RECONSTRUCTION OF NUMERICAL DERIVATIVES FROM SCATTERED NOISY DATA. 1. Introduction

Transcription

1 RECONSTRUCTION OF NUMERICAL DERIVATIVES FROM SCATTERED NOISY DATA T. WEI, Y. C. HON, AND Y. B. WANG Abstract. Based on the thin plate spline approximation theory, we propose in this paper an efficient regularization algorithm for the reconstruction of numerical derivatives from two-dimensional scattered noisy data. An error estimation that deduces a good regularization parameter is given. Numerical results show that the proposed method is efficient and stable.. Introduction The problem to reconstruct numerical derivatives from noisy observational data arises from several research disciplines, for instances, image processing [5], Volterra integral equation [4, 6], analyzing photoelectric response data [4] and identification [8]. This numerical differentiation problem is well known to be ill-posed in the sense that a small noise in measurement data can induce a huge error in the approximated derivatives. There are some computational methods proposed recently for one dimensional case [, 7, 5, 6, 7] but so far only a very few results on higher dimensional cases have been reported []. These papers fall into three categories: difference methods [, 7, 5, 7], interpolation methods [6] and regularization methods [, 9, 9]. The finite difference method is simple and effective for precise data. For noisy data, the chosen stepsize, as a regularization parameter, must depend on the level of noise. This restricts the choice of specified data points used in the computation. Lagrangian numerical differentiation technique is optimally stable [6] but suffers from the limited choice on the interpolation nodes in order to minimize the error. To solve effectively the ill-posed problems, regularization methods play an important role [8]. Most of the regularization procedures for Key words and phrases. Numerical derivatives, Thin plate spline, Tikhonov regularization. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 78/P). The third author was also supported by the NSF of China (7).

2 T. WEI, Y. C. HON, AND Y. B. WANG numerical differentiation [, 9] make use of the variational approach to solve a regularized functional. Once the regularization parameter is chosen, the approximation to the unknown function and its derivatives are then obtained. Unfortunately, the determination of optimal parameter is generally a nontrivial task. Recently, Wang et. al [9] gave an effective method for treating the numerical differentiation by Tikhonov regularization in one dimension. With a simple strategy to choose the regularization parameter, the numerical results are computed much faster than using the discrepancy principle in [9]. Based on their work we extend the result to two-dimensional case and give an error estimation from which a good regularization parameter can be devised. Following the idea given in [, ], we use the thin plate spline approximation to obtain the numerical derivatives for noisy scattered data in two- dimension. The minimizer of the objective functional can be expressed in terms of a linear combination of thin plate spline functions and a linear polynomial (see [,,, ] for details). Furthermore, we apply the idea given in [9] and the results obtained in [] to give an error estimation for computing the numerical derivatives from which a priori strategy for choosing a good regularization parameter can be derived. Numerical results for various cases show that this selection method for the regularization parameter is effective and stable.. Formulation of problem and solution Let y = y(x) be a function in R. Suppose that the noisy scattered data ỹ i are collected at points x i R, i =,,, n. The problem is to find a function such that its partial derivatives approximate the exact partial derivatives of the function y(x). Let {x, x,, x n } be a finite set of distinct points in R which are not collinear and Ω be its convex hull. Suppose T h be a triangular mesh of Ω with nodes x, x,, x n and denote h = max T T h diam(t ). Let s i be the total area of the triangles that connect with node i and s be the area of the domain Ω. Suppose that the given noisy samples ỹ i of the values y(x i ) satisfy ỹ i y(x i ) σ i δ, i =,,, n,

