An iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities

Transcription

1 * Manuscript An iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities Vincent R. Durante a, Jae-Hun Jung b a Department of Mathematics, University of Massachusetts at Lowell Lowell, MA 8, USA b Department of Mathematics, University of Massachusetts at Dartmouth North Dartmouth, MA 77-, USA Abstract In [J.-H. Jung, Appl. Numer. Math. 7 (7) -9], an adaptive multiquadric radial basis function method has been proposed for the reconstruction of discontinuous functions. Utilizing the vanishing shape parameters near the local jump discontinuity, the adaptive method considerably reduces the Gibbs oscillations and enhances convergence. In this paper, a new jump discontinuity detection method is developed based on the adaptive method. The global maimum of the epansion coefficients, λ i, eists at the strongest jump discontinuity and its magnitude increases eponentially with the number of the center points, N. The adaptive method, however, dynamically reduces its magnitude to O(N) once applied. The global maimum of λ i then eists at the net strongest jump discontinuity and its magnitude is eponentially large. In such a way, the local jump discontinues are successively detected with the adaptive method applied iteratively. Numerical eamples are provided using the piecewise analytic functions and the numerical solution of the shock interaction equations. Numerical results verify that the proposed method is efficient and accurate in finding local jump discontinuities. Key words: Multiquadric radial basis functions, Jump discontinuity, Gibbs phenomenon, Adaptive method, Iterative method, Local jump detection. Corresponding author. Tel:+89998; fa: address: jjung@umassd.edu (Jae-Hun Jung b ). Preprint submitted to Elsevier April 8

2 Introduction Radial basis functions(rbfs) have been actively investigated in the last decades in various areas of applications [6,9,,,,,] (for the general review see also []). The main advantage of RBF methods is that they do not necessarily require any particular grid system consequently yielding more fleibility when approimating the function in an irregular domain. RBF methods also yield a fast convergence if the function to be considered is smooth enough [,8,9,]. In order to eplain RBF methods briefly, suppose that the one-dimensional input data point set or the center set X = { i i Ω, i =,, N} in the given domain Ω, i X Ω R is given. The RBF interpolation of f() denoted by s f,x () is then given by the linear combination of RBFs, ψ i ( i, ɛ i ), i =,, N; N s f,x () = λ i ψ i ( i, ɛ i ). () i= Here λ i and ɛ i are the epansion coefficients and the shape parameters, respectively and denotes the metric for which the Euclidean norm is commonly used, that is, i = i. Commonly used RBFs, include multi-quadric RBFs (MQ RBFs), ( i ) + ɛ i, inverse multi-quadric RBFs (IMQ RBFs),, and Gaussian RBFs, ep( ɛ i ( i ) ). The shape parameters ɛ i ( i ) +ɛ i are given or prescribed for the interpolation and the epansion coefficients λ i are to be determined by the interpolation condition below s f,x () = f(), X. () The interpolation condition Eq. () yields a linear system Mλ = f, () where λ = (λ,, λ N ) T, f = (f( ),, f( N )) T, and the interpolation matri M is given by M ij = ψ( i j, ɛ j ). () The epansion coefficient vector λ is obtained by solving Eq. () λ = M f, where M is square and nonsingular [].

3 Suppose that Y = {y i i =,, N R } Ω is the reconstruction point set, then using the above notation, we obtain s Y = M R λ = M R M f, where s Y = (s f,x (y ),, s f,x (y NR )) T and the reconstruction matri M R is given by M Rij = (y i j ) + ɛ j. If Y = Ω, the interpolation matri M R is -by-n matri. The inde j denotes the center point in the given X and the center set X is not necessarily structured, that is, it can have an arbitrary distribution. The arbitrary grid structure is one of the major differences between the RBF method and other global methods. Such a mesh-free grid structure yields higher fleibility especially when the domain is irregular. The basis functions ψ i ( i, ɛ i ) are defined in the entire Ω for i and the RBF method is a global method. As the RBF method is global, the approimation with RBFs yields a fast convergence if the function is smooth. If the function has a jump discontinuity, however, such fast convergence is deteriorated and some additional conditions are needed to improve the convergence. This is so-called the Gibbs phenomenon commonly observed with the global methods. In our previous work [], the Gibbs phenomenon of the MQ-RBF method has been investigated. In [], it has been shown that the Gibbs phenomenon is inevitable with evenly structured center set X and the non-zero shape parameters ɛ i. It is also shown that the over/under-shoots, δ N [f] of the Gibbs oscillations have different patterns depending on the values of ɛ i and N. For eample, if ɛ i /N, i, δ N is invariant for N while if ɛ i = ɛ, i and N, δ N increases with N. Here [f] denotes the jump at the discontinuity, i.e. [f] = f + f. Once there eists the Gibbs phenomenon, the approimation s f,x () is oscillatory near the local discontinuity. Several techniques have been developed to deal with the stiff functions for RBF approimations [,7]. These methods restructure the distribution of the given center set X. When restructuring the center set X, it is important that the shape parameters ɛ i also vary accordingly. There is an empirical relation between the distribution of X and the shape parameters ɛ i, such as ɛ i =.8 mean(d i ) where d i = min j X i j [8]. In [], the ɛ-adaptive method has been proposed with which the grid distribution is not changed but only the adaptation of ɛ i is applied. It has been also shown that by having ɛ-adaptation only near the jump discontinuity, the Gibbs oscillations can be considerably reduced and convergence is highly improved. We also note that there has been research on the optimization of the shape parameters to enhance accuracy such as [,] both in D and multi-dimensions for smooth problems. In order to apply the adaptive method, one should first know the locations of the local jump discontinuities. In this paper, we develop a new jump disconti-

