the test function with a constant value δ , it follows that E rs error is also not very sensitive to the ratio of the length scales δ r, where δ is used here as a convenient length scale representing the severity of the function curvature. Subsection Suary For the MLS fits tested with either uniforly spaced calculation points or randoized points: (i) there is a iniu nuber of nearest neighbors N such that N in N aintains error E rs below 10 and a aiu nuber of nearest neighbors N a such that N a N in addition avoids an ill-conditioned MLS coefficient atri even for uniforly spaced calculation points, (ii) the nuber of nearest neighbors N in the range N N increases with increasing order of MLS fit, (iii) using SVD with in N a N in N the error E rs increases either abruptly or quickly by about orders of agnitude greater than the error with Nin N, and (iv) E rs errors are nearly insensitive to the size of the MLS window as represented by the ratio of length scales δ r. The condition nubers in 8

29 C a (v) increase abruptly at N Na for uniforly spaced points and increase gradually, are soothed, for randoized points; for orders of MLS fit greater than unity, this occurs at N considerably less than N a ; (vi) this soothing is attributed to the randoized calculation points aking possible unique contributions of the basis function to the MLS coefficient atri. (vii) The values of error E rs and condition nuber C a are not sensitive to the randoization within the range 0.05 ε (viii) The SVD solver does not fail over the tested range 1 N 100 of nearest neighbors used in the MLS fits. (i) The GJE solver produces highly inaccurate results for a poorly conditioned MLS coefficient atri and fails for an ill-conditioned atri. () The E rs errors are fairly insensitive to the size of the MLS window. (i) The MLS approach, a representative of residual iniization approaches, calculates values on randoized calculation points with equal or greater accuracy as easured by E rs and with greater stability as easured by C a than on uniforly spaced points. 9

30 3.. esolution N s. The orders of accuracy of the 1 st, nd and 3 rd order MLS fits versus N are not 1 st, nd and 3 rd 3 order as ight be epected. In this subsection, the total nuber of points N N s resolving the three-diensional functions is varied fro 961 N 13651, nearly 1.5 orders of agnitude, by varying the nuber of points per side N s of the 1 11 test space in the range 1 N 51. s Using nearest neighbors N 7, error E rs results for the 1 st order MLS fit are shown in Figure 10. The order of accuracy is strongly dependent on whether the calculation points are uniforly spaced or randoized. The uniforly spaced points show a nd order accuracy and the randoized points show only a 1 st order accuracy. The sall effect of resolution N s on the condition nuber C a is shown in Figure 11. The condition nuber C a reains constant as the resolution increases using uniforly spaced calculation points and it increases slightly as the resolution increases using randoized points. For the nd order MLS fit using N N in 14, the order of accuracy for the gradient is again nd order, shown in Figure 1. The Laplacian shows slightly better than a 1 st order fit. These results hold for both unifor and randoized calculation points. The condition nubers C a for the unifor and randoized cases, shown in Figure 13, are fairly constant. However, the condition nubers C a for the uniforly spaced points are about 15 orders of agnitude greater than those for randoized points, consistent with Figure 8 in the last subsection. Using N N a 3, Figures 14(a) and 14(b) show a soewhat reduced sensitivity to calculation point position randoization: the gradient calculation continues to ehibit nearly nd order accuracy independent of randoization and only the Laplacian calculation shows sensitivity to randoization with 1.6 order accuracy for uniforly spaced points but only 0.4 order accuracy for randoized points. To see if this order of accuracy trend of decreasing sensitivity to randoization correlates to the increase in the nuber of nearest neighbors N, Figure 14(c) is presented on the following page. The single distinction between Figures 14(b) and 14(c) is the increase in N fro 3 to 3. Indeed, for N N in, the order of accuracy decrease in sensitivity to calculation point randoization is a function of N. The condition nuber values change greatly when varying the nuber of nearest neighbors used in the nd order MLS fit fro N N in 14 to N N a 3 as shown in Figure 15. The condition nubers C a for the uniforly spaced points drop greater than 15 orders of agnitude to a values now lower than that for the randoized calculation points. Furtherore, the randoized case now ehibits soe, though insignificant, sensitivity to resolution, again increasing slightly with increasing resolution. s 30

31 Slope = -1.0 E rs 10 - Slope = N s Figure 10. The error E rs for the gradient with uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a 1 st order MLS fit. The nuber of nearest neighbors used is N C a N s Figure 11. The condition nuber C a for the gradient with uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a 1 st order MLS fit. The nuber of nearest neighbors used is N 7. 31

32 Slope = -1.1 Slope = -.0 E rs N s Figure 1. The error E rs for the gradient ( ) and Laplacian ( ) using a nd order MLS fit. The nuber of nearest neighbors used is N N in 14. esults apply to calculation points either uniforly spaced or randoized with ε C a N s Figure 13. The condition nuber C a for uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a nd order MLS fit. The nuber of nearest neighbors used is N N 14. in 3