3 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA where δ is a positive constant denoting the maximum noise level. Let I e = { i σ i = } be an index set of points with exact measurement data and I a = {,,, n} \ I e be an index set of the noisy data. We aim at finding an approximate function f (x) of y(x) so that f y L (Ω) converges to zero as δ and h tend to zero. Denote x = (ξ, η) and define a functional ( ) () R(f, g) = ( f g ξ )( ξ ) + ( f ξ η )( g ξ η ) + f ( η )( g η ) dξdη and R () I(f) = R(f, f). Define X = {f f C (R ), D f L (R ), Df < } where Df and D f denote the first and the second derivatives of f respectively; and Y = {f f X, f(x j ) = y j, j I e }. Analogous to the approach in [, ], the problem is changed to seek a solution f α in space Y that minimizes the following functional () Φ(f) = ( ) s i f(xi ) ỹ i + αi(f), s σ i i I a where ασ i are regularization parameters in the sense of Tikhonov [8]. In this paper, we consider the following two problems: () How to find f α Y such that Φ(f α ) Φ(f) for all f Y? () Assume that such f α has been obtained. How to choose a good parameter α so that f α (x) converges to the exact function y(x) at a convergence order as high as possible? The first problem has been solved in [, ] by using the reproducing kernel theory in Hilbert space. In this section we give a simpler proof by using mathematical analysis. For the second problem, we give a new convergence order estimate in the next section.

4 4 T. WEI, Y. C. HON, AND Y. B. WANG Theorem.. Suppose f α is a linear combination of thin plate splines and a linear polynomial: n (4) f α (x) = c j x x j log x x j + d + d ξ + d η, j= where x = (ξ, η), x x j is Euclidean distance in R and {c j } n, {d j } are constants satisfying the following linear system of equations (5) f α (x i ) + 4πα s s i σ i c i = ỹ i, i I a, (6) f α (x i ) = y(x i ), i I e, n n (7) c j = c j x j =. j= j= Then f α is the unique minimizer of the functional (), i.e. Φ(f α ) = min f Y Φ(f) Φ(y). We first need the following Lemmas. Lemma.. Let g be any function of the form (4) whose coefficients satisfy equation (7). Then, for sufficiently large x, g satisfies (8) g(x) c x and g(x) c, where c and c are positive constants. Furthermore, for sufficiently large x, the moduli of all second and third derivatives of g are bounded above by c x and c 4 x, respectively, where c and c 4 are also positive constants. Proof. See Powell s paper []. Lemma.. Let f be any function of the form (4) whose coefficients satisfy (7). Then for any g X, we have n (9) R(f, g) = 8π c j g(x j ). Proof. From Lemma., we know that the second derivative of f is square integrable over the part of R which is outside the circle {x : x M }, where M is a sufficiently large constant. Since f is infinitely differentiable on any region of R that excludes the points x j, j =,,, n, its second derivatives are square integrable on those regions which are closed and bounded. Since the second derivative j=

5 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 5 of f is O(log x x i ) near the point x i, it is square integrable on any bounded region including point x i. It is then easy to see that f C (R ), and hence f X and R(f, g) is well defined. Since f is a function of the form (4), f = 8π n j= c jδ(x x j ) in the sense of distribution. That is, for every ϕ(x) C (R ), we have () R(f, ϕ) = ( f, ϕ) L = 8π n c j ϕ(x j ). From the results on Sobolev space [], we know that H (R ) C (R ) and j= C (R ) is dense in H (R ). Since the second derivative of f is also square integrable, equation () holds for any function in H (R ). Let ψ C (R ) satisfy ψ(x) = if x and ψ(x) = if x. Suppose that D α ψ M for any multi-index α whose degree α and M is a constant. For any g X, let g m = g(x)ψ m (x) where ψ m = ψ( x m ), m is a positive number, then g m H (R ). Thus we have () R(f, g m ) = 8π n c j g m (x j ), for m =,,. j= It is now clear that for sufficiently large m, n j= c jg m (x j ) = n j= c jg(x j ). In the following, we show that R(f, g m ) converges to R(f, g) as m tends to. Since () R(f, g m ) ( ) = ( f g R ξ )( ξ ) + ( f ξ η )( g ξ η ) + f ( η )( g η ) ( + ( f ψ m ξ )( ξ ) + ( f ψ m ξ η )( ξ η ) + f ψ m ( η )( η ) + R R ( ( f ξ )( g +( f g )( η η )( ψ m η ) ξ )( ψ m ξ ) + ( f ξ η )( g ξ ) dξdη. ψ m η + g ψ m η ξ ) ψ m dξdη ) gdξdη It suffices to show that the first part of integrations in the right-hand side of () converges to R(f, g) as m tends to and the second and third parts both converge to. Note that for any multi-index β, β, () D β ψ m = ( m ) β D β ψ( x m ) M( m ) β.

6 6 T. WEI, Y. C. HON, AND Y. B. WANG Each integrand in the first part of integrations in the right-hand side of () is bounded by M D f(x)d g(x) and hence is integrable in R. Using the dominant convergence theorem, we know that this first part of integrations converges to R(f, g) as m. (4) For sufficiently large m, from Lemma. and (), we have R f(x) ξ g(x) ξ ψ m (x) ξ dx CM m m x m x dx, as m. All the other terms in the third part of integration of the right-hand side of () have the same convergence. The bound of g in X implies that g(x) is at most a constant multiple of x when x is large. Similarly, we have (5) R f(x) ψ m ξ ξ g dx CM m m x m x x dx, as m. All the remained terms in the second integration of the right-hand side of () have the limit zero. Therefore, by letting m tend to in equation (), we then obtain (6) R(f, g) = 8π n c j g(x j ). Proof of Theorem.: For any f Y and f α given by (4)-(7), denote function j= g = f f α. From Lemma., we have n (7) R(f α, g) = 8π c i g(x i ) i= = 8π i I a c i g(x i ) = s i α s i I a (f α (x i ) ỹ i )g(x i ) σi. Note that Φ(f) Φ(f α ) = ( ) s i f(xi ) f α (x i ) + α (I(f) I(f α )) s σ i i I a + ( ) s i g(xi )(f α (x i ) ỹ i ) s σ i I a i = I + αi,

7 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 7 and I = I(f) I(f α ) = I(f f α ) + R(f α, g) = I(f f α ) ( (fα (x i ) ỹ i )g(x i ) α i I a s i s σ i ). Thus, we have (8) Φ(f) Φ(f α ) = ( ) s i f(xi ) f α (x i ) + αi(f f α ), s σ i i I a which indicates that f α is a minimizer of Φ. Furthermore, if we assume that there exists another function f Y such that Φ(f ) = Φ(f α ), then substituting f into (8) and note that (6), we have I(f f α ) =, f (x i ) = f α (x i ), i =,,, n. Hence f f α is a linear polynomial. Since the points x, x,, x n are not collinear, we have f = f α, i.e. the minimizer of functional () is only given by (4) and (5)-(7). Theorem.4. There is a unique solution for the linear system of equations (5),(6) and (7). Proof. Let A = ( x i x j log x i x j ) n n and D = diag(β,, β n ), y = (y,, y n ) T in which β i and y i are given by follows 4παss i σi β i =, i I a, ỹ i, i I a,, y i =, i I e, y i, i I e. Denote the transpose of matrix P to be P T = ξ ξ ξ n η η η n, then the linear system of equations (5), (6) and (7) can be rewritten as (9) A + D P c = y, P T d

8 8 T. WEI, Y. C. HON, AND Y. B. WANG where c = (c, c,, c n ) T and d = (d, d,, d k ) T are vectors of coefficients in (4). Suppose (a T, b T ) = (a, a,, a n, b,, b k ) is any solution for the homogeneous equations of (9), i.e. (A + D)a + P b = and P T a =. This implies a T (A + D)a =. In the following we show that a T Aa. Since a T Da, we have a T Da = and hence a i = for all i I a. Denote g(x) = n i= a j x x j log x x j. From P T a = and Lemma., we have () I(g) = n a i g(x i ) = i= n a i a j x i x j log( x i x j ) = a T Aa. i,j= Combining these results gives () A P a = A + D P P T b P T a b =. When points x, x,, x n are not collinear, the (n + ) (n + ) matrix A P P T is nonsingular (see Powell []). Thus (a T, b T ) = and the matrix A + D P P T is nonsingular. The proof is completed. Note that a special case I a =, that is, formula (4) and (6), (7) give a thin plate splines interpolation on the data {(x i, y i ), i =,,, n}. Then from Theorem., we know the interpolation function minimizes the functional I(f) in space Y. In fact, we have Corollary.5. Suppose that I a = and y X and s is a thin plate spine interpolation function given by (4), (6) and (7). Then we have () I(s y) + I(s) = I(y) Proof. It is easy to complete the proof by using the fact R(s, s y) = that comes from Lemma..