4 nuity detection method based on the adaptive MQ-RBF method proposed in []. We will show that the adaptive method can be used as a robust method for the detection of the local jump discontinuity if applied iteratively. The iterative adaptive MQ-RBF method utilizes the epansion coefficients as the local jump discontinuity indicator. The local maimum of the epansion coefficients eists at the centers where the local jump discontinuity eists and its magnitude grows eponentially with N. The adaptive method, however, reduces its magnitude to O(N). If there eist multiple jump discontinuities, the global maimum of the epansion coefficients eists at the strongest jump discontinuity. The adaptive method is applied at those centers where the global maimum eists in order to reduce its magnitude to O(N). Then the global maimum of the epansion coefficients eists at the centers where the net strongest jump discontinuity eists and its magnitude is eponentially large. If the adaptive method is applied iteratively in such a way, all the jump discontinuities are detected successively. Furthermore, the proposed method does not require any additional condition to determine the threshold of the jump discontinuity at each iteration stage. The numerical eamples provided in this paper verify that the proposed iterative adaptive MQ-RBF method detects the multiple jump discontinuities accurately and efficiently. The structure of the paper is as follows. In Section, the iterative ɛ-adaptive method is eplained. In this section, we first briefly eplain the ɛ-adaptive method and then show how the ɛ-adaptive method can be used as a jump discontinuity detector. In Section, the numerical eamples are provided. The piecewise analytic function with multiple jumps are considered. The numerical simulation data of the shock density wave interaction is also used for the detection of the local jump discontinuities. In Section, a brief summary and conclusion are given. Iterative ɛ-adaptive method for jump discontinuity detection In this work, we consider the MQ-RBF method, that is, the RBF basis functions are ψ i (, i ) = ( i ) + ɛ i. The MQ-RBF approimation is then given by N s f,x () = λ i ( i ) + ɛ i, i X Ω, Ω. () i=. ɛ-adaptive method The ɛ-adaptive method changes the values of the shape parameters depending on the smoothness of f(). With the fied grid structure, accuracy of

5 the MQ-RBF approimation is solely determined by ɛ i. Theoretically better convergence can be obtained as ɛ i increase when f() is smooth. In practice, however, ɛ i can not be arbitrarily large due to round-off errors. If ɛ i =, the local MQ-RBF basis functions become ψ i = i, a piecewise linear function. If ɛ i =, i, the overall approimation yields linear convergence as all the basis functions are linear. The main idea of the ɛ-adaptive method is that ɛ i vanish only at the centers where the local discontinuity eists or in the small neighborhood of the local discontinuity. By letting ɛ i vanish near the local discontinuity, the local basis functions used in this region become linear. Consequently the Gibbs oscillations do not appear in the approimation and convergence away from the discontinuity is also enhanced []. The simple local ɛ-adaptive method is ɛ i, i X\S ɛ =, i S, (6) where i are the centers and the boundary set S denotes the set composed of the centers in the neighborhood of the local discontinuity. In [], it has been shown that the boundary set S composed of only two centers at the local discontinuity can considerably enhance convergence. For eample, if f() = f +, > and f() = f, < with even N on the evenly distributed grid for which f + and f are constants and f + f, the minimal boundary set is S = { N/, N/+ } for the reduction of the Gibbs oscillations.. Iterative ɛ-adaptive method In order to use the ɛ-adaptive method, one should locate the jump discontinuity first. The simplest method to locate the jump discontinuity is to use the first order derivatives at the centers. As we will show in this paper, the first order derivatives alone are not efficient for the detection of the local jump discontinuity. Instead, the epansion coefficients λ i can be efficiently used to locate the local jump discontinuities of the function as investigated in []... Single jump discontinuity For simplicity, consider the simple step function f(), [, ] with the single jump discontinuity at = such as, < f() =, >. (7)

6 Suppose that the evenly distributed center set X is given for various N. Also assume that the shape parameters are given as ɛ i = ɛ >, i =,, N and N. Without loss of generality, assume that N is even. Then the jump discontinuity eists in ( N/, N/+ ). Let = /(N ). Then i = + (i ) for i =,, N and the interpolation matri M is well structured as M = ɛ + ɛ.... ɛ ((N ) ) + ɛ... + ɛ ((N ) ) + ɛ.. (8) ɛ The interpolation matri M is a symmetric Toeplitz matri. Thus the inversion of M can be conducted efficiently. The minimum of the elements of M eists in the diagonal of M. Moreover, each ith off-diagonal has the same value and this value increases with the same rate as (i ) + ɛ. We consider the properties of the epansion coefficients. First of all, the epansion coefficients λ i are anti-symmetric and have the maimum absolute values at i = N/ and i = N/ +, that is, λ i = λ N i+ for i =,, N/ and ma i λ i = λ N/ = λ N/+ for N >. Since M is symmetric, the inverse matri M is symmetric and also symmetric around its cross-diagonal, (MN,, MN). Due to the given assumption that f( i ) = f( N i+ ) for i =,, N/, it is obvious that the epansion coefficients λ i are also antisymmetric. Furthermore the absolute maimum of M eists at M N N and M N + N + with M N N =M N + N + due to the symmetry. The antisymmetric property of λ i is also obvious since the interpolation matri is nonsingular. Using the property of f(), we have (,,,,, ) T = M (λ,, λ N/, λ N/+,, λ N ) T. Since M is symmetric, we also have (,,,,, ) T = M (λ N,, λ N/+, λ N/,, λ ) T. Adding the above two equations yields (,, ) T = M (λ + λ N,, λ N/ + λ N/+, λ N/+ + λ N/,, λ N + λ ) T. Since M is non-singular, λ i + λ N i+ =, for i =,, N/. First we consider the growth rate of λ i with the assumption that ɛ N/ = ɛ N/+ = and ɛ i = ɛ > for all i ecept N/ and N/ +. Under this assumption, the maimum absolute of λ i is given by 6

7 ma λ i = (N )/. (9) i Here we note that the shape parameters are not uniform for Eq. (9). According to Eq. (9), we know that the growth rate of ma i λ i is only linear with these non-uniform shape parameters. In order to verify Eq. (9), suppose that λ N/ = and λ N/+ =. Then with ɛ N/ = and ɛ N/+ =, s f,x () can be written as s f,x () = N/ N/+ + Then s f,x () N/ + N/+ = N λ i i=,i N, N + ( i ) + ɛ i. N λ i i=,i N, N + ( i ) + ɛ i. Let z() = s f,x () N/ + N/+. Using the given function values at the center points, we know that Consequently we obtain z( i ) =, i, i =,, N. N λ i ( j i ) + ɛ i =, j =,, N. i=,i N, N + Then we know that λ i = for all i =,, N ecept i = N/ and i = N/+. Since the interpolation matri M with ɛ i = for i = N/ and i = N/ + is still nonsingular, the epansion coefficient vector λ is uniquely determined. Since λ = (,,,,,,, )T satisfies the interpolation condition as shown above, we know that λ i = for i N, N +. Since ɛ N/ and ɛ N/+ are taken to be zero, the approimation s f,x () becomes s f,x () = λ N/ N/ + λ N/+ N/+. Since N/ =, N/+ = and λ N/ = λ N/+ λ, we have for s f,x () = λ. Since s f, ( i ) = for i, this confirms that λ = / = (N )/. Here note that λ is determined independently of the given shape parameter ɛ. Now we consider the growth rate of ma i λ i with the uniform shape parameters, i.e. ɛ i = ɛ >, i. The numerical eperiments imply that the maimum absolute value of λ i increases eponentially with N such as 7

8 ma λ i Cq αn, () i where q > and C and α are the positive constants independent of N. The constants C and α can be ɛ-dependent. The detailed analysis will follow in our future work. Figures and confirm the eponential growth of the maimum absolute value of λ i with N for the uniform shape parameters. In Figures and, the absolute values of the epansion coefficients λ i are shown in logarithmic scale for the interpolation of f() in Eq. (7). The left figure of Figure shows λ i versus i for various N, N =,, 6, 8, and. As described above, ma i λ i eists at the centers around =. λ i also decays eponentially as i increases as shown in the figure. Notice that ma i λ i eists at the two center points, i.e. at = and = for all N. λ i have the same values at these two center points. The right figure of Figure shows the absolute values of the epansion coefficients λ i versus i in logarithmic scale with a simple adaptation for N =,, 6, 8, and. For the adaptation, we use ɛ i = at i = and i = and ɛ i = ɛ =. elsewhere. As Eq. (9) says, all the epansion coefficients λ i vanish if, and λ i = (N )/ at i =,. Due to round-off errors, however, the figure shows that λ i at i, also grow eponentially as N increases. But we note that these values are still small.the left figure of Figure shows λ i versus i in logarithmic scale for N = with both the non-adaptive (the upper line) and adaptive (the lower line) methods. This figure shows that the adaptive method dramatically changes the overall shape of λ i. The right figure of Figure shows the growth of ma i λ i with N in logarithmic scale. The upper line shows the growth of ma i λ i with the non-adaptive method and the lower line with the adaptive method. As shown in the figure, without the adaptation ma i λ i grows eponentially with N. The growth of ma i λ i is, however, only linear as the growth rate is only first order as ma i λ i = (N )/ with the adaptive method. Remark The result obtained above is valid for odd N as well. Furthermore it is also valid when the discontinuity lies on a center point. For eample, suppose N is odd with i = + (i ) and f() = C, for <, f() = C for > and f() = for = where C is a non-zero real constant. Then ma i λ i also eist at = nc = and = nc+ = and λ i are antisymmetric around = nc =. Consequently, λ nc = λ nc+ and λ nc =. Here = and n N c = N+. As in the case with even N, ma i λ i eists around the discontinuity. If f() = C for > and f() = C for = and C C where C and C are real constants, ma i λ i eists eactly at =. The etension to the case of the multiple discontinuities eplained in the net sections is also valid for the case with odd N and for the case that the edge lies on a center point. 8

9 Log λ i Log λ i Fig.. The absolute values of the epansion coefficients, λ i versus in logarithmic scale for N =,,6,8, and for Eq. (7). Left: ɛ i =., for i =,,N and N. As N increases, λ i also increase. Thus the top line represents λ i for N =. Right: ɛ N/ = ɛ N/+ = and ɛ i =. for i N/,N/+. λ i become as small as machine zero for i N/,N/+ for N, but they increase as N increases due to round-off errors. Note, however, that these values are still small. Also notice that the global maimum of λ i eists at = N/ and = N/+ for any N Log λ i Log Ma( λ ) N Fig.. Left: The absolute values of the epansion coefficients λ i versus i in logarithmic scale for N =. The upper line represents log λ i with ɛ i =., i and the lower line with ɛ N/ = ɛ N/+ = and ɛ i =., i N/,N/+. Right: ma i λ i versus N in logarithmic scale. The upper line represents the case with ɛ i =., i,n and the lower line with ɛ N/ = ɛ N/+ =, N and ɛ i =. i N/,N/ +. Notice the eponential growth of ma i λ i of the upper line and the algebraic growth of the lower line... Multiple jump discontinuities Consider the case that there eist multiple jump discontinuities in the given domain. For simplicity, consider the function f() which has two jump discontinuities in Ω = [ b, b], b > such as 9

10 , f() =, < D + α, D <, () where = /(N ), D a positive integer and α a positive real number. The jumps are [f] = and [f] = α at = and = D, respectively. We assume that b D so that we can ignore the boundary effects on the epansion coefficients. If α =, these jump magnitudes are the same. We consider the case of α > so that one jump is larger than the other. We denote the jump at = D as the strong jump and the one at = as the weak jump. The function f() can be rewritten as f() = g() + h() + k(), where, g() =,, > h() = αg( D ), and k() = α. Define the epansion coefficient vectors λ g, λ h and λ α corresponding to the functions g(), h() and k(), respectively as λ g = M g, λ h = M h, λ α = M k, λ g = (λ g,, λ g N) T, λ h = (λ h,, λ h N) T, λ α = (λ α,, λ α N) T, g = (g( ),, g( N )) T, h = (h( ),, h( N )) T and k = (k( ),, k( N )) T. Then the epansion coefficients λ i are given by Furthermore λ i = λ g i + λ h i + λ α i. ma i λ h i = α ma λ g i. () i This is obvious due to the fact that f() is a linear combination of g(), h() and k(). With the assumption that α > and ɛ i = ɛ >, i, the global maimum of λ i eists at the strong jump, i.e. = D and = D +. By Eq. () and the fact that λ α i has almost the same order of magnitude for all i as k() is a constant function, it is obvious that the global maimum of λ i eists at the strong jump. Here we use the fact that λ g i and λ h i have the same decay rate. Also note that the second maimum of λ i does not necessarily eist at the weak jump. If D is large enough, the second maimum eists at the weak jump. Based on the properties of the epansion coefficients described above, we propose the iterative adaptive MQ-RBF method. With the proposed method the

11 strong and weak jumps in Eq. () can be eactly found. Let ɛ i vanish at the center point i where, for eample, λ i > ma λ i. () i Since the λ h i decay eponentially by Eq. () for given N, the first jump is detected at = D and = D + with Eq. () where the shape parameters vanish. Once ɛ i vanish at the strong jump, Eq. (9) implies that ma i λ h i O(N) and that the global maimum of λ i eists at the weak jump where ma i λ g i eists for the second reconstruction of f(). By applying Eq. () again, we locate the weak jump eactly and all the jumps are now found.. Iterative detection of the boundary set S The boundary set S in Eq. (6) is determined by the iterative adaptive method. Once the first strongest jump is located, the shape parameters at those jumps are taken to vanish. Based on the new reconstruction at the center points with the adapted shape parameters, the epansion coefficients λ i at this jump become O(N) while the epansion coefficients λ i at the second strongest jump now become relatively larger than those λ i of O(N). In such a way, the second strongest jump is located. Here we note that the first order derivative s f,x () at the first strongest jump will vanish by definition after the first iteration. If the jump indicator is defined as C i = λ i s f,x ( i), C i vanish at the jump discontinuity after the first iteration. This provides clearer contrast among C i and we use C i as the jump indicator to find the boundary set S. By Eq. () the first derivative of s f,x (y) at = y Y Ω is given by s f,x (y) = N i= λ i (y i )/ (y i ) + ɛ i, y Ω. () Define the derivative operator D such as D ij = (y i j )/ (y i j ) + ɛ j, i =,, N R, j =,, N, () where N R is the total number of the reconstruction points y i in Y. If we define the derivative vector s = (s f,x(y ),, s f,x(y NR )) T, we obtain s = D M f. (6)

12 If ɛ i = at i, s f,x ( i) is not defined as lim s f,x () = and i lim + i s f,x () =. To define s f,x ( i) at every point, we let s f,x ( i) =, if ɛ i =. Then ψ (, ɛ), X has a jump discontinuity at = i where ɛ i =. Define the concentration set C such that each element of C is defined as N C i = λ i D ij λ j, i =,, N, (7) j= where the indices i and j run from through N, not N R. We adopt the terminology of the concentration set C from the paper by Gelb and Tadmor [7]. The element of the concentration set C is the absolute value of the first order derivative of s f,x () weighted by λ i at i, that is C i = λ i s f,x ( i). By multiplying s by λ, the derivative is weighted and yields a good local nonsmoothness indicator as C i vanish where λ i become O(N) at the local jump discontinuity. With C, we define the boundary set S such as S = { i i X, C i η > }, (8) where η is a real non-zero positive tolerance level. Eq. () indicates one of the possible choices of η can be η = ma C i. i In Table, the iterative adaptive algorithm to find the local discontinuities is given. Numerical eamples. Piecewise analytic functions The first eample is the piecewise linear function with multiple jumps [f] in [, ],

13 Table The iterative adaptive algorithm to find the boundary set S based on the given initial shape parameters ɛ i and the given tolerance level η. In Step, the superscripts new and old denote the updated and previous sets respectively. In Step, denotes any matri norm and the positive constant δ is a given tolerance level to stop the iteration. Given : {ɛ i },η >,δ > Step : Compute C using Eq. (7) with {ɛ i } Step : Find S,S = { i i X,C( i ) η > } Step : ɛ new i = ɛ old i, ɛ new i = at i S and ɛ i = ɛ new i Step : Repeat Step through Step if C new C old > δ,.7,.7 <.,. < f() =, <.6,.6 <.8,.8 <, [f] = 6, =.8, =., =, =.6, =.7 (9) For the given f(), there are local jump discontinuities at =.7,.,,.6, and.8. The smallest jump [f] is [f] = at =.7 and the biggest jump is [f] = 6 at =.8. For the numerical eperiments, the initial shape parameters ɛ i = ɛ =., for i =,, N, the number of the center points N = and the number of the reconstruction points N R = are used. For the adaptation, we use the normalized concentration map Ĉ defined as Ĉ = C/ ma C i, i where recall that ma i C i is positive. With this definition we have always ma i Ĉ i =. Use the fact that λ i decay eponentially away from the local jump discontinuity in order to have the adaptation criteria η used at any iteration step such as η =. () In order to show that the first order derivative s f,x () is not efficient to locate the local jump discontinuities, the concentration map is first defined as

14 f R () C i ε i Fig.. The iterative adaptive method based on the first order derivative with C i = N, j= D ijλ j i =,,N for the first iterations for f() in Eq. (9). Top: The reconstruction f R (). Middle: The normalized concentration map Ĉ. Bottom: The shape parameters ɛ i. The left figures in each row are the results of the first reconstruction and adaptation and the far right figures the results of the th reconstruction and adaptation. For the adaptation η =. is used with N = and N R =. Note that only the strongest jump discontinuity is detected. N C i = D ij λ j, i =,, N. () j= Figure shows the results with this concentration map C, Eq. (). In the figure, the top figures show the reconstruction f R (). The figures in this row show the results at each reconstruction stage, from left to right. That is, the first figure is the first reconstruction f R () with ɛ i =., i. The second figure is the reconstruction after the first adaptation using Eqs. (6) and () and the third and fourth figures are the reconstructions after the second and third adaptations, respectively. The middle figures, from left to right, show the normalized concentration maps Ĉ at each reconstruction stage. The bottom figures show the shape parameters ɛ i versus the center points i at each adaptation stage. That is, the first figure in the bottom shows ɛ i adapted using the adaptation criteria Eq. (6) based on the first reconstruction f R () at the centers. Note that the second figure of the top figures is the reconstruction f R () based on the shape parameters ɛ i given in the first figure of the bottom figures. The

