![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
Abstract
In this article, we consider a class of ill-posed problems in two-dimensional setting which can be formulated as the problem of solving ill-posed operator equation. The spectral method is a simple and effective regularization method for such a class of ill-posed problems. For the spectral method, we obtain the error estimates with optimal order under an adaptive a posteriori parameter choice rule called balancing principle. Two ill-posed examples including an inverse diffraction problem and a deblurring problem are provided.
1 Introduction
Many PDE-based ill-posed problems in mathematical physics can be formulated as the problem for solving an operator equation, e.g. the inverse diffraction problem [Citation1–Citation4] and the deblurring problem.[Citation5, Citation6] Therefore, theory analysis, such as error estimate for the spectral method, is possible for this kind of ill-posed problems. In this paper, we discuss this kind of ill-posed problems in two-dimensional space, which involves the numerical computation of pseudodifferential operator.
Let1.1
1.1 be the Fourier transform of the function
. The Sobolev function space
is defined by
1.2
1.2 where
is the Fourier transform of function
and
denotes the norm in
.
Consider the numerical computation of the pseudodifferential operator with an unbounded symbol given by
1.3
1.3 where
is a fixed constant and
is the Fourier transform of the exact data
; the data
is only given approximately by
satisfying
1.4
1.4 where
is the noise level and is assumed to be known.
We assume that the symbol satisfies the following growth condition:
1.5
1.5 where the function
is strictly increasing with
, the function
with
and
are constants.
If satisfies
as
with
, then we call the problem of numerical computation of the above pseudodifferential operators as a severely ill-posed problem. Likewise, if
satisfies
as
with
, then we call the problem of numerical computation of above pseudodifferential operators as a mildly ill-posed problem. In this paper, we only consider the severely ill-posed problem since the mildly ill-posed problems can be solved more easily. The problem (Equation1.3
1.3
1.3 ) in one-dimensional space has been discussed in [Citation7] by the a posteriori Fourier method based on Morozov’s discrepancy principle. But, many ill-posed problems with important application backgrounds such as inverse diffraction and deblurring problems are set in two-dimensional case. This motivates us to consider the two-dimensional inverse problems. Although the authors in [Citation7] analysed the Fourier method with a posteriori parameter choice rule by using a direct method, here we stress that we give a rule for selecting regularization parameter based on the theory of regularization framework. The analysis involved is more general.
A priori information about unknown solution has been proved to be essential in the analysis of ill-posed problems in mathematical physics. Otherwise, without the a priori information, the convergence rate of the constructed regularization method will be arbitrarily slow.[Citation8] For analysis, we assume that there holds the a-priori bound for the unknown solution1.6
1.6 Let
denote a regularization solution. If
, we call the
as interior inversion. If
, we call the
as boundary inversion.
Obviously, the ill-posedness of problem (Equation1.31.3
1.3 ) is caused by components of high frequency. If
, we only need to regularize the solution in the case of
, because for the case of
the problem (Equation1.3
1.3
1.3 ) is well posed which can be observed by the following discussion:
Let the sets and
, and we have
.
Consider the differencewhere
denotes the solution with noisy data
.
The same technique has been used in.[Citation9, Citation10] Throughout this paper, we mainly deal with the case of .
By the balance principle,[Citation11] we can derive a Hölder-type error estimate like under the a priori bound (Equation1.6
1.6
1.6 ) with
. However, when
, in which we should be more interested, we cannot get any convergence. To overcome the difficulty, usually a stronger a priori bound on the unknown solution is added, i.e. we should require
for (Equation1.6
1.6
1.6 ) to obtain the convergence rate at
. In the following, we will construct the interior and boundary inversions by a spectral regularization method.
The outline of this paper is as follows. In Section 2, we present the spectral method. In Section 3, the spectral method is applied to solve problem (Equation1.31.3
1.3 ); some applications are given in Section 4. In Section 5, some numerical results are presented. Finally, Section 6 concludes the paper.
2 The spectral method [Citation12, Citation13]
2.1 Preliminaries
Ill-posed operator equations arise in several contexts and various aspects have been treated in the literature.[Citation8, Citation14, Citation15] We cannot give here an exhaustive survey. In this paper, we are interested in solving the solution of linear ill-posed problems by spectral cut-off method. Consider
2.1
2.1 where
is a linear injective, closed operator between infinite-dimensional Hilbert spaces
and
with non-closed range
. We suppose that
are the noisy data with
2.2
2.2 and known noise level
.
If we consider the ill-posed problem where only noisy data
are available,
is a compact operator with singular system
since the ill-posedness is associated with the small singular values, an obvious idea is to truncate or damp the smaller singular values. That is the well-known truncated singular value decomposition method. These methods are simple and effective. In practical computation, we can implement the spectral cut-off method by mollification method discovered by Hào [Citation16].
For problem (Equation2.12.1
2.1 ), most regularization operators can be written in the form,
2.3
2.3 with some function
satisfying
where the operator function
is well defined via the spectral representation
. Here,
,
denotes the spectral family of the operator
and
is a constant satisfying
with
if
is unbounded.
Then, for the regularization solution with unperturbed data, we have and
with
. For example, for spectral cut-off method,
2.4
2.4 In general, the exact solution
is required to satisfy a so-called source condition, otherwise the convergence of the regularization method approximating the problem can be arbitrarily slow. For ill-posed problems, the source condition is chosen as
2.5
2.5 i.e.
belongs to the source set
2.6
2.6 where
satisfies some properties:
Assumption 1
and
is strict monotonically increasing,
is convex.
For the stable approximate solution of problem (Equation2.12.1
2.1 ) some regularization technique has to be applied, which provides regularized approximations
with property
as
.
Any operator can be considered as a special method for solving problem (Equation2.1
2.1
2.1 ). the approximate solution to (Equation2.1
2.1
2.1 ) is then given by
. Consider the worst case error
for identifying the solution
of problem (Equation2.1
2.1
2.1 ) from noisy data
under the assumptions
and
belongs to a source set
which is defined by
2.7
2.7 This worst case error characterizes the maximal error of the method
if the solution
of problem (Equation2.1
2.1
2.1 ) varies in the set
. An optimal method
is characterized by
. It can easily be realized that
2.8
2.8 where
The following theorem and definition can be found in [Citation15].
Theorem 1.1
Let is given by (Equation2.6
2.6
2.6 ), and Assumption 1 be satisfied. If
, then
2.9
2.9 where
is given by
.
Definition 1.1
Let Assumption 1 be satisfied. Any regularization method for problem (Equation2.1
2.1
2.1 ) with noisy data is called
(i) | Optimal on the set | ||||
(ii) | Order optimal on the set |
2.2 The balancing principle
Now, we prove some estimates for the spectral regularization method under the a posteriori parameter choice rule based on balance principle.
First we have [Citation12]:
Theorem 2.1
Let , Assumption 1 is satisfied, then there holds
2.10
2.10
Now denote ,
. Since
increases with
but
decreases with
, let us rewrite (Equation2.10
2.10
2.10 ) as
2.11
2.11 By virtue of the behaviour of the functions
and
, we can choose an
such that
. Therefore,
2.12
2.12 However, since the a priori bound
of the unknown solution is seldom known such that
is unknown, the a priori choice of the theoretically value of
is impossible. Therefore it is necessary to use some a-posteriori rule of
.
For this purpose, we use the balancing principle.
Consider a discrete set of possible values of regularization parameter2.13
2.13 where
,
sufficiently large such that
.
Denote the set2.14
2.14 Choose the value of regularization parameter by the rule
2.15
2.15 Below we show that the choice of
gives the error estimate, which differs from (Equation2.12
2.12
2.12 ) only by a factor of
.
To prove this conclusion, denote the auxiliary set2.16
2.16 and the auxiliary quantity
2.17
2.17 Without loss of generality, we assume that
Theorem 2.2
Let the parameter be chosen by (Equation2.152.15
2.15 ), then there holds
2.18
2.18
Proof
As the process is similar to [Citation11], we omit it.
3 Spectral regularization for problem (Equation1.3
1.3
1.3 )
Now, in order to apply the spectral regularization method, we formulate problem (Equation1.31.3
1.3 ) as an operator equation in the frequency domain:
where
is a multiplication operator. Obviously
, and the adjoint operator of
is
, where the symbol
denotes the complex conjugate of
. Therefore,
. In this section, we deal with problem (Equation1.3
1.3
1.3 ) for two cases separately:
Case I (the case of the interior inversion): in (Equation1.6
1.6
1.6 ).
Case II (the case of the boundary inversion): in (Equation1.6
1.6
1.6 ).
Now we investigate the spectral method for two cases.
First, we can write the spectral regularization according to (Equation2.42.4
2.4 ):
3.1
3.1 In order to obtain the explicit expression of the error bound, we need to know the expression of
in (Equation2.5
2.5
2.5 ).
3.1 Interior inversion
In this subsection, we note that in (Equation1.6
1.6
1.6 ) and we want to recover the solution
with
. First we use the equality
Now (Equation1.6
1.6
1.6 ) reads
Denote the sets
3.2
3.2
3.3
3.3 In fact, by
,
is equivalent to
given in the form of (Equation2.6
2.6
2.6 )
3.4
3.4 where
3.5
3.5 Now we only need to show that every element from
belongs to the set
. For any element
, we have
3.6
3.6 By (Equation1.5
1.5
1.5 ), this implies
3.7
3.7 On the other hand, for any element
, we have two-side estimates
Thus, we have shown that every element from
belongs to the set
. Likewise, we can also show that every element from
belongs to the set
by constructing another set
similar to
. We have another
which differs from (Equation3.5
3.5
3.5 ) only by a constant factor. Now we summarize what we have:
Conclusion 1
If the a-priori bound (Equation1.61.6
1.6 ) with
holds, for problem (Equation1.3
1.3
1.3 ), the function
in the source set (Equation3.5
3.5
3.5 ) has the form of
with
is a constant which only depends on
and
. Thus, the function
and
. Therefore for problem (Equation1.3
1.3
1.3 ) for solving
, the optimal convergence order for error estimates is given by
3.8
3.8
3.2 Boundary inversion
In this subsection, we note that in (Equation1.6
1.6
1.6 ) and we want to recover the solution
. Now (Equation1.6
1.6
1.6 ) with
reads
Denote the sets
3.9
3.9 where the
denotes the inverse function of
and
is a positive constant depending on
. Similar to Conclusion 1, we can prove that every element from
belongs to the set
. Thus, we can obtain the expression of
:
3.10
3.10 By an elementary calculation, it yields
3.11
3.11 where
is a constant depends on
.
Conclusion 2
If the a-priori bound (Equation1.61.6
1.6 ) with
holds, the optimal convergence order for solving
is given by
3.12
3.12
4 Applications
Now, we give some applications of Theorem 2.3. Throughout this section, the are the positive constants which are not dependent on
and
.
Example 4.1
The reconstruction problem of aperture in the plane from their diffraction patterns arises in acoustics and optics. Let us consider the following inverse diffraction problem. Let be a bounded aperture in an infinite perfectly soft screen which is located in the plane
in
. A harmonic plane wave with wave-number
propagates along with the positive
direction. It hits the screen and escapes through the aperture
. The measured data at receiving screen
are given. The problem is to reconstruct the shape (domain) of the aperture
. By Kirchhoff approximation, the mathematical modelling can be formulated as follows.
Let be the solution of the following problem:
4.1
4.1
4.2
4.2
4.3
4.3
Here we wish to find the from the measured data
, where
is the characteristic function of the domain
.
As usual, we assume that there exists a priori bound:4.4
4.4 where
denotes the norm of Sobolev space
with
and
is a positive constant.
We have the solution in the frequency domain:4.5
4.5 Therefore, the symbol of pseudodifferential operator is given by
4.6
4.6 Now let
and its inverse function
,
; by (Equation3.11
3.11
3.11 ), we have
4.7
4.7 Therefore, by Theorem 2.3, we have the following error estimate:
4.8
4.8 Now we need to estimate
. At
, we have
. Therefore,
i.e.
Since
holds for any
, we have
i.e.
4.9
4.9 As a result, we have the error estimate
4.10
4.10 The convergence order is optimal.
Example 4.2
Consider the linear diffusion final value problem in ,
4.11
4.11 where
is the Laplacian operator, usually
are the data with approximation
. As usual, we assume that there exists a priori bound:
4.12
4.12 where
denotes the norm of Sobolev space
with
and
is a positive constant.
This problem can be associated with a deblurring problem. Consider a space invariant point spread functions and formulate the deblurring problem as
4.13
4.13 where
is the desired unblurred image and
is the blurred image that would have been recorded in the absence of noise. In general, the image
is viewed as originally defined on a bounded rectangle-like domain; we could extend
periodically to all of
.
We can easily get the unique Fourier space solution for problem (Equation4.114.11
4.11 )
4.14
4.14 If we take
, we may identity the blurred image
with the temperature distribution at time
in an infinite two-dimensional medium whose initial temperature distribution is given by the unblurred image
. This involves solving the diffusion equation backwards in time. For a fixed
, we call the solution
a partial restoration. For the case
, we call the solution
a full restoration.
Therefore, the symbol of pseudodifferential operator is given by4.15
4.15 Now let
,
,
and its inverse function
,
.
Boundary inversion. By (Equation3.113.11
3.11 ), we have
4.16
4.16 Therefore, similar to Example 4.1, by Theorem 2.3, we have the following error estimate:
4.17
4.17 Interior inversion. By (Equation3.5
3.5
3.5 ), we have
4.18
4.18 Therefore, by Theorem 2.3, we have the following error estimate:
4.19
4.19 Now we need to estimate
. At
, we have
. Therefore,
i.e.
As a result, we have the error estimate
4.20
4.20 The convergence order in (Equation4.17
4.17
4.17 ) and (Equation4.20
4.20
4.20 ) are optimal.
5 Numerical results
In this section, we only consider the numerical results of Example 4.1. The numerical results of Example 4.2 can be found in [Citation6], we do not repeat them here. Although the examples are described in an unbounded domain in the plane , we are interested in the domain
. This is reasonable because the problem can be solved by periodic extension in the
plane.
Define the discrete norm of
by
5.1
5.1 where
is the total number of sampled points. In this section, we take
.
In order to measure the accuracy of numerical results, we define the discrete -norm error for the exact solution
between the approximate solution
as follows:
5.2
5.2 The noise was added to
by setting
, where
denotes the noise level and
is a random number drawn from a uniform distribution in the range
. In numerical experiment, we use (Equation5.2
5.2
5.2 ) to compute
.
We shall illustrate the reconstruction method (Equation3.13.1
3.1 ) with different numerical examples although we have used some ‘inverse crimes’.[Citation17] The Fourier formula (Equation3.1
3.1
3.1 ) is based on FFT algorithm.
Example 1
We choose the aperture in the shape of the letter ‘L’ for our test. The characteristic function, as shown in Figure (a) on the domain
, is to be constructed. And Figure (b) displays data
which is a very blurred picture of the original aperture.
In this example, relative error is added to the data
at the sampled points in the simulation and we can get
. In the computation, we used
,
. Figure corresponds to the results obtained by the spectral method, where the distance from the receiving plane to the screen is
with the wave-number
.
Example 2
The aperture consists of three independent squares. The characteristic function to be constructed and the data
are displayed in Figure .
In this example, relative error is added to the data
at the sampled points in the simulation as the Example 5.1. And we can get
. In the computation, we used
,
.
Figure correspond to the results by the spectral method, where the distance from receiving plane to the screen is with the wave-number
.
In the numerical tests, we find that the reconstruction gets more severe as the receiving plane is moved away from the aperture and as the wave-number is decreased. Because if
is decreased and
is increased, the degree of ill-posedness of the problem increases.
6 Concluding remark
Numerical solutions for many ill-posed problems can be recast as the numerical computation of a class of pseudodifferential operators. In this paper, for the numerical computation of a class of pseudodifferential operators, we investigate the spectral method. Under the balancing principle, we derived the error bounds for the numerical computation of a class of pseudodifferential operators. The obtained error bounds have the optimal convergence order. Two numerical examples are provided and the numerical experiment shows that the method works well.
Acknowledgments
The reviewers are thanked for their very careful reading and for pointing out several mistakes as well as for their useful comments and suggestions.
Notes
1 The work described in this paper was partially supported by a grant from the National Natural Science Foundation of China [grant number 11001223]; the Research Fund for the Doctoral Program of Higher Education of China [grant number 20106203120001]; the Key (Keygrant) Project of Chinese Ministry of Education [grant number 212179]; and the Doctoral Foundation of Northwest Normal University, China [grant number 5002–577].
References
- Magnanini R, Papi G. An inverse problem for the Helmholtz equation. Inverse Probl. 1985;1:357–370.
- Sondhi M. Reconstruction of objects from their sound-diffraction patterns. J. Acoust. Soc. Am. 1969;46:1158–1164.
- Bertero M, De CM. Stability problems in inverse diffraction. IEEE Trans. Antenn. Propag. 1981;AP-29:368–372.
- Santosa F. A level-set approach for inverse problems involving obstacles. Control Optim. Cal. Var. 1996;1:17–33.
- Carasso AS. Overcoming Hölder continuity in ill-posed continuation problems. SIAM J. Numer. Anal. 1994;31:1535–1557.
- Xiong XT, Wang JX, Li M. An optimal method for fractional heat conduction problem backward in time. Appl. Anal. 2012;91:823–840.
- Fu CL, Zhang YX, Cheng H, Ma YJ. The a posteriori Fourier method for solving ill-posed problems. Inverse Probl. 2012;28:095002. 26 p.
- Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Dordrecht: Kluwer Academic Publisher; 1996.
- Xiong XT, Fu CL. Two approximate methods of a Cauchy problem for the Helmholtz equation. Comput. Appl. Math. 2007;26:285–307.
- Reginska T, Tautenhahn U. Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation. Numer. Funct. Anal. Optim. 2009;30:1065–1097.
- Pereverzev S, Shock E. On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J. Numer. Anal. 2006;75:2060–2076.
- Xiong XT, Zhao XC, Wang JX. Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation. Numer. Algor. 2013;63:691–711.
- Xiong XT, Fu CL, Qian Z. On three spectral regularization methods for a backward heat conduction problem. J. Korean Math. Soc. 2007;44:1281–1290.
- Nair MT. Linear operator equations: approximation and regularization. Singapore: World Scientific; 2009.
- Tautenhahn U. Optimality for linear ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 1998;19:377–398.
- Hào DN. Methods for inverse heat conduction problems. Frankfurt am Main, New York (NY): Peter Lang; 1998.
- Kaipio JP, Somersalo E. Statistical and computational inverse problems. New York (NY): Springer, Applied Mathematical Sciences; 2005.