2009 International Joint Conference on Computational Sciences and Optimization
A New Mollification Method for Numerical Differentiation of 2D Periodic Functions Zhenyu Zhao Guangdong Ocean University College of Science 524088 Zhanjiang, China
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Zehong Meng Zhejiang University Of Finance & Economics School of Mathematics and Statistics 310018 Hangzhou, China
[email protected]
Li Xu Tianjin Univerity of Commerce College of Science 300134 Tianjin, China
[email protected]
Junfeng Liu Military Traffic Institute Department of Fundamental Courses Tianjin 300161 China
[email protected]
Abstract
follows. In section 2, we will give the method to construct approximate function. The convergence result will be found in section 3. Some numerical results are given in section 5, which show the new methods work very well.
In this paper, we present a new method for numerical differentiation of bivariate periodic functions when a set of noisy data is given. TSVD is chosen as the needed regularization technique. It turns out the new method coincides with some type of truncated Fourier series approach. A numerical example is also given to show the efficiency of the method.
2. Define of approximate function Suppose that Ω = [0, 1] × [0, 1], the following notations will be used in this paper. 1/p Lp (Ω) = {f | Ω f p (x, y)dxdy < ∞} W m,p (Ω) = {f |Dα f ∈ Lp (Ω), |α| ≤ m},
1. Introduction
where α = (α1 , α2 ), α1 , α2 are nonnegative integers and |α| = α1 + α2 .
n this paper, we deal with the problem of approximating the gradient of a bivariate periodic function when noisy data is given. The gradient knowledge is crucial in many applications of engineering, physics, and in all those applied problems in which we need to recover functions satisfying some fairness properties[3], [5]. However, It is well known that numerical differentiation is an ill-posed problem, which means, the small error of measurement may cause huge error in the computed derivatives[4], [6]. A number of techniques have been well developed for numerical differentiation in one-dimensional case[6], [7], [9], [10], [14]. Besides, as far as we know, the literatures on noisy data in two dimensions are relatively poor[2], [12], [13]. In [14], we present a mollification method to deal with numerical differentiation in one-dimensional case. In this paper, we will generalize the method to deal with 2D periodic functions. We will show that the method leads to FFT, so the method can be realized easily and fast. This paper is organized as 978-0-7695-3605-7/09 $25.00 © 2009 IEEE DOI 10.1109/CSO.2009.174
H k (Ω) f p
= W k,2 (Ω), 1/p = |f (x, y)|p dxdy Ω
especially, f = f 2 . Let C1 =C1 (R) denote the space of continuous functions with period 1 and L21 =L21 (R) denote the space of square integrable functions on the interval [0, 1] with period 1, the associated norm is given by f = f L21 =
0
1
2
|f (t)| dt
1/2 .
Finally, let Ck1 = C1 ∩ Ck (R),
Hk1 = {f |f (j) ∈ L21 , 0 ≤ j ≤ k}.
We will discuss the following questions. Problem 1. Suppose g(x, y) ∈ Hn1 (Ω) and the perturbed 205
function g δ (x, y) ∈ L2 (Ω), satisfying
where δ > 0 is a given constant called the error level, to find a function f δ (x, y) which is close to g(x, y) in the sense that
then we have the singular system{σj , vj , uj }∞ j=1 of operator I is[4] 1 σj = 2 (aj (1) + aj (2)2 )π 2 + 1 if j = 1, i.e. a1 (1) = a2 (2) = 0
lim Dα f δ (x, y) − Dα g(x, y) = 0
u1 = ψ1 (x)ϕ1 (y),
g δ − g ≤ δ
(1)
δ→0
(2)
if aj (1) = 0, aj (2) = 0, √ √ uj1 = 2ψ1 (x)ϕaj (2) (y), uj2 = 2ψ1 (x)ϕaj (2)+1 (y),
here α = (α1 , α2 ), |α| < n. First, we introduce the auxiliary equation If = g
if aj (1) = 0, aj (2) = 0,
(3)
uj1 uj2 uj3 uj4
where I : X = L2 (Ω) → L2 (Ω) = Y is an unit operator. Let operator L is defined as: ⎡ ⎤ ∂f ⎢ ∂x ⎥ (Lf )(x, y) = ⎣ ∂f (4) ⎦ ∂y
and vj = σ j u j Using the singular system of operator I, we can give the approximate function of g(x, y) as following
and the domain of L is D(L) = {f (x, y) | f ∈ H11 (Ω)}. We introduce a new inner product f1 , f2 ∗ = If1 , If2 + Lf1 , Lf2
f δ (x, y) =
(5)
(10)
where m = m(δ)is determined by the discrepancy principle ∞
g δ , ui 2 ≤ τ 2 δ 2
1, f δ (x, y) defined by (10). Suppose f † = I † g ∈ R((I ∗ I)ν ), ν ≥ 0, then
I ∗ Iφ = λ(I ∗ I + L∗ L)φ Obviously, 0 < λ ≤ 1. It can be changed into the following Helmholtz equation ⎧ 2 ⎨ ∂ φ ∂2φ + 2 = −γφ(x, y) (8) ∂x2 ∂y ⎩ φ ∈ D(L∗ L)
sup
2ν
g δ −g≤δ
