Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system

Jiang, Daijun; Feng, Hui; Zou, Jun

doi:10.1088/0266-5611/28/10/104002

Cited by 20 publications

(16 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, some type of regularization has to be applied to stabilize such problem. In Jiang et al, the inverse problem is transformed into an optimization problem and the convergence of regularized solution is proved. Our major interest in this work is to study the existence, uniqueness, and stability of the optimal solution.…”

Section: Optimal Control Problemmentioning

confidence: 99%

“…It should be mentioned that the forward problem / is well‐posed in the sense of Hadamard. More precisely speaking, for any

q \in scriptA

, there exists a unique weak solution u ( x , t ), which satisfies the following regularity property (see Stiemer and Jiang et al)

u \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false), u_{t} \in L^{2} false(false(0, T false); L^{2} false(normalΩ false) false) .

…”

Section: Optimal Control Problemmentioning

confidence: 99%

“…Using the boundedness of

scriptQ

, we have

\begin{matrix} right \int_{ω} | Q {false|}^{4} d x & left \leq \max | Q {false|}^{2} \int_{ω} | Q {false|}^{2} d x & right \\ right & left \leq C \int_{ω} | Q {false|}^{2} d x \\ right & left \leq C \int_{ω} | \nabla Q {false|}^{2} d x, \end{matrix}

where we have used Lemma . Moreover, u ( q ) and v ( q ) have the following regularity property (see Jiang et al):

u false(q false), v false(q false) \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false) .

By using the Sobolev imbedding theorem (see Adams) and , we have for u 1 the following estimate:

false‖ \nabla u_{1} ‖_{L^{2} false(false(0, T false); L^{4} false(ω false) false)} \leq C false‖ \nabla u_{1} ‖_{L^{2} false(false(0, T false); H^{1} false(ω false) false)} \leq C false‖ u_{1} ‖_{L^{2} false(false(0, T false); H^{2} false(ω false) false)} \leq C .

Analogously, we have for u 2 and v 2 the following estimates:

false‖ \nabla u_{2} ‖_{...}

…”

Section: Uniqueness and Stabilitymentioning

confidence: 99%

See 2 more Smart Citations

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Deng

Cai

Yang

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic-elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic-elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper. DENG ET AL 3415 coefficient problem of identification of the diffusion coefficient from the final overspecified data has been considered by several authors (see, for instance, Demir and Hasanov 22,23 ).Compared with other papers, which also deal with parameter identification problems, the results to be given in this paper have the following features. Firstly, the mathematical model proposed in this paper is a coupled parabolic-elliptic system rather than a single parabolic or elliptic equation. So this kind of problem indeed belongs to the boundary value problems of mixed type. Secondly, our task is to identify the diffusion coefficient in the coupled system, and this inverse problem generally belongs to the mathematical class of severely ill-posed problems. Furthermore, though the underlying mathematical model is linear, the corresponding inverse problem is completely nonlinear. Thirdly, since the exchange of matter and energy occurs on the interface, and after all, the connective condition is concerned with the diffusivity, the underlying mathematical model can be viewed as a nonlinear boundary-value problem combined with a nonlinear parameter-identification problem. From this point of view, such problem is rather difficult than those of single equation. Finally, the extra condition used in the paper is only imposed on the solution in the small domain rather than on the whole domain Ω. Because of the lack of measurement data, we shall carefully treat every integral to avoid using more information outside . In summary, the complexity of the inverse problem 1.2 to be introduced in following Section 3 is higher than those that are governed by single equations.In this work, we use an optimal control framework (see, eg, recent works 7,21,24,25 ) to discuss Problem 1.1 mainly from the theoretical analysis angle. As mentioned above, since the mathematical model in this paper is a coupled parabolic-elliptic system rather than the single parabolic equation in previous studies, 7,21,24,25 the major differences between them should be noted. It is well known that the properties of the solution for elliptic or parabolic system are quite different f...

show abstract

Section: Optimal Control Problemmentioning

confidence: 99%

“…It should be mentioned that the forward problem / is well‐posed in the sense of Hadamard. More precisely speaking, for any

q \in scriptA

, there exists a unique weak solution u ( x , t ), which satisfies the following regularity property (see Stiemer and Jiang et al)

u \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false), u_{t} \in L^{2} false(false(0, T false); L^{2} false(normalΩ false) false) .