9 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 9. Error estimation In this section, we give an error estimation on the proposed method. The result will use the following lemma given by Powell s paper []. Lemma.. Let {x, x,, x m } be any finite set of distinct points in R that are not collinear and {c, c,, c m } be any real coefficients that satisfy the conditions () m m c i = c i x i =. i= i= Furthermore, let g be any function from R to R that has square integrable second derivatives. Then the linear functional m (4) L(g) = c i g(x i ) has the property m m (5) L(g) (8π) I(g) c i c j φ( x i x j ) where φ(r) = r log(r). i= i= j= /, Let e = f α y be the error function between the thin plate spline approximation f α and the exact function y. The error estimates on e L (Ω) and e L (Ω) are given in the following theorems. Hereafter e L (Ω) denotes the L (Ω) norm of function e(x). Theorem.. For any y X and f α given in Theorem., we have (6) f α y L (Ω) (6π) / (s log ) / ( α + I(y) ) / h +6s / ( + αi(y)) / δ. Proof. For every x Ω, suppose that x is in a triangle ijk with vertices x i, x j, x k. Then x = λ x i + λ x j + λ x k, λ + λ + λ = and λ, λ, λ. The conditions of Lemma. are satisfied if we let m = 4, c = λ, c = λ, c = λ, c 4 =, x = x i, x = x j, x = x k, x 4 = x and let g be the error function e = f α y. We have (7) 4 L(e) = c i e(x i ) i= = λ e(x i ) + λ e(x j ) + λ e(x k ) e(x),

10 T. WEI, Y. C. HON, AND Y. B. WANG and hence, (8) e(x) L(e) + λ e(x i ) + λ e(x j ) + λ e(x k ) [ (8π) I(e)C(λ) ] / + [ e (x i ) + e (x j ) + e (x k ) ] /, where (see Powell s paper []) (9) 4 4 C(λ) = λ i λ j φ( x i x j ), i= j= (h /) log. Hence, for every x ijk, we have () e (x) (4π) I(e)C(λ) + 6 [ e (x i ) + e (x j ) + e (x k ) ]. Integrating the above inequality on the triangle ijk, we get () ijk e dx (4π) I(e)C(λ)vol( ijk ) + 6 [ e (x i ) + e (x j ) + e (x k ) ] vol( ijk ). Summing over all triangles of Ω then gives n () e dx (4π) I(e)C(λ)s + 6 s i e (x i ). Furthermore, () n i= Ω s i s e (x i ) s i δ s i I a δ ( + Φ(y)), i= (f α (x i ) ỹ i ) + (ỹ i y i ), σ i δ ( + αi(y)). From (4) αi(f α ) Φ(f α ) Φ(y) + αi(y), we then have (5) I(f α ) α + I(y) and (6) I(e) (I(f α ) + I(y)) (α + I(y))

11 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA Substituting equations (), (6) and (9) into () yields (7) e dx (6π) s log ( α + I(y) ) h Ω +6sδ ( + αi(y)). Hence, we have (8) e L (Ω) (6π) / (s log ) / ( α + I(y) ) / h +6s / ( + αi(y)) / δ. Lemma.. Let Ω be a bounded domain in R satisfying the cone property. There exists a constant K = K(Ω) such that for < ε and for any u H (Ω), (9) u L (Ω) Kε(I(u)) / + Kε u L (Ω). Proof. see Adams []. Theorem.4. Suppose that f α is a thin plate spline approximation given by Theorem.. If y X, then for sufficiently small δ and h we have ( (4) f α y L (Ω) Ks/4 D h / + D δ /), where D and D are constants depending on parameter α and I(y). Proof. Denote e(x) = f α (x) y(x). For sufficiently small δ and h, from Theorem. we can get e L (Ω) Theorem., we have (4) e L (Ω) K e / where D =. By putting ε = e / L (Ω) L (Ω) [ ] (I(e)) / + [ K (6π) /4 (s log ) /4 ( α + I(y) ) /4 h / in Lemma. and using + 6 / s /4 ( + αi(y)) /4 δ /] [ ] / α / + (I(y)) / + = Ks /4 D h / + Ks /4 D δ /, [ (6π) /4 (log ) /4 ( α + I(y) ) ] [ ] /4 / α / + (I(y)) / +