f δ − f † ∗ = o(δ 2ν+1 ),
and
as δ → 0, 1
2ν
f δ − f † ∗ ≤ Cν,τ f † ν2ν+1 δ 2ν+1 , 2ν 2ν+1
(13)
(14)
1 − 2ν+1
Hence, if we define {an }∞ n=1 as a sequence which is constituted by (z1 , z2 ), where z1 , z2 are integer, satisfies
where Cν,τ = [(τ + 1) + (τ − 1) ]. Theorem 2. Suppose f (x, y) ∈ Hn1 (Ω). Then we have
an (1)2 + an (2)2 ≤ an+1 (1)2 + an+2 (2)2
f ∈ R((I ∗ I)
(9)
n−1 2
)
(15)
Obviously, we can obtain the following conclusion . Theorem 3. Suppose g(x, y) ∈ Hn1 (Ω), (1) is hold and f δ (x, y) is defined by (10), then we have
and we let β1 = 0, βj = 2j, j = 1, 2, · · · ψ1 (t) = ϕ1 (t) = 1, ψ2j (t) = ϕ2j (t) = sin(2jπt) ψ2j+1 (t) = ϕ2j+1 (t) = cos(2jπt)
Dα f δ − Dα g = o(δ
206
n−|α| n
), ∀|α| ≤ n
(16)
4. Numerical Implementation
5. Conclusion
In practical problems, the perturbed data of functions are usually given at the scattered points. In this case, our approach naturally calls for a fast fourier transform(FFT) of the data. Give knots 1 tij = (ih, jh), h = , i, j = 0, 1, · · · , N N and the noisy matrix ⎛ δ ⎞ δ δ g00 g01 , · · · , g0N δ δ δ ⎜ g10 ⎟ g11 , · · · , g1N ⎜ ⎟ δ g =⎜ . . . .. .. ⎟ ⎝ .. ⎠ δ δ δ gN 0 gN 1 , · · · , gN N
A new mollification method of to the numerical differentiation of 2D functions from noisy data is proposed and analyzed. The test numerical example presented in the paper shows that the new method works quite well.
δ is given, where gij is the perturbed data of gij = g(tij ) and the condition ⎛ ⎞ 12 N N δ ⎝ (gij − gij )2 ⎠ ≤ δ (17)
References
Acknowledgement This work is supported by the Science Foundation of Guangdong Ocean University, (No. 0812279) and youth cultivation Found of Tianjin University of Commerce (No.070117).
[1] R. A. Adams. Sobolev Spaces (Pure and Applied Mathematics). Academic Press, New York, 1975. [2] M. Bozzini and M. Rossini. Numerical differentiation of 2d functions from noisy data. Comput. Math. Appl., 45:309– 327, 2003. [3] S. R. Deans. The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley&Sons Inc, New York, 1983. [4] H. W. Engl. Regularization of Inverse Problems. Kluwer Academic, Dordrecht, 1996. [5] R. Gorenflo. Analysis and applications Abel Integral Equations (Lecture Notes in Mathematics ). Springer, Berlin, 1991. [6] M. Hanke and O. Scherzer. Inverse problems light:numerical differentiation. Am. Math. Mon., 108(6):512–521, 2001. [7] I. R. Khan and R. Ohba. New finite difference formulas for numerical differentiation. J. Comput. Appl. Math, 126:269– 276, 2000. [8] J. Locker and P. M. Prenter. Regularization with differential operators i. general theory. J. Math. Anal. Appl, 74:504–529, 1980. [9] D. A. Murio. The Mollification Method and the Numerical Solution of Ill-posed Problems. A Wiley-Interscience Publication, John Wiley&Sons Inc, New York, 1993. [10] A. G. Ramm and A. B. Smirnova. On stable numerical differentiation. Math. Comput., 70:1131–1153, 2001. [11] G. M. Vainikko. The discrepancy principle for a class of regularization methods. USSR Comp. Math. Math. Phys., 22:1–19, 1982. [12] Y. B. Wang and T. Wei. Numerical differentiation for twodimensional scattered data. J.Math.A-nal.Appl., 312:121– 137, 2001. [13] T. Wei, Y. C. Hon, and Y. B. Wang. Reconstruction of numerical derivatives from scattered noisy data. Inverse Problems, 21:657–672, 2001. [14] Z. Y. Zhao and G. Q. He. Reconstruction of high order derivatives by a new mollification method. Appl. Math. Mech-Engl., 29(6):352–373, 2008.
i=0 j=0
is assumed. One numerical examples will be given to verify the effort of our proposed scheme. We let N = 128, the perturbed discrete data are given by g δ (xi , yj ) = g(xi , yj ) + ij ,
|ij | < δ1 ,
{ij }N i,j=0
are generated by Function (2 × rand(N + where 1) − 1) × δ1 in Matlab. Example 1. g(x, y) = − exp(cos(4πx)) exp(sin(4πy)) Obviously, g(x, y) ∈ H∞ 1 (Ω), So we have the convergence 2ν ) of the method is O(δ) by theorem 3. The rates (μ = 2ν+1 numerical results can be found in Table 1. The numbers Table 1. Results of example 1 δ1 m ξ fxδ − gx 2 gx 2 fxδ − gx 2 δ fyδ − gy 2 gy 2 fyδ − gy 2 δ f δ − g2 g2 f δ − g2 δ
1e − 1 115 0.4276
1e − 2 199 0.3115
1e − 3 319 0.7805
0.0762
0.0126
0.0037
21.9929
31.4714
39.2506
0.0764
0.0126
0.0037
22.5036
32.1153
40.1486
0.1818
0.0394
0.0124
1.2708e+003
2.3872e+003
3.2182e+003
in rows 1, 3 and 5 show that the method work very well. Moreover, the numbers in rows 2, 4 and 6 exhibit the ratios are stable when δ → 0.
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