Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations

Mophou, Gisèle; Warma, Mahamadi

doi:10.1016/j.jmaa.2022.126618

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…In ( 14), (15), and in what follows, comma in the bottom indices separates harmonic number from the actual indices of the tensor or vector components. Using (10), (12), and (13), the displacements can be given as…”

Section: Solution Constructionmentioning

confidence: 99%

“…Such information about the "side" or "secondary" parameters along with the constitutive equations of the mathematical model for the considered coupled processes allows for formulating inverse problems on the identification of unknown loadings acting over the "forbidden" area [6]. Quite often such problems appear to be ill-posed in the Hadamard sense which requires advanced approaches towards their formulation and solution [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

Postolaki

Tokovyy

2023

Z Angew Math Mech

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An inverse problem is solved for identifying unknown force loadings on the inner surface of a finite‐length hollow cylinder using the variational method of homogeneous solutions. The problem is considered within the axisymmetric formulation, and the radial and axial displacements of the outer surface of the cylinder are used as the auxiliary data for solving the inverse problem. The accessible surfaces of the cylinder (the end‐faces and the outer surface) are assumed to be free of force loading. By making use of the variational method of homogeneous solutions, the problem is reduced to an infinite system of linear algebraic equations. The solution is verified numerically and its stability with respect to small errors in the input data is analyzed.

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Section: Solution Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

Postolaki

Tokovyy

2023

Z Angew Math Mech

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“…Since the work in [21], several researchers have analysed the QRM in a wide range of problems. The application of QRM can be found, for example, in [3] for the elliptic setting, [7,24,27,32,36] for the parabolic setting and [22,26] for the hyperbolic setting. Also, with the increasing work in the backward time fractional diffusion equation, the QRM has found its application in several recent works related to time/space fractional setting, see for instance [8,37].…”

Section: Introductionmentioning

confidence: 99%

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

Mondal

Nair

2021

Computational Methods in Applied Mathematics

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An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

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Stabilization estimates for the Brinkman–Forchheimer–Kelvin–Voigt equation backward in time

Gentile

Straughan

2023

Acta Mech

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The final value value problem for the Brinkman–Forchheimer–Kelvin–Voigt equations is analysed for quadratic and cubic types of Forchheimer nonlinearity. The main term in the Forchheimer equations is allowed to be fully anisotropic. It is shown that the solution depends continuously on the final data provided the solution satisfies an a priori bound in $$L^3.$$ L 3 . The technique employed avoids the use of a specialist method for an improperly posed problem such as logarithmic convexity.

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Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations

Cited by 4 publications

References 22 publications

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

Stabilization estimates for the Brinkman–Forchheimer–Kelvin–Voigt equation backward in time

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