12 T. WEI, Y. C. HON, AND Y. B. WANG and D = [ ] [ ] 6 / ( + αi(y)) /4 / α / + I / (y)) +. Remark.5. If the value of α = is chosen as one in the paper [9], then D and D are constants depending only on I(y). This implies that this choice of the parameter α is admissible in our proposed method. Remark.6. We can rewrite the right hand side of (4) as f α y L (Ω) Ks /4 [M α /4 + M + M α /4 + M 4 α / + M 5 α /4 ], where M = 6 / δ(i(y)) /4 + 6 / δ(i(y)) /4, M = /4 π /4 (log()) /4 h(i(y)) /4 +6 / 5/4 δ(i(y)) / + 6 / /4 δ +π /4 /4 (log()) /4 h(i(y)) /4, M = /4 π /4 /4 (log()) /4 h(i(y)) / +π /4 6 /4 (log()) /4 h + / δ(i(y)) /4, M 4 = /4 6 / δ + π /4 /4 / (log()) /4 h(i(y)) /4, M 5 = π /4 /4 / (log()) /4 h. By choosing a suitable parameter α > such that [M α /4 + M + M α /4 + M 4 α /4 + M 5 α + M 6 α 5/4 ] = min, the corresponding solution f α provides a more accurate approximation to the solution. In numerical computations, we can replace I(y) by I( s) in which s is the interpolation function of form (4) under the requirement of (7) on the noisy data {(x i, ỹ i )} n i=. If the domain Ω is a square and all measurement data are given on points {x j } n j= which generate a square partition on Ω, then we can modify the functional () to be (4) Φ(f) = ( ) f(xi ) ỹ i + αi(f). n i I a σ i

13 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA The minimizer f α of functional (4) is determined by (4), (6), (7) and following substitution of (5): (4) f α (x i ) + 8πnασi c i = ỹ i, i I a. Noting that the formula (9) under this special case will be changed to the following form as given in []: 5 5 (44) C(λ) = λ i λ j φ( x i x j ), i= j= 4 (log )h, where h is the side length of each small square. By the similar proof, we can then give an error estimation for the function and its derivatives in the following theorem. Theorem.7. If Ω is a square [a, b] [a, b] and all points {x j } n j= generate a square mesh with size h h, h = (b a)/( n ), then we have the following estimations (45) f α y L (Ω) (8π) / ( log ) / (b a) ( α + I(y) ) / h +6(b a) ( + αi(y)) / δ, and (46) f α y) L (Ω) K(b a) / [ / α / + (I(y)) / + ] [ (8π) /4 ( log ) /4 ( α + I(y) ) /4 h / + 4 ( + αi(y)) /4 δ /], where f α is given by (4), (6), (7) and (4). Remark.8. By choosing a suitable parameter α > such that the right hand side of (46) except constant K attains its minimum, the corresponding solution f α is a more accurate approximation. 4. Numerical results To verify the effectiveness of the proposed algorithm, we given in this section several tests under various cases. We take a smooth function y = sin(πξ)sin(πη)e (ξ +η ) for illustration.

14 4 T. WEI, Y. C. HON, AND Y. B. WANG To estimate the computational error of the proposed approximation, we choose N test points {t i } N i= over domain Ω and then compute the Root Mean Square Errors by the following formula (47) E(f α ) = N and (48) E( f α ) = N where is the -norm in R. N (f α (t i ) y(t i )), i= N f α (t i ) y(t i ), i= Case : Ω is a square domain [, ] [, ] and the points are uniformly distributed on Ω and generate a partition of h h square grids. The noisy data are generated by adding random numbers uniformly distributed over [ δ, δ]. In our computation we take h =. and δ =.5. The numerical results are shown in Figures -. Figure (a) displays the surface of exact function y and Figure (b) is the plot of its approximation f α while α =.54 given by Remark.8. In this case E(f α ) =.5, E( f α ) =.8. In Figure, we display five curves of function y(ξ, η and f α (ξ, η) for η =,,,, respectively. The solid line represents the exact function and the dotted line represents its numerical approximation. We can observe from the figures that the approximation f α matches the exact solution y well which indicates that the proposed priori choice of the regularization parameter works fine. Case : we partition the square domain [, ] [, ] into small square grids like Case and further divide each square grid into two triangles by using their diagonal. The length of the longest triangle side is h =.. The noisy data are generated by adding random numbers uniformly distributed over [ δ, δ]. In this case we take δ =.5. In Figure, we plot the curves for y(ξ, η) and f α (ξ, η) while η =,,,, respectively. In Figure (a), we choose α = and the root mean square errors are E(f ) =.86, E( f ) =.45. In Figure (b), we use the parameter α =.46 given in Remark.6 and the root mean square errors are then E(f α ) =., E( f α ) =.75. The numerical results in this case show that even for large noise, the proposed regularized parameter α gives a more accurate approximation than the constant parameter α =.

15 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 5 Parameter α= y.5 f α.5.5 η ξ η ξ (a) (b) Figure. The exact gradient function and its approximation. (a) y (b) f α for α =.54. α=.54, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure. The curves of functions y(ξ, η) and f α (ξ, η) for η =,,,, respectively. Case : we choose 665 scattered points on [, ] [, ] as shown in Figure 4. The length of longest triangle side is given by h =.58. The noisy data are generated by adding random numbers uniformly distributed over [.5,.5]. The computational results are displayed in Figure 5 in which curves have the same sense with Figure 4. In Figure 5(a), we take α = and RMSEs are E(f ) =.8, E( f ) =.96. In Figure 5(b), α =.4 and E(f α ) =.6, E( f α ) =.. All the numerical results show that for small noise f α match y well for two different parameters α. But the choice strategy given in Remark.6 works more accurate. Our proposed method is much more simple compared with other methods that use the discrepancy principle in [9].

16 6 T. WEI, Y. C. HON, AND Y. B. WANG α=, η=,,,, α=.466, η=,,,,.5.5 y(ξ,η) and f α (ξ,η).5 y(ξ,η) and f α (ξ,η) ξ ξ (a) (b) Figure. The curves of function y(ξ, η) and its approximation f α (ξ, η) for η =,,,,. (a) α = (b) α =.46. Figure 4. A triangular subdivision of the scattered points. Modifying the number n of specified points in Case and other parameters are fixed as δ =., ỹ i = y i +δsin(/πξ i )sin(/πη i ), i =,,, n. The root mean square errors for the function y and its gradient y with respect to n are shown in Table. The upper value is E(f α ) and the lower one is E( f α ) in every sheet of tables. Numerical results show that even for dense specified points, the proposed algorithm can given an accurate approximation when the finite difference method doesn t work. Table gives the root mean square errors for the function y and its gradient y under different noise levels δ for Case. In the computation, we take n =, ỹ i = y i + δsin(/πξ i )sin(/πη i ), i =,,, n. Numerical results shown in

17 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 7 α=, η=,,,, α=.44, η=,,,,.5.5 y(ξ,η) and f α (ξ,η).5 y(ξ,η) and f α (ξ,η) ξ ξ (a) (b) Figure 5. The plots of the function y and its approximation f α for η =,,,,. (a) α = (b) α =.4. n = n = n = α = α = α α Table. Root mean square errors with respect to the numbers of total specified points. δ =. δ =. δ =.5 δ =. δ =.5 δ =. α =.69e α = α α Table. Root mean square errors with respect to noise levels δ. Table indicate that the proposed algorithm is stable for the choice of regularization parameters and effective for the data with these noise levels.

18 8 T. WEI, Y. C. HON, AND Y. B. WANG 5. Conclusion In this paper, we provide a thin plate spline approximation to a classical illposed problem numerical derivatives of two dimensional scattered data. A simple algorithm for the choice of a suitable regularization parameter is proposed. Proof on the convergence rate is given for the case when the second order derivative of the exact solution is square integrable. Numerical results for various cases show that the proposed method is effective and stable. References [] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 975. [] R. S. Anderssen and M. Hegland, For numerical differentiation, dimensionality can be a blessing!, Mathematics of Computation, 68(999), -4. [] J. Cullum, Numerical differentiation and regularization, SIAM Journal on Numerical Analysis, 8(97), no., [4] J. Cheng, Y.C. Hon and Y.B. Wang, A numerical method for the discontinuous solutions of Abel intergral equations.(preprint) [5] S. R. Deans, Radon transform and its applications, A Wiley-Interscience Publication, 98. [6] R. Gorenflo and S. Vessella, Abel integral equations, Analysis and applications, Lecture Notes in Mathematics, 46, Springer-Verlag, Berlin, 99. [7] C. W. Groetsch, Diferentiation of approximately specified functions, The American Mathematical Monthly, 98(99), [8] M. Hanke and O. Scherzer, Error analysis of an equation error method for the identification of the diffusion coefficient in a quasi-linear parabolic differential equation, SIAM Journal on Applied Mathematics, 59(999), no., -7. [9] M. Hanke and O. Scherzer, Inverse problems light: numerical differentiation, The American Mathematical Monthly, 8(), no. 6, 5-5. [] F. J. Hickernell and Y.C. Hon, Radial basis function approximation of the surface wind filed from scattered data, International Journal of Applied Science and Computations, 4(998), no., -47. [] F. J. Hickernell and Y. C. Hon, Radial basis function approximations as smoothing splines, Applied Mathematics and Computation, (999), -4. [] W. Light, Advances in numerical analysis, Vol.II, Wavelets, Subdivision Algorithms, and Radial Basis Fuctions, Oxford Science Publication, 99. [] M. J. D. Powell, The uniform convergence of thin plate spline interpolation in two dimensions, Numerische mathematik, 68(994), 7-8.

19 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 9 [4] C. K. Pallaghy and U. Luttge, Light-induced and H ion fluxes and bioelectric phenomena in mesophyll cells of atriplex spongiosa, Zeit. fuer Pflanz., 6(97), [5] R. Qu, A new approach to numerical differentiation and integration, Mathematical and Computer Modelling, 4(996), no., [6] T. J. Rivlin, Optimally stable Lagrangian numerical differentiation, SIAM Journal on Numerical Analysis, (975), no. 5, [7] A. G. Ramm and A. B. Smirnova, On stable numerical differentiation, Mathematics of Computation, 7 (), -5. [8] A. N. Tikhonov and V. Y. Arsenin, Solutions of ill-posed problems, Winston and Sons, Washington 977. [9] Y. B. Wang, X. Z. Jia and J. Cheng, A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problem, 8(), [] G. Wahba and J. Wendelberger, Some new mathematical methods for variational objective analysis using splines and cross validation, Monthly Weather Review, 8(98), -4. [] G.Wahba, Spline Models for Observational Data, SIAM, Philadelphia, 99.

20 T. WEI, Y. C. HON, AND Y. B. WANG.5 y.5.5 η ξ Figure (a)

21 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA Parameter α= f α.5 η ξ Figure (b)

22 T. WEI, Y. C. HON, AND Y. B. WANG α=.54, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure

23 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA α=, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure (a)

24 4 T. WEI, Y. C. HON, AND Y. B. WANG α=.466, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure (b)

25 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 5 Figure 4

26 6 T. WEI, Y. C. HON, AND Y. B. WANG α=, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure 5(a)

27 NUMERICAL DERIVATIVES FOR SCATTERED NOISY DATA 7 α=.44, η=,,,,.5 y(ξ,η) and f α (ξ,η) ξ Figure 5(b)

28 8 T. WEI, Y. C. HON, AND Y. B. WANG Department of Mathematics, City University of Hong Kong and department of Mathematics, Lanzhou University, China address: Department of Mathematics, City University, Hong Kong, China address: Department of Mathematics, Fudan University, China address: