- Research
- Open Access
On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data
- Nguyen Huy Tuan^{1},
- Vo Anh Khoa^{2, 3}Email author,
- Le Trong Lan^{3} and
- Tran The Hung^{3}
https://doi.org/10.1186/s13660-014-0526-y
© Tuan et al.; licensee Springer 2015
- Received: 20 September 2014
- Accepted: 13 December 2014
- Published: 14 January 2015
Abstract
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a regularization method to stabilize the solution, then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends the earlier results in Zouyed and Rebbani (J. Inverse Ill-Posed Probl. 22(4):449-466, 2014).
Keywords
- ultraparabolic equation
- ill-posed problem
- semi-group method
- stability
- error estimate
MSC
- 47A52
- 20M17
- 26D15
1 Introduction
Ultraparabolic equations arise in several areas of science, such as mathematical biology in population dynamics [1] and probability in connection with multi-parameter Brownian motion [2], and in the theory of boundary layers [3]. From those applications, ultraparabolic equations have gained considerable attention in many mathematical aspects (see, e.g., [1, 4–9] and the references therein).
In the mathematical literature, various types of ultraparabolic problems have been solved. There have been some papers dealing with the existence and uniqueness of solution for many kinds of ultraparabolic equations, e.g., [1, 10, 11]. As the pioneer in numerical methods for such equations, Akrivis et al. [5] numerically approximated the solution of a prototype ultraparabolic equation by applying a fixed-step backward Euler scheme and a second-order box-type finite difference method. Some extension works for the numerical angle that should be mentioned are [12, 13] by Ashyralyev and Yilmaz, and Marcozzi, respectively. We also remark that, in general, ultraparabolic equations do not possess properties that are closely fundamental to many kinds of parabolic equations including strong maximum principles, a priori estimates, and so on.
In the phase of ill-posed ultraparabolic problems, the authors Zouyed and Rebbani very recently proposed in [7] the modified quasi-boundary value method to regularize the solution of problem (1) in the homogeneous backward case \(f\equiv0\). In particular, via the instability terms in the form of the solution of (1) (cf. [4, Theorem 1.1]), they established an approximate problem by replacing \(\mathcal{A}_{\alpha}=\mathcal{A} (I+\alpha\mathcal{A}^{-1} )\) for the operator \(\mathcal{A}\) and taking the perturbation α into final conditions of the ill-posed problem, and obtained the convergence order \(\alpha^{\theta}\), \(\theta\in(0,1 )\). Motivated by that work, this paper is devoted to investigating a new regularization method.
In the past, many approaches have been studied for solving ill-posed problems, especially the backward heat problems. For example, Lattès and Lions [14], Showalter [15] and Boussetila and Rebbani [16] used the quasi-reversibility method; in [17] Ames and Epperson applied the least squares method with Tikhonov-type regularization; Clark and Oppenheimer [18], Denche and Bessila [19] and Trong et al. [20] used the quasi-boundary value method. Moreover, some other methods that should be listed are the mollification method by Hao [21] and the operator-splitting method studied by Kirkup and Wadsworth [22]. To the best of the authors’ knowledge, although there are many works on several types of parabolic backward problems, the theoretical literature on regularizing the inverse problems for ultraparabolic equations is very scarce. Therefore, proposing a regularization method for problem (1) is the scope of this paper.
Our work presented in this paper has the following features. At first, for ease of the reading, we summarize in Section 2 some well-known facts in a semi-group of operators and present the formula of the solution of (1). Secondly, in Section 3 we construct the regularized solution based on our method, then obtain the error estimate. Finally, a numerical example is given in Section 4 to illustrate the efficiency of the result.
2 Preliminaries
The operator −Δ is a positive self-adjoint unbounded linear operator on \(L^{2} (0,\pi)\). Therefore, it can be applied to some elementary results in [4, 7, 15, 23–26]. Particularly, the formula of the solution of problem (1) can be obtained by Lorenzi [4], and the authors in [23, 24] gave a detailed description on fundamental properties of the generalized operator. In this section, we thus recall those results which we want to apply to our main results in this paper. We list them and skip their proofs for conciseness.
- 1.
\(\Vert S (t )\Vert \le1\) for all \(t\ge0\);
- 2.
the function \(t\mapsto S (t )\), \(t>0\) is analytic;
- 3.
for every real \(r\ge0\) and \(t>0\), the operator \(S (t )\in\mathcal{L} (\mathcal{H},\mathcal{D} (\mathcal{A}^{r} ) )\);
- 4.
for every integer \(k\ge0\) and \(t>0\), \(\Vert S^{k} (t )\Vert =\Vert \mathcal{A}^{k}S (t )\Vert \le c (k )t^{-k}\);
- 5.
for every \(x\in\mathcal{D} (\mathcal{A}^{r} )\), \(r\ge0\), we have \(S (t )\mathcal{A}^{r}x=\mathcal{A}^{r}S (t )x\).
Remark 1
- (A1)
\(\varphi\in C ( [0,T ];\mathcal{D} (\mathcal{A} ) )\cap C^{1} ( [0,T ];\mathcal{H} )\);
- (A2)
\(\psi\in C ( [0,T ];\mathcal{D} (\mathcal{A} ) )\cap C^{1} ( [0,T ];\mathcal{H} )\);
- (A3)
\(\varphi(0 )=\psi(0 )\);
- (A4)
\(f\in C ( [0,T ]\times[0,T ];\mathcal{H} )\cap C^{1} (D_{1}\times D_{2};\mathcal{H} )\).
In the following theorems, we show the formula of the solution of problem (2) by employing Theorem 1.1 in [4] with \(a_{1} (t )=a_{2} (s )=1\) and following the steps in [7].
Theorem 2
Theorem 3
Proof
Theorem 4
Proof
We can see that the instability is caused by all of the exponential functions. In fact, let us see the case \((t,s )\in D_{1}\) in (8). Since the discrete spectrum increases monotonically as n tends to infinity, the rapid escalation of \(e^{ (T-t )n^{2}}\) and \(e^{ (T-\eta)n^{2}}\) is mainly the instability cause. Even though these exact given functions \((\psi_{n},f_{n} )\) may tend to zero very fast, performing classical calculation is impossible. It is because the given data may be diffused by a variety of reasons such as round-off errors, measurement errors. A small perturbation in the data can arbitrarily generate a large error in the solution. A regularization method is thus required.
3 Theoretical results
In this section, assuming that the problem has an exact solution u satisfying various corresponding assumptions, we construct the regularized solution depending continuously on the data such that it converges to the exact solution u in some sense. Moreover, the accuracy of regularized solution is estimated.
Now we shall show two elementary inequalities in the following lemmas.
Lemma 5
Proof
It is obvious that \((\varepsilon+e^{-n^{2}p} )^{\frac{t-T}{p}}\leq \varepsilon^{\frac{t-T}{p}}\) since \(\varepsilon+e^{-n^{2}p}\ge\varepsilon\). □
Lemma 6
Proof
In the sequel, we only prove the case \((t,s )\in D_{1}\) in our main result because of the similarity. The results are about the regularized solution depending continuously on the corresponding data and the convergence of that solution to the exact solution. Now we shall use two elementary lemmas above to support the proof of the main results.
Lemma 7
Under conditions (A1), (A2), (A4) and assuming that \(\varphi(T )=\psi (T )\), then the function \(u^{\varepsilon}\) given by (10)-(11) depends continuously on \((\varphi,\psi)\) in \(L^{2} (0,\pi)\).
Proof
Theorem 8
Proof
Hence, we complete the proof. □
Remark 9
Moreover, the error is of order \(\mathcal{O} (\varepsilon^{\frac {p-T}{p}} )\) for all \((t,s )\in[0,T ]\times[0,T ]\). If \(p>T\), this error is faster than the order \(\ln(\varepsilon^{-1} )^{-q}\), \(q>0\) as \(\varepsilon\to0\) which is studied in many works, such as [18–20, 23]. Combining the strong points above, the reader can infer that our method is feasible.
4 A numerical example
We thus seek the discrete solutions \(u_{ex}^{j,k,l}=u_{ex} (x_{j},t_{k},s_{l} )\) and \(v_{m}^{j,k,l}=v_{m} (x_{j},t_{k},s_{l} )\) given by (18) and (19)-(20), respectively.
Comparison of absolute errors between the regularized solutions \(\pmb{v_{m}}\) of \(\pmb{m=10^{2}}\) and \(\pmb{m=10^{10}}\)
( x , t , s ) | Exact value | App. value 1 ( \(\boldsymbol {m=10^{2}}\) ) | App. value 2 ( \(\boldsymbol {m=10^{10}}\) ) | Abs. error 1 | Abs. error 2 |
---|---|---|---|---|---|
\((\frac{\pi}{2},0.75,0.75 )\) | 0.1053992246 | 0.0915741799 | 0.1053992172 | 0.0138250446 | 7.4E−09 |
\((\frac{\pi}{2},0.5,0.5 )\) | 0.2231301601 | 0.1684339068 | 0.2231301293 | 0.0546962533 | 3.08E−08 |
\((\frac{\pi}{2},0.25,0.25 )\) | 0.4723665527 | 0.3098032761 | 0.4723664549 | 0.1625632766 | 9.87E−08 |
\((\frac{\pi}{2},0.125,0.125 )\) | 0.6872892788 | 0.4201595585 | 0.6872891127 | 0.2671297203 | 1.661E−07 |
\((\frac{\pi}{2},0,0 )\) | 1 | 0.5698263001 | 0.9999997239 | 0.4301736999 | 2.761E−07 |
5 Conclusion
In this work, a regularization method has been successfully applied to the inverse ultraparabolic problem. This method is to replace the instability terms appearing in the formula of the solution which is employed by semi-group theory. Therefore, such a way forms the so-called regularized solution which strongly converges to the exact solution in \(L^{2}\)-norm. We also obtain the error estimate which is of order \(\varepsilon^{\frac{p-T}{p}}\), \(p>T\). By a numerical example, application of the method is flexible and calculation of successive approximations is direct and straightforward. This work is more general than [7], a recent work of Zouyed and Rebbani, in both error estimate and the considered problem.
Declarations
Acknowledgements
The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments leading to the improved version of this paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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