an optimal filtering method for stable analytic continuation

Journal of Computational and Applied Mathematics 236 (2012) 2582–2589

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Journal of Computational and AppliedMathematics

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An optimal filtering method for stable analytic continuation

Hao Cheng, Chu-Li Fu ∗, Xiao-Li FengSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China

a r t i c l e i n f o

Article history:Received 15 March 2011Received in revised form 29 October 2011

Keywords:Analytic functionNumerical analytic continuationIll-posed problemFiltering regularization

a b s t r a c t

In this paper, we consider the problem of numerical analytic continuation of an analyticfunction f (z) = f (x + iy) on a strip domain Ω+

= z = x + iy ∈ C | x ∈ R, 0 < y < y0,where the data is given approximately only on the real axis y = 0. This problem is severelyill-posed: the solution does not depend continuously on the given data. A novel method(filtering) is used to solve this problem and an optimal error estimate with Hölder type isproved. Numerical examples show that this method works effectively.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The problem of numerical analytic continuation is a classical problem in the theory of the complex analysis, whicharises from many real applications [1,2]. Some uniqueness results for this problem have been obtained in the 19th centuryand have been presented in the majority of textbooks. However, the stable numerical analytic continuation is a ratherdifficult problem. In general, this problem is severely ill-posed, i.e., the solution does not depend continuously on the data,a small perturbation in the data may cause dramatically large error in the solution, so only stable methods are of practicalinterest. To the authors’ knowledge, except for some theoretical results for the conditional stability [3] and rather complexcomputational techniques [4–6], there are few results both in theory and algorithm based on the modern regularizationtheory which has been developed intensively in the last few decades.

In recent years, the problem of analytic continuation in the following strip domains Ω and Ω+ has been considered[7,8], where

Ω = z = x + iy ∈ C| x ∈ R, 0 < |y| < y0

and

Ω+= z = x + iy ∈ C| x ∈ R, 0 < y < y0,

i is the imaginary unit and y0 is a positive constant. In [7,8] the authors give the Fourier and modified Tikhonov methods,respectively. They obtain order optimal error estimates with an effective algorithm. However, these error estimates are notoptimal in theory.

In the present paper, we will consider the problem of analytic continuation on domain Ω+ again.

Problem 1.1. Assume the function f (z) = f (x + iy) is an analytic function on Ω+, and the data is given approximatelyon the real axis y = 0, i.e., f (z)|y=0 = f (x) is known approximately, and the noisy data is denoted by fδ(x). We want todetermine the value of function f (z) on Ω+ by using data fδ(x).

The project is supported by the NNSF of China (Nos. 11171136, 11126187).∗ Corresponding author.

E-mail address: [email protected] (C.-L. Fu).

0377-0427/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.cam.2011.12.016

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H. Cheng et al. / Journal of Computational and Applied Mathematics 236 (2012) 2582–2589 2583

Another new regularization method (optimal filtering) will be given, which is motivated by some earlier works [9–11]for the inverse problem of heat conduction. The benefit of this method is that the optimal error estimate can be obtained,which is the fastest convergence velocity, i.e., any improvement of this result is impossible in theory. We will also providesome numerical examples to verify the effectiveness of this method.

2. The filtering regularization and error estimate

Let g(ξ) denote the Fourier transform of the function g(x), which defined by

g(ξ) =1

√2π

∞

−∞

e−ixξg(x)dx, i =√

−1. (2.1)

Assume function f (z) = f (x + iy) be analytic in domain Ω+ and

f (· + iy) ∈ L2(R) for 0 ≤ y ≤ y0. (2.2)

Due to [7,8], we know

f (z) = f (x + iy) =1

√2π

∞

−∞

eixξ e−yξ f (ξ)dξ . (2.3)

The ill-posedness of the analytic continuation problem can be seen from (2.3), the details can be found in [7,8].Assume the exact data f (x) and the measured data fδ(x) both belong to L2(R) and satisfy

∥f − fδ∥ ≤ δ, (2.4)

where δ > 0 denotes the noise level, and ∥ · ∥ denotes the L2-norm. Moreover, assume there hold the following a prioribound

∥f (· + iy0)∥ ≤ E, (2.5)

where E is a fixed positive constant.Due to (2.3) we know

f (· + iy)(ξ , y) = e−yξ f (ξ), (2.6)

and therefore

f (· + iy0)(ξ , y) = e−y0ξ f (ξ), f (ξ) = ey0ξ f (· + iy0)(ξ , y). (2.7)

Define the filtering function ρ(ξ, y) as

ρ(ξ, y) =

e−yξ , ξ ≥ 0,ρ1(ξ , y), ξ < 0, (2.8)

where

ρ1(ξ , y) =

e−yξ , e−yξ

≤ β(y),β(y), e−yξ

≥ β(y),(2.9)

with

β(y) =

1 −

yy0

Eδ

yy0

. (2.10)

Let the approximate solution v(z) = v(x + iy) on Ω+ be defined by its Fourier transform with respect to variable x:

v(ξ + iy) = ρ(ξ, y)fδ(ξ), (2.11)

or equivalently

v(x + iy) =1

√2π

∞

−∞

eixξρ(ξ, y)fδ(ξ)dξ, (2.12)

where ρ(ξ, y) is the filtering function given by the above formulas (2.8)–(2.11).The following lemma is known, which gives the optimal error bound for Problem 1.1.

Lemma 2.1 ([8]). Suppose conditions (2.4) and (2.5) hold. Then the optimal error bound for solving Problem 1.1 is

ω(δ, E) = δ + Eyy0 δ

1− yy0 , 0 < y < y0. (2.13)

2584 H. Cheng et al. / Journal of Computational and Applied Mathematics 236 (2012) 2582–2589

Theorem 2.2. Let f (x + iy) be the exact solution of Problem 1.1, and v(x + iy) be its regularization approximation definedby (2.12). Assume the conditions (2.4) and (2.5) be satisfied, then there holds the estimate

∥f (· + iy) − v(· + iy)∥ ≤ Eyy0 δ

1− yy0 + δ. (2.14)

Proof. Due to the Parseval formula and (2.6), (2.11), (2.7), (2.5), (2.4), we have

∥f (· + iy) − v(· + iy)∥ = ∥ f (· + iy) − v(· + iy)∥

= ∥e−yξ f (ξ) − ρ(ξ, y)fδ(ξ)∥

≤ ∥e−yξ f (ξ) − ρ(ξ, y)f (ξ)∥ + ∥ρ(ξ, y)f (ξ) − ρ(ξ, y)fδ(ξ)∥

= ∥e(y0−y)ξ f (ξ)e−y0ξ − ρ(ξ, y)ey0ξ f (ξ)e−y0ξ∥ + ∥ρ(ξ, y)(f (ξ) − fδ(ξ))∥

= ∥(e(y0−y)ξ− ρ(ξ, y)ey0ξ ) f (· + iy0)∥ + ∥ρ(ξ, y)(f (ξ) − fδ(ξ))∥

≤ E supξ∈R

|e(y0−y)ξ− ρ(ξ, y)ey0ξ | + δ sup

ξ∈R|ρ(ξ, y)|. (2.15)

For ξ ≥ 0 from (2.8) it is easy to see that

∥f (· + iy) − v(· + iy)∥ ≤ δ supξ≥0

e−yξ≤ δ. (2.16)

Therefore, we only need to consider the case ξ < 0. From (2.9) and (2.10), we know

|ρ(ξ, y)| ≤ β(y) uniformly for ξ ∈ R, (2.17)

and therefore

∥ f (· + iy) − v(· + iy)∥ ≤ E supξ<0

|e(y0−y)ξ− β(y)ey0ξ | + δβ(y)

= E supξ<0

(e(y0−y)ξ− β(y)ey0ξ ) + δβ(y). (2.18)

Let

g(ξ) := e(y0−y)ξ− β(y)ey0ξ , (2.19)

we can easily get

g ′(ξ) = (y0 − y)e(y0−y)ξ− y0β(y)ey0ξ , (2.20)

and the maximum point of function g(ξ) is

ξ0 =

ln

y0−yy0β(y)

y

, (2.21)

therefore

supξ<0

g(ξ) = supξ<0

(e(y0−y)ξ− β(y)ey0ξ )

= g(ξ0) =

y0 − yy0β(y)

y0−yy

− β(y)

y0 − yy0β(y)

y0y

. (2.22)

Combining (2.22) with (2.18), we obtain

∥ f (· + iy) − v(· + iy)∥ ≤ E

y0 − yy0β(y)

y0−yy

− β(y)

y0 − yy0β(y)

y0y

+ δβ(y). (2.23)

Plugging β(y) given by (2.10) into the right-hand side of (2.23), the estimate (2.14) is obtained.Due to Lemma 2.1, we know the result of Theorem 2.2 is just the optimal error bound of Problem 1.1.

Remark 2.3. In general, the a priori bound E in (2.5) is unknown exactly in practice. In this case, replace β(y) in (2.10)by

β(y) =

1 −

yy0

1δ

yy0

, (2.24)


then there holds

∥f (· + iy) − v(· + iy)∥ ≤ E(1 + o(1))δy0−yy0 for (x, y) ∈ Ω+ and δ → 0, (2.25)

where E is just a positive constant and it is not necessary to be know exactly.

3. Numerical examples

In this section we give some simple numerical examples to verify the validity of the filtering regularization method. Inthe numerical experiments, we use the fast Fourier and inverse Fourier transform. We always take y0 = 1, i.e.,

Ω+= z = x + iy ∈ C| |x| ≤ 10, 0 < y < 1.

Suppose the vector F represents samples from the function f (x), then we obtain the perturbation data by

F ϵ= F + ϵrandn(size(F)), (3.1)

where the function ‘‘randn(·)’’ generates arrays of random numbers whose elements are normally distributed with mean 0,variance σ 2

= 1. The error is given by

δ = ∥F ϵ− F∥l2 :=

1M + 1

M+1n=1

|F ϵ(n) − F(n)|2, (3.2)

here we usually chooseM = 100. In the following, we will use V (x+ iy) to represent the discrete regularization solution ofF(x + iy).

In all numerical experiments, we compute the approximation V (x + iy) according to Theorem 2.2, and take E = ∥f ∥l2 .To compare the regularization effect with other methods, we use the same examples as in [7,8].

The absolute error ea(F(· + iy)) and the relative error er(F(· + iy)) are defined by

ea(F(·i y)) := ∥V (· + iy) − F(· + iy)∥l2 ,

er(F(· + iy)) :=∥V (· + iy) − F(· + iy)∥l2

∥F(· + iy)∥l2,

respectively. For simplicity, themodified Tikhonovmethod [8], the Fourier method [7], and the optimal filteringmethod areabbreviated to MTM, FM, and OFM respectively.

Example 3.1. The function

f (z) = e−z2= e−(x+iy)2

= ey2−x2(cos 2xy − i sin 2xy)

is analytic in the domain

Ω = z = x + iy ∈ C| x ∈ R, 0 < y < 1

with

f (z)|y=0 = e−x2∈ L2(R),

and

Ref (z) = ey2−x2 cos 2xy,

Imf (z) = ey2−x2 sin 2xy.

Fig. 1 is the comparison of the real and imaginary parts of the exact f (z) and the approximate solution v(z) at y = 0.1 fordifferent noise level ϵ = 10−2, 10−3. Fig. 2 is the same process as Fig. 1 at y = 0.9. From Figs. 1 and 2, we can find that thesmaller the ϵ is, the better the computed approximation is. And the bigger the y is, the worse the computed approximationis.

Tables 1–3 are the comparisons of three regularization methods. From these tables, we know that the numerical resultsof OFM are as well as those of MTM and FM.

Example 3.2. The function

f (z) = cos z = cos(x + iy) = cosh y cos x − i sinh y sin x

is also an analytic function in the domain

Ω = z = x + iy ∈ C|x ∈ R, 0 < y < 1


Table 1Example 3.1: errors with ϵ = 10−1 .

y MTM FM OFMea er ea er ea er

0.3 0.0820 0.3010 0.1265 0.5086 0.0779 0.29860.6 0.1298 0.3636 0.1698 0.5211 0.1505 0.48890.9 0.3486 0.6227 0.3786 0.7406 0.3663 0.8453



0.3 0.0068 0.0250 0.0201 0.0809 0.0132 0.05050.6 0.0268 0.0751 0.0330 0.1013 0.0495 0.16080.9 0.0837 0.1494 0.1373 0.2686 0.2497 0.5763



0.3 0.0018 0.0066 0.0110 0.0444 0.0016 0.00620.6 0.0076 0.0212 0.0170 0.0523 0.0056 0.01800.9 0.0732 0.1308 0.0210 0.0411 0.0533 0.1230

Fig. 1. Example 1: (a) the real part at y = 0.1, (b) the imaginary part at y = 0.1.





0.3 0.0659 0.0839 0.2453 0.3231 0.0672 0.08870.6 0.1540 0.1598 0.1264 0.1341 0.1766 0.20580.9 0.4042 0.3217 0.1611 0.1299 0.6636 0.6395



0.3 0.0287 0.0366 0.0252 0.0332 0.0181 0.02380.6 0.0912 0.0946 0.0638 0.0677 0.0630 0.07340.9 0.2968 0.2363 0.1290 0.1040 0.1049 0.1010


with

f (z)|y=0 = cos x,

and

Ref (z) = cosh y cos x,Imf (z) = − sinh y sin x.

Fig. 3 shows that the real and imaginary parts of the exact f (z) and the approximation v(z) at y = 0.1 for different noiselevel ϵ = 10−2, 10−3. Although cos x ∈ L2(R), Fig. 3 shows that the computed approximation is also well. In Fig. 4, the sameprocess for y = 0.9 is used.

From Tables 4 and 5, we also know that the numerical results of OFM are as well as those of MTM and FM.

Example 3.3. If the function f is a ‘‘piece’’ of an analytic function then this approximation is often possible. This example istypical. Let

f (z) =

36 − z2 =

36 − (x + iy)2, |x| < 6,

0, |x| ≥ 6.

Fig. 5 gives the real and imaginary parts of the exact f (z) and the approximation v(z) at y = 0.1 for the same noise levelϵ = 10−2.

Table 6 shows the comparison of the results of three regularization methods. From these tables, we know OFM is betterthan MTM and FM for Example 3.3.





0.3 0.2372 0.0623 0.3468 0.0911 0.1222 0.03220.6 0.4110 0.1064 0.5243 0.1353 0.2846 0.07430.9 0.6726 0.1696 0.9393 0.2348 0.3263 0.0839


4. Conclusion

In this paper, we use the optimal filtering regularization method to solve the problem of analytic continuation on stripdomain. By using this method, we obtain the Hölder type’s optimal error estimate. Although the optimal filtering regular-ization method is optimal in theory, its numerical effect is not better than some of other methods. The reasons may be thatnew errors must appear in the computational process and we can approximately adjust the regularization parameter forthe other methods to obtain a better result. While for the optimal filtering method, the parameter β(y) is completely fixedand cannot be corrected.

References

[1] C.L. Epstein, Introduction to the Mathematics of Medical Imaging, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.[2] A.G. Ramm, The ground-penetrating radar problem, III, J. Inverse III-Posed Probl. 8 (2000) 23–30.[3] M.M. Lavrent’ev, V.G. Romanov, S.P. Shishat-skıı, Ill-Posed Problems ofMathematical Physics and Analysis, in: Transl. Math.Monogr., vol. 64, American

Mathematical Society, Providence, RI, 1986.[4] R.G. Airapetyan, A.G. Ramm, Numerical inversion of the Laplace transform from the real axis, J. Math. Anal. Appl. 248 (2000) 572–587.[5] J. Franklin, Analytic continuation by the fast Fourier transform, SIAM J. Sci. Stat. Comput. 11 (1990) 112–122.


[6] P. Henrici, Applied and Computational Complex Analysis, vol. 1, John Wiley & Sons, Inc., New York, 1988.[7] C.L. Fu, F.F. Dou, X.L. Feng, Z. Qian, A simple regularization method for stable analytic continuation, Inverse Problems 24 (2008) 065003. 15 pp.[8] C.L. Fu, Z.L. Deng, X.L. Feng, F.F. Dou, A modified Tikhonov regularization for stable analytic continuation, SIAM J. Numer. Anal. 47 (2009) 2982–3000.[9] T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal. 33 (1996) 162–170.

[10] T.I. Seidman, L. Eldén, An Optimal filtering method for the sideways heat equation, Inverse Problems 6 (1990) 681–696.[11] H. Cheng, Two regularization methods for solving some inverse problems of elliptic equations, Master thesis, Lanzhou University, 2009.

an optimal filtering method for stable analytic continuation

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