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Regularization of an initial inverse problem for a biharmonic equation
Advances in Difference Equations volume 2019, Article number: 255 (2019)
Abstract
In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous biharmonic equation. The problem is severely ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose two regularization methods to solve the problem. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in \(\mathcal{L}^{2}\) uniformly with respect to the space coordinate under some a priori assumptions on the solution. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability.
1 Introduction
In this paper, we consider the non-homogeneous biharmonic equation
where \(\varOmega \subset \mathbb{R}^{d}\), \(d\geq 1\) is an open bounded domain with a smooth boundary ∂Ω, and the linear source function \(\rho \in \mathcal{L}^{\infty } (0,L;\mathcal{L}^{2}( \varOmega ) )\). In practice, the data \(g,h \in \mathcal{L}^{2}( \varOmega )\) are noisy and are represented by the observation data \(g^{\alpha },h^{\alpha }\in \mathcal{L}^{2}(\varOmega )\) satisfying
here \(\alpha >0\) is a small positive number representing the level of noise.
There are many papers on different methods for approximating solutions to boundary value problems for elliptic partial differential equations and most are centered on second order equations where maximum principles are used to obtain asymptotic estimates for the error [1,2,3,4,5, 7, 8, 11, 13,14,15, 17,18,19]. The theory for elliptic equations of order greater than two is less developed [8] (note that such equations arise in physics and in engineering design and they also appear naturally in many areas of mathematics, including conformal geometry, and nonlinear elasticity [1, 4, 5]).
The prototypical example of a higher-order elliptic operator, well known from the theory of elasticity, is the biharmonic \(\Delta ^{2} = \Delta (\Delta ) = \nabla ^{4}\), and a more general example is the polyharmonic operator \(\Delta ^{p}=\underbrace{\Delta (\Delta ... \Delta )}_{p \text{ times}}\), \(p>2\). The biharmonic equation arises in many engineering applications such as the deformation of thin plates, the motion of fluids, free boundary problems, nonlinear elasticity and for historical details we refer the reader to [2, 3, 7, 14] (for a more elaborate history of the biharmonic problem and the relation with elasticity from an engineering point of view we refer the reader to the survey of Meleshko [11]).
In 1928, Covrant et al. [6] posed a difference analog for the first boundary value problem for the homogeneous biharmonic equation
and proved that the approximate solutions converge to the exact solution as the mesh is refined (however, no estimates for the error were given). In [10], the authors obtained necessary and sufficient conditions for existence of a solution for the biharmonic equation (1.3) in a rectangular domain \([0,\pi ] \times [0,L]\) in the space \(\mathcal{L}^{2}(0,\pi )\). In [9] using a nonlocal boundary value problem method, convergence of regularized approximation with a priori parameter choice was proven, provided data noise level tends to zero (however, the authors did not investigate error estimates). The method of nonlocal boundary value problems for second order elliptic equations was used by several authors (see [13, 15, 17,18,19]). There are many papers on the linear homogeneous case for the biharmonic equation, but, however, very little is known on regularization theory and numerical simulation for the linear inhomogeneous case. Our main aim in this paper is to discuss regularized solutions for problem (1.1). Using the Fourier truncation method introduced in [16], we propose the regularized solution and give an error estimate.
The paper is organized as follows. In Sect. 2, the formulation of the problem and its ill-posed property are given. In Sect. 3, stability estimates are proved under a priori conditions on the solution. Numerical results are presented and discussed in Sect. 4 and, finally, conclusions are summarized in Sect. 5.
2 Preliminaries
2.1 Notations and assumptions
We begin this section by introducing some notations and assumptions that are needed for our analysis in the next sections.
Definition 2.1
Without loss of generality, we assume that −Δ has the eigenvalues \(\lambda _{m} \) (\(m \in \mathbb{N}^{*}\)):
and the corresponding eigenelements \(\xi _{m}(x)\), which form an orthonormal basis in \(\mathcal{L}^{2}(\varOmega )\).
Definition 2.2
(Hilbert scale space, see [12])
The Hilbert scale space \(\mathbb{H}^{p}\), \((p >0)\) defined by
is equipped with the norm defined by
For a Hilbert space X, we denote by \(\mathcal{L}^{p} (0,L;X )\) (respectively, \(C ( [0,L ];X )\)) the Banach space of measurable (respectively, continuous) functions \(f:[0,L]\to X\), such that
respectively,
Throughout this paper, the function ρ is perturbed so as to contain errors in the form of noisy \(\rho ^{\alpha }\in \mathcal{L}^{\infty } (0,L;\mathcal{L}^{2}(\varOmega ) )\) satisfying
2.2 Mild solution and ill-posed of problem (1.1)
The solution to problem (1.1) can be represented in the form of an expansion in the orthogonal series
By considering that the series (2.5) converges and allows a term by term differentiation (the required number of times), we construct a formal solution to the problem. We obtain the problems
Here \(g_{m}\), \(h_{m}\) and \(\rho _{m}(y)\) are Fourier coefficients of the expansion according to the orthonormal basis \(\{\xi _{m}(x)\}_{m\in \mathbb{N}^{*}}\) of the functions \(g(x)\), \(h(x)\) and \(\rho (y,x)\), respectively:
By direct calculation, the solution to problem (2.6) has the form
Substituting the result into (2.5), we obtain the formal solution to problem (1.1).
Next, we give an example which shows that the solution of problem (1.1) does not depend continuously on the final data.
Example
For any \(j \in \mathbb{N}^{*}\), let \(\widetilde{g} _{j}\), \(\widetilde{h}_{j}\) and \(\widetilde{\rho }_{j}\) be as follows:
Let \(\widetilde{u}_{j}\) be the solution of (1.1) with \(\widetilde{g}_{j}\), \(\widetilde{h}_{j}\) and \(\widetilde{\rho }_{j}\). One has
Since \(z>0\), we have
and it follows that
Hence, we deduce that
as \(j \to \infty \), we see that (for \(0 \leq y < L\))
Thus our problem is ill-posed in the Hadamard sense in the \(\mathcal{L}^{2}(\varOmega )\)-norm.
3 Regularization and error estimate
In order to obtain stable numerical solutions, we propose two regularization methods to solve the problem. As was shown in the previous section, for the linear biharmonic problem (1.1), its solution (true solution) can be represented as an integral equation which contains some instability terms. Indeed, we find that the four functions
in (2.7) are unbounded, as functions of the variable m, for \(y \in (0,L)\). Consequently, small errors in high frequency components can blow up and completely destroy the solution for \(y \in (0,L)\). A natural idea to stabilize the problem is to eliminate all high frequencies (truncation method) or to replace them by a bounded approximation (quasi-boundary value method). We introduce two bounded operators as follows:
-
For \(f \in C([0,L];\mathcal{L}^{2}(\varOmega ))\), we define
$$\begin{aligned} \mathbf{\widehat{Q}}^{\gamma (\alpha )} f (y,x) = \sum_{m=1}^{\infty} \mathcal{I}^{\gamma (\alpha )}_{L,m} \bigl\langle f(y,x),\xi _{m}(x)\bigr\rangle _{\mathcal{L}^{2}(\varOmega )}\xi _{m}(x), \end{aligned}$$(3.1)where
$$ \mathcal{I}^{\gamma (\alpha )}_{L,m} = \bigl(1+\gamma (\alpha )\sqrt{ \lambda _{m}} e^{\sqrt{\lambda _{m}} L} \bigr)^{-1}, \quad \forall m \in \mathbb{N}^{*}, $$and \(\gamma (\alpha ) > 0\) is the parameter regularization which satisfies
$$\begin{aligned} \lim_{\alpha \to 0^{+}} \gamma (\alpha ) = 0. \end{aligned}$$(3.2) -
For \(f \in C([0,L];\mathcal{L}^{2}(\varOmega ))\), we define
$$\begin{aligned} \mathbf{\widehat{B}}^{M_{\alpha }} f (y,x) = \sum_{m \in \mathbb{T}_{\alpha }^{\dagger }} \bigl\langle f(y,x),\xi _{m}(x)\bigr\rangle _{\mathcal{L}^{2}(\varOmega )}\xi _{m}(x), \end{aligned}$$(3.3)where
$$\begin{aligned} \mathbb{T}_{\alpha }^{\dagger } &:= \bigl\{ m \in \mathbb{N}^{*} | \lambda _{m} \leq M_{\alpha } \bigr\} , \end{aligned}$$and \(M_{\alpha }> 0\) is the parameter regularization which satisfies
$$\begin{aligned} \lim_{\alpha \to 0^{+}} M_{\alpha }= +\infty . \end{aligned}$$(3.4)
3.1 The main results
3.1.1 Result for quasi-boundary value method
Let us consider the following well-posed problem:
Theorem 3.1
((QBV) method)
Assume that the exact solution u of (1.1) satisfies
where p, \(E_{1}\) are positive constants. Choose \(\gamma (\alpha ) \in (0,1)\) such that
Then the estimate
holds.
Remark 3.1
From condition (3.7), if we choose \(\gamma (\alpha )=\alpha ^{k}\) for some \(k \in (0,1)\), then the error estimate in (3.8) is of order \(\frac{\alpha ^{1-k}}{\log (\frac{L}{\alpha ^{k}} )}\), which tends to zero as \(\alpha \to 0^{+}\).
3.1.2 Result for truncation method
Next, we propose a second regularized solution \(u^{\alpha }\) solving the following problem:
Theorem 3.2
((TR) method). Suppose that the problem (1.1) has a solution u satisfying
for some known constant \(E_{2} >0\). Assume that we can choose \(M_{\alpha }>0\) such that
Then
Remark 3.2
Let us choose \(M_{\alpha }= \frac{1}{L^{2}}\log ^{2} (\alpha ^{- \ell } )\), for some \(\ell \in (0,1)\). Then the hypothesis
is fulfilled and (3.12) is of order
Theorem 3.3
(Estimate \(\mathbb{H}^{p}\))
Let us choose \(M_{\alpha }>0\) such that \(\lim_{\alpha \to 0^{+}}M_{\alpha }= \infty \) and
Assume further that the problem (1.1) has a unique exact solution u satisfying \(u\in \mathcal{L}^{\infty } (0,L;\mathbb{H}^{p+q} )\), for \(p,q>0\). Then, for all \(y \in [0,L]\), we have
Remark 3.3
Let any \(\chi \in (0,1)\). We choose
Then condition (3.14) is satisfied as \(\alpha \to 0^{+}\) and the right-hand side of (3.15) is of order
3.2 Proof of Theorem 3.1
Problem (3.5) can be rewritten as the following integral equation:
where we define the operators for \(z>0\)
and
First, we shall prove some inequalities which will be used in the main part of our proof. The following lemma is proved directly (we omit the proof).
Lemma 3.1
For \(z\geq 0\), we have
We need the following lemma.
Lemma 3.2
For \(z \in [0,L]\). The following estimates hold
Proof
(a) We have
On other hand, it is easy to see that
for \(0< c < Le\). Hence if \(\gamma (\alpha ) < Le\), then we obtain
It follows from (3.22) that
The proof of (b) is similar. This completes the proof of the lemma. □
We are now in a position to prove the theorem.
Proof of Theorem 3.1
Using the triangle inequality, we have
We observe that
We first estimate the term \(\widetilde{\mathcal{A}^{\alpha }}\). Combining with (2.7) and (3.17) we obtain
From Parseval’s relation we obtain
Using (3.21a) we have
where we have used the elementary inequality \(e^{z} \geq z\), for \(z>0\) which leads to
and thus it follows that
It follows from (3.21b) that
Using Hölder’s inequality, (3.21a) and (2.4), one has
Thus from (2.4) and (3.21b), by the Hölder inequality, we have
Combining (3.28)–(3.32) yields
where \(C(\lambda _{1},L)\) is a positive constant that depends on \(\lambda _{1}\), L but it is independent of y and m. Next we have
It follows from Parseval’s relation that
Using inequality (3.23), we get
for \(C(\lambda _{1},L)\) a positive constant which depends on L and \(\lambda _{1}\). Hence, we get
Combining (3.25), (3.33) and (3.35), we deduce that
which leads to (3.8). The proof of Theorem 3.1 is completed. □
3.3 Proof of Theorem 3.2
It is easy to verify the following result.
Lemma 3.3
For \(z\geq 0\) and \(\lambda _{m} \leq M^{\alpha }\), we have
The solution of the regularized problem (3.9) is given by
By the triangle inequality, one has
It is straightforward to see that
Observe that, from (2.7) and (3.2), we get
Using Parseval’s relation coupled with the basic inequality \((a+b+c+d)^{4} \leq 4(a^{2}+b^{2}+c^{2}+c^{2})\), we have
We first estimate the term \(J_{1}^{\alpha }\). Using (3.37a) and (1.2), one has
It follows from (3.37b) and (1.2) that
For \(J_{3}^{\alpha }\), applying Hölder’s inequality and using (3.37a) coupled with (2.4) we have
Similarly, from (3.37b), (2.4) and Hölder’s inequality, we deduce that
Combining (3.42)–(3.46), we conclude that
Also we have
Combining (3.39), (3.47) and (3.48), we get
This completes the proof of the theorem.
3.4 Proof of Theorem 3.3
Proof
Using the triangle inequality, we deduce that
From (3.41), we have
Using similar arguments to obtaining (3.47), we deduce that
Similarly, we infer from (3.48) that
Combining (3.50), (3.52) and (3.53), we get
leading as a result to (3.15). □
4 Numerical results
In this section, we provide an example to illustrate how the proposed regularized solution approximates the exact solution for the biharmonic elliptic problem. Let \(Q_{L} := (0, 1) \times (0,\pi )\), and the problem has the following form:
subject to the conditions given by
The eigenvalues and eigenvectors of the operator −Δ depend on the specified boundary conditions. For the Dirichlet boundary conditions, the eigenvalues are \(\lambda _{m} = m^{2}\) and the corresponding eigenelements \(\xi _{m}(x) = \sqrt{\frac{2}{\pi }} \sin (mx)\) which form an orthonormal basis in \(\mathcal{L}^{2}(0, \pi )\).
Then we have the exact solution
Next, we generate the final measurement data with noise by
For the discretization, a uniform grid of mesh points \((x_{i},y_{j})\) is used to discretize the space and time intervals for \(i = \overline{1,N _{x} +1}\), \(j = \overline{1,N_{y} +1}\),
The inner product in \(\mathcal{L}^{2}(0, \pi )\) can be approximated by the 1-D composite Simpson rule of numerical integration as
where \(x_{k} = \frac{k\pi }{N_{x}+1}\), \(x_{0} =0\), \(x_{N_{x}+1} = \pi \).
The regularized solution of the problem (4.1)–(4.2) is as follows:
where we define the operators for \(z>0\)
and
The relative errors are evaluated by
Here, we present graphs illustrating the numerical example, which we are considering. Table 1 shows that the smaller α, the smaller the error between the exact solution and the regularized solution, and the errors are acceptable. Specifically, in Fig. 1, we can see the evaluation results at \(y = 0.1\) and \(y = 0.3\). Moreover, we also show 3D graphs of the exact and regularized solutions throughout the domain \((0,1) \times (0,\pi )\) in Fig. 2. Seen from that point of view, the result of the method of correction is effective.
5 Conclusions
Problem (1.1) was solved using two regularization methods based on problems (3.1) and (3.2). Convergence and stability estimates, as the noise level tends to zero, are formulated and proved. Numerical examples support the theoretical findings of the paper. In future work we hope to consider extending the current study to nonlinear sources to allow for an even wider range of physical applications in for example nonlinear elasticity.
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Danh, H.Q.N., O’Regan, D., Vo, V.A. et al. Regularization of an initial inverse problem for a biharmonic equation. Adv Differ Equ 2019, 255 (2019). https://doi.org/10.1186/s13662-019-2191-4
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DOI: https://doi.org/10.1186/s13662-019-2191-4