15 first figure in the top figures clearly shows the Gibbs oscillations at each jump discontinuity. The first figure in the normalized concentration map Ĉ shows that Ĉ detects the strongest jump at =.8. With the adaptation criteria, Eq. (), the shape parameters ɛ i vanish near =.8 as also shown in the first figure of the bottom figures. As shown in the following figures, however, the iterative method with the first order derivatives alone is not efficient to detect the other local jumps. With the adaptation criteria Eq. () fied at all iteration stages, the method only detects the strongest jump at =.8. In fact, the reconstruction f R () are not improved at all after the first iteration. This results clearly show that the first order derivatives are not recommended to be used alone as the jump discontinuity indicator for the proposed iterative adaptive MQ-RBF method. Remark Note that the reconstruction is not necessary at every reconstruction points y i to find the jump discontinuities. The reconstruction can be done once at the final stage. At each iteration step, the matri used to find λ i is not M R but M, that is, only the reconstruction at the given center set X is enough for the detection of the jump discontinuities. Now we use the proposed concentration map C defined in Eq. (7). With this map, the first order derivatives and the epansion coefficients are both used to locate the local jump discontinuities. Figure shows the same quantities as in Figure for the first iterations. The bottom figures of the shape parameters ɛ i also show that the iterative adaptive method detects the local jump discontinuities in the order of the jump magnitude. The figures clearly show that the shape parameters ɛ i are switched to from. at the centers near =.8([f] = 6) after the first iteration. It is interesting that the second and third strongest jumps are found simultaneously near =.([f] = ) and = ([f] = ) after the second iteration. Then the method detects the net jump near =.6([f] = ). Finally the method detects the weakest jump near =.7([f] = ). Consequently, in the top figures, it is clearly seen that the Gibbs oscillations are considerably reduced in the reconstruction f R () at each reconstruction after the adaptation method is applied. Notice that the iterative adaptive method uses only two center points around each jump discontinuity for the adaptation. In Figure, the pointwise errors of f R (), log f R () f() are shown. The top figures are the pointwise errors with the concentration map C i = Nj=,i D ij λ j =,, N based on the first order derivatives as in Figure and the bottom figures with the concentration map C i = λ Nj=,i i D ij λ j =,, N based on both the first order derivatives and the epansion coefficients. The figures show that the iterative adaptive MQ-RBF method yields highly accurate results with the concentration map C based on the epansion coefficients and the first order derivatives. We also consider the following piecewise analytic function f() with local jumps [f] given by

16 f R () C i ε i Fig.. The iterative adaptive method with the concentration map C i = N, λ i j= D ijλ j i =,,N for the first iterations for f() in Eq. (9). Top: The reconstruction f R (). Middle: The normalized concentration map Ĉ. Bottom: the shape parameters ɛ i. The left figure in each row is the result of the first reconstruction and adaptation and the far right figure the result of the th reconstruction and adaptation. For the adaptation, η =. is used with N = and N R =. Note that all the five local jump discontinuities are detected. sin(.),.7 f() = 7.cos(),.7 <.,. <.6.sin(.)),.6 <.8 cos(.) +,.8 < For the given f() there are local jump discontinuities ()., =.8.7, =.7 [f] =.76, =.6.6, =. () 6

17 Log f R () f() Log f R () f() Fig.. Pointwise errors of the reconstruction, log f R () f() for f() in Eq. (9). Top: The pointwise errors with the concentration map C based on the first order derivatives only. Bottom: The pointwise errors with the concentration map C based on the first order derivatives and the epansion coefficients. Each figure from left to right is the result after each iteration. Here we note that there eists a jump discontinuity in the first order derivative of f() at =. Figure 6 shows the results. As in Figures and, the top, middle and bottom figures show the reconstruction f R (), the normalized concentration map Ĉ and the adapted shape parameters ɛ i, respectively with the same adaptation criteria η = with N = and N R =. The same concentration map Ĉ as in Figure is used. The figures show that the iterative adaptive MQ- RBF method detects all the local jump discontinuities in the order of the jump magnitude. The bottom figures show that the iterative adaptive MQ- RBF method first detects the strongest jump near =.8([f].) after the first iteration. It detects the second strongest jump near =.7([f].7). Then it detects two other jumps simultaneously near =. and =.6. The reconstructions f R () in the top figures clearly show that the Gibbs oscillations are also considerably removed. Figure 7 shows the pointwise errors of f R (), log f R () f(). The error behavior shows the similar results as in the bottom figures of Figure. After the th iteration, all the jump discontinuities are detected and accuracy away from the jump discontinuities has been enhanced. Overall these eamples show that the iterative adaptive MQ-RBF method is very efficient and robust to 7

18 f R () C i ε i Fig. 6. The iterative adaptive method with the concentration map C i = N, λ i j= D ijλ j i =,,N for the first iterations for f() in Eq. (). Top: The reconstruction f R (). Middle: The normalized concentration map Ĉ. Bottom: The shape parameters ɛ i. The left figure in each row is the result of the first reconstruction and adaptation and the far right figure the result of the th reconstruction and adaptation. For the adaptation, the criteria η = is used with N = and N R =. Note that all local jump discontinuities are detected. Fig. 7. Pointwise errors of the reconstruction f R (), log f R () f() for f() in Eq. (). The concentration map C is defined based on the first order derivatives and the epansion coefficients. Each figure from left to right is the result after each iteration. detect the local jump discontinuities. Remark One can define the concentration map using only the epansion 8

19 coefficients λ i such as C i = λ i, i =,, N. () We find that the results with Eq. () are almost the same as those with C i = Nj= λi D ij λ j, i =,, N, but the method with Ci = Nj= λi D ij λ j yields a better performance for the eample in Section... Shock-density wave interaction We consider the D shock-wave interaction equations [] with the following initial condition (.87,.6969,.), < (ρ, u, P) = ( +. sin(k),, ),, () where k = and ρ, u and P denote the density, the velocity and the pressure, respectively. The Mach number, M = is used. In Figure 8, the solid line in each figure is the numerical solution at t = with N = with WENO- Z scheme []. The figure shows the first 8th iterations with the iterative adaptive method. The same conditions are used, that is, η = and C i = λ Nj=, i D ij λ j i =,, N. The iteration number increases from top to bottom and left to right. In each figure, the solid line represents the numerical solution of ρ and the red circles denote the possible jump locations detected. As shown in the figures, the strongest jump near =. is detected first at the first iteration. At the second iteration more center points near the strongest shock are detected. At the third iteration, the three more jumps in the left region are detected simultaneously as their jump magnitudes are almost the same. At the fifth iteration, totally 6 jumps are detected. After the fifth iteration, no more jumps are detected. The figures show that the iterative adaptive method detects the jumps very accurately with the small number of iterations. For comparison, five different edge detection methods are considered. These methods are the Prewitt, Canny, Marr/Hildretch, Fourier conjugate partial sum, and filtered Fourier gradient methods and these methods are applied for the same shock-density wave interactions as in Figure 8. For the Fourier conjugate partial sum method and the filtered Fourier gradient method, the even etension of the given data is used in order to avoid the boundary effect The authors gratefully acknowledge that the numerical data has been provided by W.-S. Don. The numerical solution is obtained with the WENO-Z method. 9

20 6 f(), ε= f(), ε= 6 f(), ε= f(), ε= 6 6 f(), ε= f(), ε= 6 6 f(), ε= f(), ε= 6 6 Fig. 8. Shock-density wave interaction and shock detection. The solid line is the numerical solution of the shock-density wave interaction equations at t = with N = computed with the WENO-Z method. From top to bottom, left to right, each figure shows the detection of the possible jumps in the numerical solution denoted by the red circle at each iteration step. Notice that at the th iteration, 6 jumps are detected and no more jumps are found after the th iteration.

21 due to the non-periodicity of the given data. The generalized conjugate partial sum is given by [7] S σ N[f](y) = N/ k= N/ i σ(k/n)sign(k) ˆf k ep(iky), where ˆf k are the Fourier coefficients, σ(k/n) the concentration function and y [, π]. For the discussion of the concentration function to enhance the convergence, see [7]. For the numerical eperiment, the Gaussian concentration function is used, i.e. σ(k/n) = ep( ɛ M ( k/n) p ), where ɛ M and p are positive constants and p is even. The values of ɛ M = and p = 8 are used. The filtered Fourier partial sum is given by f σ N () = N/ k= N/ iσ(k/n)k ˆf k ep(ik), where we use the eponential filter for the filter function σ(k/n) such as σ(k/n) = ep( ɛ M (k/n) p ). The same values of ɛ M = and p = 8 are used. For the Prewitt, Canny and Marr/Hildretch methods, the one-dimensional WENO-Z data is cast into two dimensional array such that the D array has its variation in one direction only. Figures 9A-F show the edge detections (red circles) with the iterative adaptive RBF method, Prewitt, Canny, Marr/Hildretch, Fourier conjugate partial sum and filtered Fourier gradient methods, respectively. For Figure 9E the edges are defined as the points where the concentration C() satisfies the criteria C() C o where C() = S σ N [f]()/ ma ( S σ N [f]()), and C o =. (% of the maimum of C()). For Figure 9F the edges are defined as the points where the concentration C() satisfies the criteria C() C o where C() = f σ N () / ma ( f σ N () ), and C o =. (% of the maimum of C()). For the Prewitt, Canny and Marr/Hildretch methods, the MATLAB R built-in subroutine is used, i.e. edge(, ) with the second arguments prewitt (Prewitt), canny (Canny), and log (Marr/Hildretch). As shown in the figures, all these methods find the strongest shock near. successfully. The Prewitt method, however, fails to find other shocks rather than the strongest. The Canny, Marr/Hildretch and filtered Fourier gradient methods find all 6 shocks, but these methods

22 A B f() f() 6 6 C D f() f() 6 6 E F f() f() 6 6 Fig. 9. Edge detections of shock-density wave interaction, Eq. () with different edge detection methods. A) The iterative adaptive RBF method. B) The Prewitt method. C) The Canny method. D) The Marr/Hildretch method. E) The Fourier conjugate sum method. F) The filtered Fourier gradient method. mistakenly identify the false shocks in the interval around [.6,.] where the solution is oscillatory but smooth without any discontinuity. The Fourier conjugate partial sum method finds only shocks and fails to find the weakest jump near.6. As shown in Figure 9, among these methods, the iterative adaptive RBF method is obviously the most accurate method.. Iterative method for functions without jump Finally we show how the proposed method works for the smooth function without any local jump discontinuity inside the given domain. For the smooth function, we want no jump discontinuity inside the domain to be found. In

23 .... f R ().... C i ε i Fig.. Iterative adaptive method for f() = sin(). Top: The reconstruction f R (). Middle: The normalized concentration map Ĉ. Bottom: The shape parameter ɛ i. Notice that only few boundary center points are adapted. [], it has been shown that ma i λ i eist at the domain boundaries if the given function is smooth. Furthermore, its growth is not eponential but only linear, i.e. ma i λ i O(N) if the given function is smooth. Consider the test function f() such as f() = sin(kπ), [, ]. Assume that the given function is numerically smooth, that is, N is large enough such that the modal behavior of f() is well resolved with the given N and k. For simplicity, we assume that k = and N =. For the numerical eperiment, we also use the same adaptation criteria η = and the same normalized concentration factor C i = Nj=,i λi D ij λ j =,, N. Figure shows the same quantities as in Figures, and 6. The figure shows the results for the first iterations. As shown in the figure, the adaptation occurs at the domain boundaries and no jump is found inside the domain. It is also shown that after the first iteration, the global maimum of Ĉ eists at the two center points at = and =. The result implies that the proposed method also successfully detects that there is no jump discontinuity inside the domain if the function is smooth.

24 Summary In this work, the iterative adaptive MQ-RBF method is proposed for the detection of local jump discontinuities. We show that the epansion coefficients λ i can be used as the indicator of the local jump discontinuities as the local maimum of λ i eists at the jump discontinuities and its magnitude is eponentially large. By adapting the vanishing shape parameters ɛ i at those jumps, the magnitude of λ i considerably decreases to O(N). By repeating this procedure, the proposed method detects all the jump discontinuities successively. We use the product of the epansion coefficients and the first order derivatives as the jump indicator for the numerical eamples. According to our numerical eperiments, the product of the epansion coefficients and the first order derivatives provides an accurate performance. For numerical eperiments, we use both the piecewise analytic functions and the numerical simulation data of the shock-density interaction wave equations. The numerical results show that the proposed method is efficient and accurate. For the current work, the evenly distributed center set has been used. The jump discontinuity detection with other distributions of the center set such as a random distribution will be investigated in future work. The etension of the proposed method to two dimensional problems will be also investigated in future work. Acknowledgements JHJ gratefully acknowledges the support from the NSF under Grant No. DMS JHJ also acknowledges useful communication with Edward Kansa and Wai-Sun Don. Authors also thank the anonymous referees for their helpful suggestions. References [] R. Borges, M. Carmona, B. Costa and W.-S. Don, An improved Weighted Essentially Non-Oscillatory scheme for hyperbolic conservation laws, preprint, 7. [] M. D. Buhmann, Radial Basis Functions, Cambridge UP, Cambridge,. [] M. D. Buhmann and N. Dyn, Spectral convergence of multiquadric interpolation, Proc. Edinburgh Math. Soc. () 6 (99) 9-. [] T. A. Driscoll and A. R. H. Heryudono, Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. Math. Appl. (7)

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