Variational method for multiple parameter identification in elliptic PDEs

Quyen, Tran Nhan Tam

doi:10.1016/j.jmaa.2018.01.030

Cited by 9 publications

(4 citation statements)

References 33 publications

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“…such a sequence does exist by the definition of the infimum. We immediately notice that in view of [57,Theorem 1.7] we can conclude that Q is a weakly ˚compact subset of L 8 pΩq; see also [54,Remark 2.1] and [7,Theorem 3.16]. Consequently, we deduce the existence of q δ,ρ P Q and a subsequence pq n k q kPN Ă pq n q nPN such that pq n k q kPN converges weakly ˚to q δ,ρ in L 8 pΩq, i.e., lim…”

Section: Notation and Preliminariesmentioning

confidence: 78%

A reaction coefficient identification problem for fractional diffusion

Otárola¹,

Quyen²

2019

Inverse Problems

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We analyze a reaction coefficient identification problem for the spectral fractional powers of a symmetric, coercive, linear, elliptic, second-order operator in a bounded domain Ω. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder Ωˆp0, 8q. We thus consider an equivalent coefficient identification problem, where the coefficient to be identified appears explicitly. We derive existence of local solutions, optimality conditions, regularity estimates, and a rapid decay of solutions on the extended domain p0, 8q. The latter property suggests a truncation that is suitable for numerical approximation. We thus propose and analyze a fully discrete scheme that discretizes the set of admissible coefficients with piecewise constant functions. The discretization of the state equation relies on the tensorization of a first-degree FEM in Ω with a suitable hp-FEM in the extended dimension. We derive convergence results and obtain, under the assumption that in neighborhood of a local solution the second derivative of the reduced cost functional is coercive, a priori error estimates.

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Section: Notation and Preliminariesmentioning

confidence: 78%

A reaction coefficient identification problem for fractional diffusion

Otárola¹,

Quyen²

2019

Inverse Problems

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“…Note that the gradient type data ∇u δ can be constructed via a mollification procedure by the Clément interpolation [12] if only the observation data of u is available, see, e.g. [20,31,43]. In some literatures such as [11,19,51], it has been proposed to recover the discontinuous coefficient q(x) in (1.1) with ∇u δ ∈ (L 2 (Ω)) d via the model min…”

Section: Tv Modelmentioning

confidence: 99%

An ADMM-Newton-CNN numerical approach to a TV model for identifying discontinuous diffusion coefficients in elliptic equations: convex case with gradient observations

Tian¹,

Yuan²,

Yue³

2021

Inverse Problems

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Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known alternating direction method of multipliers (ADMM) is applied to the TV-regularized models, hence it is meaningful to consider designing numerical schemes based on the ADMM. Moreover, we show that one of the ADMM subproblems can be well solved by the active-set Newton method along with the Schur complement reduction method, and the other one can be efficiently solved by the deep convolutional neural network (CNN). The resulting ADMM-Newton-CNN approach is demonstrated to be easily implementable and very efficient even for higher-dimensional spaces with fine mesh discretization.

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“…In case some priori knowledge of the identified source is available, such as a point source, a characteristic function or a harmonic function, numerical methods treating the problem have been obtained in [5,6,30,39]. A survey of the problem of simultaneously identifying the source term and coefficients in elliptic systems from distributed observations can be found in [38], where further references can be found.…”

Section: Introductionmentioning

confidence: 99%

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

Hinze

Hofmann

Quyen³

2019

Numerical Functional Analysis and Optimization

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In this paper we investigate the problem of identifying the source term f in the elliptic system −∇ · Q∇Φ = f in Ω ⊂ R d , d ∈ {2, 3}, Q∇Φ · n = j on ∂Ω and Φ = g on ∂Ω from a single noisy measurement couple (j δ , g δ ) of the Neumann and Dirichlet data (j, g) with noise level δ > 0. In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn-Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem.The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to solutions of the regularized problem. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

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Variational method for multiple parameter identification in elliptic PDEs

Cited by 9 publications

References 33 publications

A reaction coefficient identification problem for fractional diffusion

A reaction coefficient identification problem for fractional diffusion

An ADMM-Newton-CNN numerical approach to a TV model for identifying discontinuous diffusion coefficients in elliptic equations: convex case with gradient observations

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

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