33 10 - E rs 10-3 Slope = -1.9 Slope = -1.6 (a) uniforly spaced points N s 10 - Slope = -0.4 E rs 10-3 Slope = -1.9 (b) randoized points ε N s Figure 14. The error E rs for the gradient ( ) and Laplacian ( ) using a nd order MLS fit. The nuber of nearest neighbors used is N N 3. a 33

34 10-1 E rs 10 - Slope = -1. Slope = N s Figure 14(c). The error E rs for the gradient ( ) and Laplacian ( ) using the sae paraeters as those of Figure 14(b) with one eception: the nuber of nearest neighbors used is N C a N s Figure 15. The condition nuber C a for uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a nd order MLS fit. The nuber of nearest neighbors used is N N 3. a 34

35 For the 3 rd order MLS fit using N N in 40, the order or accuracy has no eaningful sensitivity to whether the calculation points are randoized. esults for the uniforly spaced calculation points and for randoized points are shown in Figure 16. Coparing the 1 st, nd and 3 rd order MLS fits, for N N in the order of accuracy of all derivative calculations generally becoes less sensitive to randoization of calculation point positions as the nuber of nearest neighbors N increases. The condition nubers C a for the unifor and randoized cases, shown in Figure 16, are fairly constant. Siilar to the nd order MLS fit results, the condition nubers for the uniforly spaced calculation points are about 13 orders of agnitude greater than that for the randoized points, shown in Figure 17. This difference in condition nubers C a is consistent with Figure 9 of the previous subsection. Using N N a 76 the error E rs results change slightly, as shown in Figure 18. For uniforly spaced calculation points, the gradient calculations have a 1.8 order accuracy and the randoized points have a slightly lower 1.5 order accuracy. The Laplacian ehibits nearly nd order accuracy independent of calculation point randoization. The 3 rd order gradient ehibits orders of accuracy of 3.6 and 3., respectively, for uniforly spaced and randoized points. The condition nuber values again change greatly when varying the nuber of nearest neighbors used in the 3 rd order MLS fit fro N N 40 to N N a 76, as shown in Figure 19. The condition nuber C a for the unifor case drops about 14 orders of agnitude to values now lower than that for the randoized calculation point case; in this 3 rd order MLS fit, however, C a does not ehibit sensitivity to resolution for either uniforly spaced or randoized calculation points. As discussed in Subsection 3..1 (and will be further discussed in Subsection 3..4), r E rs error is not very sensitive to the ratio of the length scales including those δ corresponding to the changes in resolution N s. For all MLS fits tested, the order of accuracy 1 is, therefore, attributed to the relative changes in the average spacing given by N s 1, where δ is used here as a convenient length scale representing the severity of the δ function curvature. in 35

36 10-1 Slope = -1. E rs Slope = -1.9 Slope = -.9 (a) uniforly spaced points N s 10-1 Slope = -1. E rs Slope = -1.9 Slope = -3.0 (b) randoized points ε N s Figure 16. The error E rs for the gradient ( ), Laplacian ( ) and 3 rd order gradient ( ) using a 3 rd order MLS fit. The nuber of nearest neighbors used is N N 40. in 36

37 C a N s Figure 17. The condition nuber C a for uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a 3 rd order MLS fit. The nuber of nearest neighbors used is N N 40. in 37

38 10-1 E rs Slope = -1.9 Slope = -3.6 Slope = -1.8 (a) uniforly spaced points N s 10-1 E rs Slope = -1.9 Slope = -3. Slope = -1.5 (b) randoized points ε N s Figure 18. The error E rs for the gradient ( ), Laplacian ( ) and 3 rd order gradient ( ) using a 3 rd order MLS fit. The nuber of nearest neighbors used is N N 76. a 38

39 C a N s Figure 19. The condition nuber C a for uniforly spaced ( ) and randoized ( ) calculation points, ε 0. 50, using a 3 rd order MLS fit. The nuber of nearest neighbors used is N N 76. a Subsection Suary (i) For an h th order MLS fit, the noinal order of accuracy for approiation of a j th order gradient is roughly h j 1. (ii) The observed orders of accuracy are due to the ratio of changes of average point spacing to a easure of the severity of function curvature. (iii) δ A broad trend is observed in which the order of accuracy of derivative calculation becoes less sensitive to randoization of points as the nuber of nearest neighbors N used in the MLS fits increases. (iv) The order of accuracy is either unchanged or slightly iproved for an MLS fit using either (a) ore nearest neighbors in the range Nin N Na or (b) 16 uniforly spaced calculation points; (v) the condition nuber C a eceeds 10 indicating the coefficient atri is ill-conditioned for the nd and 3 rd order MLS fits using uniforly spaced calculation points and N in nearest neighbors; (vi) the condition nubers C a are 1 bounded to C a 10 and are alost constant for an MLS fit using randoized calculation point locations. 39

40 3..3 Scaled isolation distance D. Inspection of the aiu condition nuber C a used throughout this report shows that this aiu consistently occurs at the corners of the 1 11 test volue. That is, the condition nuber is greatest at the ost isolated points. These corner points also have a one-sided syetry. The isolation of and syetry about a point are eplored by investigating the error E rs and the aiu condition nuber C a for various isolated points with and without function syetry about the. Eternally isolated points and internally isolated points are investigated using the test arrangeents depicted, respectively, in Figures 0 and 1. Shown in each of these figures are uniforly spaced bulk calculation points and a single isolated point placed on-center in a location of syetry. The bulk calculation points have an average local spacing ; the isolated points do not. For isolated points the scaled isolation distance is used, defined as the distance between an isolated calculation point and the nearest neighbor. This distance is easured as a ultiple D ties the average bulk calculation point spacing. The isolated points in Figures 0 and 1 are shown at a scaled isolation distance D 5. The nuber and arrangeent of bulk calculation points used in isolated eternal point tests is a block of points and for the isolated internal point tests the nuber and arrangeent of the 3 bulk calculation points is with a ( D 1) block of points reoved. In various indicated tests, all points other than the isolated point are also randoized with aiu perturbation ε as described earlier in this report. Isolated eternal points tered off-center are not placed in positions of syetry; they are offset fro the y 0. 5 line by a distance D along the -ais and D along the z-ais. Isolated internal points not placed in positions of syetry are offset fro the point (, y, z) (0.5, 0.5, 0.5) by a distance D along the -ais and D along the y-ais and 3D along the z-ais. All tests in this subsection use a nd order MLS fit with resolution given by N s 31. As with previous tests, the SVD solver is used ecept where indicated. Figures and 3 show the condition nuber C a and error E rs for the gradient and Laplacian calculations with the nuber of nearest neighbors N N a 3 used in the MLS fit. The isolated points in these figures are eternal, either on-center or off-center, and the calculation points are uniforly spaced. The scales of these figures are the sae as those of related figures presented later to provide for direct coparison. esults show the condition nuber sharply increases as the scaled isolation distance D increases beyond the value of unity. The error E rs reains fairly constant at about 10. Using the GJE solver (results not shown) rather than the SVD solver for the on-center calculations shown in Figure, yields E rs errors of the sae value at D 1. As the condition nuber C a sharply increases, the E rs errors grow for the GJE calculated values (to approiately 4 10 at D 3) and then fails (at D 4 ). 40

41 y Figure 0. Depiction of the eternal centered isolated point showing uniforly spaced bulk calculation points and scaled isolation distance D y Figure 1. Depiction of the internal centered isolated point showing uniforly spaced bulk calculation points and scaled isolation distance D 5. 41

42 C a E rs D Figure. The condition nuber C a ( ) and error E rs for the gradient ( ) and Laplacian ( ) with N N a 3. Bulk calculation points are uniforly spaced. The isolated point is eternal, on-center C a E rs D Figure 3. The condition nuber C a ( ) and error E rs for the gradient ( ) and Laplacian ( ) with N N a 3. Bulk calculation points are uniforly spaced. The isolated point is eternal, off-center. 4

43 The results for the on-center or off-center isolated points are in close agreeent. This agreeent is observed in all eternal and internal isolated point tests considered supporting a conclusion that function syetry does not contribute to C a in these or previous tests. Henceforward, only the syetrical on-center test results will be shown and these will be discussed siply as results of isolated point tests, it being understood that the presented results are very siilar to those of the off-center tests. esults for bulk calculation points greatly randoized (ε 0.50) or inially randoized (ε 0.05) are shown respectively in Figures 4 and 5. In these figures, error E rs values are siilar in agnitude (though the errors in the gradient and Laplacian calculations switch places) to those for the uniforly spaced bulk points; E rs is not sensitive to the degree of randoization. Consistent with previous tests in this report, the randoization of the bulk calculation points greatly reduces the condition nuber values 1 with the result that C a values becoe bounded to C a 10 over the tested range of D. The effects of varying the scaled isolation distance D when using a reduced nuber of nearest neighbors N N in 14 are shown in Figure 6. There is a sall increase in the error E rs of the gradient and the Laplacian calculation for unifor or randoized bulk calculation points. The condition nubers of the MLS coefficient atri becoe illconditioned for uniforly spaced calculation points beginning at scaled isolation distance D 1 using N N in 14 nearest neighbors rather than D 1 using N N a 3; this is a distinction of trivial practical iportance. Using randoized calculation points, no eaningful changes in the condition nubers C a are evident for N N in 14 versus N 3. N a Using uniforly spaced points and N N a 3, results for the internally isolated point are shown in Figure 7. The condition nuber is constant with C a 18. The error fluctuates soewhat with changing D but never departs appreciably fro E rs 10. For randoized bulk calculation points, the condition nuber and error results are fairly insensitive to randoization of the points for internally isolated points. 43