A control type method for solving the Cauchy–Stokes problem

Aboulaïch, Rajae; Abda, Amel Ben; Kallel, Moez

doi:10.1016/j.apm.2012.09.014

Cited by 12 publications

(15 citation statements)

References 7 publications

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“…In [5] the missing boundary data recovery problem is rephrased as a control problem with two states, each one satisfying one boundary condition on Γ m 0 , the control being the unknown boundary conditions on Γ a . The numerical methods proposed in [3][4][5] are iterative and therefore, for each new data, they require the resolution of many intermediate forward problem to solve the data completion problem . In the electrocardiography application, one should solve the Cauchy problem at many time steps in order to recover the dynamics of the electrical wave on the heart.…”

Section: Statement Of the Problemmentioning

confidence: 99%

Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise

Addouche

Bouarroudj

Jday

et al. 2019

Math. Model. Nat. Phenom.

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In this work, we analyze the influence of adding a body surface missing data on the solution of the electrocardiographic imaging inverse problem. The difficulty comes from the fact that the measured Cauchy data is provided only on a part of the body surface and thus a missing data boundary is adjacent to a measured boundary. In order to construct the electrical potential on the heart surface, we use an optimal control approach where the unknown potential at the external boundary is also part of the control variables. We theoretically compare this case to the case where the Dirichlet boundary condition is given on the full accessible surface. We then compare both cases and based on the distribution of noise in the measurements, we conclude whether or not it is worth to use all the data. We use the method of factorization of elliptic boundary value problems combined with the finite element method. We illustrate the theoretical results by some numerical simulations in a cylindrical domain. We numerically study the effect of the size of the missing data zone on the accuracy of the inverse solution.

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Section: Statement Of the Problemmentioning

confidence: 99%

Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise

Addouche

Bouarroudj

Jday

et al. 2019

Math. Model. Nat. Phenom.

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“…Finally, we led a comparison of our algorithm with the control-type method introduced in [1]. This method was recently extended to the Stokes system in [2]. The proposed algorithm in [1] solves the classical minimization problem (2.3) with a stopping criterion…”

Section: Two 2d Test Casesmentioning

confidence: 99%

“…Our method may be extended as it is to the linear elasticity and to the Stokes systems; see [2] where a control-type setting for Stokes system has been studied. Existence, uniqueness of a Nash equilibrium, and stability properties should hold with minimal adaption of the proofs.…”

Section: Two 3d Test Casesmentioning

confidence: 99%

Neumann--Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems

Habbal¹,

Kallel²

2013

SIAM J. Control Optim.

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We consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two-and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm.

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“…Perhaps, unfortunately, such situation occurs, for instance, in the unit disk which is a very popular domain in numerical modeling. Let us consider 12) in a unit disk domain D = {z : |z| < 1} which is biharmonic and vanishes with its normal, first and second derivatives on any nontrivial arc of ∂D not containing the boundary singularity z = 1 (see [23,24] for more details).…”

Section: Non-uniqueness Of the Sub-cauchy Stokes Problemmentioning

confidence: 99%

“…In [12], the authors propose a control-type method based on the least squares minimization problem with Cauchy-Stokes equations as constraints. Nowadays, a more appealing idea is given by the use of the energy-like formulation.…”

Section: Introductionmentioning

confidence: 99%

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

Ahmed

Abda

2018

Journal of Computational and Applied Mathematics

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In contrast to the conventional Cauchy Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius [1], we propose new Lagrange multiplier methods to solve the sub-Cauchy Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and depends on the equations used to match the values of the unknowns on the inaccessible boundary. The underlying interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and various numerical results illustrate the effectiveness and the performance of the proposed approaches.

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A control type method for solving the Cauchy–Stokes problem

Cited by 12 publications

References 7 publications

Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise

Analysis of the ECGI inverse problem solution with respect to the measurement boundary size and the distribution of noise

Neumann--Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

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