…”

Section: Optimal Control Problemmentioning

confidence: 99%

“…Using the boundedness of

scriptQ

, we have

\begin{matrix} right \int_{ω} | Q {false|}^{4} d x & left \leq \max | Q {false|}^{2} \int_{ω} | Q {false|}^{2} d x & right \\ right & left \leq C \int_{ω} | Q {false|}^{2} d x \\ right & left \leq C \int_{ω} | \nabla Q {false|}^{2} d x, \end{matrix}

where we have used Lemma . Moreover, u ( q ) and v ( q ) have the following regularity property (see Jiang et al):

u false(q false), v false(q false) \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false) .

By using the Sobolev imbedding theorem (see Adams) and , we have for u 1 the following estimate:

false‖ \nabla u_{1} ‖_{L^{2} false(false(0, T false); L^{4} false(ω false) false)} \leq C false‖ \nabla u_{1} ‖_{L^{2} false(false(0, T false); H^{1} false(ω false) false)} \leq C false‖ u_{1} ‖_{L^{2} false(false(0, T false); H^{2} false(ω false) false)} \leq C .

Analogously, we have for u 2 and v 2 the following estimates:

false‖ \nabla u_{2} ‖_{...}

…”

Section: Uniqueness and Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Deng

Cai

Yang

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…geophysics [37], imaging science [14], statistics [38] and signal processing [10], and hence has received considerable attention. In this work, motivated by the recent works on the multi-parameter Tikhonov functional [28,29,39], we consider the following multi-parameter variational problem T δ α,β (x) := 1 2 Ax − y δ 2 Y + α x ℓ 1 + β 2 x 2 ℓ 2 → min, subject to x ∈ ℓ 1 , ( 4) which is called the elastic-net regularization. The functional T δ α,β was originally used in statistics [43].…”

Section: Introductionmentioning

confidence: 99%

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

2016

Self Cite

View full text Add to dashboard Cite

We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between ℓ 2 and a Hilbert space Y , possessing the unique solution. For the cases that sparsity x † ∈ ℓ 0 is expected but often slightly violated in practice, we investigate in comparison with the ℓ 1 -regularization the elastic-net regularization, where the penalty is a weighted superposition of the ℓ 1 -norm and the ℓ 2 -norm square, under the assumption that x † ∈ ℓ 1 . There occur two positive parameters in this approach, the weight parameter η and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in ℓ 1 -regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x † k → 0 for k → ∞ and the classical smoothness properties of x † as an element in ℓ 2 .

show abstract

“…It is well known that inverse problems aimed at the identification of parameter functions in differential equations or boundary conditions from observations of state variables are in general nonlinear even if the differential equations are linear. The nonlinearity of F , however, makes the construction and the use of regularization methods more complicated and diversified; please refer to [9,26,27,33,38,45,49,56,66,71,81,87] for more details.…”

Section: Introductionmentioning

confidence: 99%

Regularization Methods for Ill-Posed Problems

Cheng

Hofmann

2014

Handbook of Mathematical Methods in Imaging

View full text Add to dashboard Cite

In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

show abstract

Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system

Cited by 20 publications

References 18 publications

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

Regularization Methods for Ill-Posed Problems

Contact Info

Product

Resources

About

Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system

Cited by 20 publications

References 18 publications

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Elastic-net regularization versusℓ1-regularization for linear inverse problems with quasi-sparse solutions

Regularization Methods for Ill-Posed Problems

Contact Info

Product

Resources

About

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions