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Vo Van Au - One of the best experts on this subject based on the ideXlab platform.
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analysis of a quasi reversibility method for a terminal value quasi linear Parabolic Problem with measurements
Siam Journal on Mathematical Analysis, 2019Co-Authors: Nguyen Huy Tuan, Vo Anh Khoa, Vo Van AuAbstract:This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value Parabolic Problem with noisy data. Based on data me...
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analysis of a quasi reversibility method for a terminal value quasi linear Parabolic Problem with measurements
arXiv: Numerical Analysis, 2018Co-Authors: Nguyen Huy Tuan, Vo Anh Khoa, Vo Van AuAbstract:This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value Parabolic Problem with noisy data. Based on data measurements, we perturb the Problem by the so-called filter regularized operator to design an approximate Problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization Problems for brain tumors and heat conduction Problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin method to study the weak solvability of the approximate Problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.
Nguyen Huy Tuan - One of the best experts on this subject based on the ideXlab platform.
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analysis of a quasi reversibility method for a terminal value quasi linear Parabolic Problem with measurements
Siam Journal on Mathematical Analysis, 2019Co-Authors: Nguyen Huy Tuan, Vo Anh Khoa, Vo Van AuAbstract:This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value Parabolic Problem with noisy data. Based on data me...
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analysis of a quasi reversibility method for a terminal value quasi linear Parabolic Problem with measurements
arXiv: Numerical Analysis, 2018Co-Authors: Nguyen Huy Tuan, Vo Anh Khoa, Vo Van AuAbstract:This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value Parabolic Problem with noisy data. Based on data measurements, we perturb the Problem by the so-called filter regularized operator to design an approximate Problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization Problems for brain tumors and heat conduction Problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin method to study the weak solvability of the approximate Problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.
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on a backward Parabolic Problem with local lipschitz source
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Nguyen Huy Tuan, Dang Duc TrongAbstract:Abstract We consider the regularization of the backward in time Problem for a nonlinear Parabolic equation in the form u t + A u ( t ) = f ( u ( t ) , t ) , u ( 1 ) = φ , where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized Problem (with a regularization parameter β > 0 ) is well-posed and that its solution U β ( t ) converges on [ 0 , 1 ] to the exact solution u ( t ) as β → 0 + . These results extend some earlier works on the nonlinear backward Problem.
Marco Picasso - One of the best experts on this subject based on the ideXlab platform.
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a posteriori error estimates and adaptive finite elements for a nonlinear Parabolic Problem related to solidification
Computer Methods in Applied Mechanics and Engineering, 2003Co-Authors: O Kruger, Marco Picasso, Jeanfrancois ScheidAbstract:A posteriori error estimates are derived for a nonlinear Parabolic Problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L-2 in time H-1 in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the effectivity index is close to one. An adaptive finite element algorithm is proposed and a solutal. dendrite is computed. (C) 2002 Elsevier Science B.V. All rights reserved.
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adaptive finite elements for a linear Parabolic Problem
Computer Methods in Applied Mechanics and Engineering, 1998Co-Authors: Marco PicassoAbstract:A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach. (C) 1998 Elsevier Science S.A. All rights reserved.
Marián Slodička - One of the best experts on this subject based on the ideXlab platform.
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determination of a solely time dependent source in a semilinear Parabolic Problem by means of boundary measurements
Journal of Computational and Applied Mathematics, 2015Co-Authors: Marián SlodičkaAbstract:A semilinear Parabolic Problem of second order with an unknown solely time-dependent source term is studied. The missing source is recovered from an additional integral measurement over the boundary. The global in time existence, uniqueness as well as the regularity of a solution are addressed. A new numerical scheme based on Rothe's method is designed and convergence of iterates towards the exact solution is shown.
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error analysis in the reconstruction of a convolution kernel in a semilinear Parabolic Problem with integral overdetermination
Journal of Computational and Applied Mathematics, 2015Co-Authors: Rob De Staelen, K Van Bockstal, Marián SlodičkaAbstract:A semilinear Parabolic Problem of second order with an unknown solely time-dependent convolution kernel is considered. An additional given global measurement (a space integral of the solution) ensures the existence of a unique weak solution. The unknown kernel function can be approximated by a time-discrete numerical scheme based on Backward Euler's method (Rothe's method). In this contribution, an error analysis for the time discretization is performed of the existing numerical algorithm. Numerical experiments support the theoretically obtained results.
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reconstruction of a convolution kernel in a semilinear Parabolic Problem based on a global measurement
arXiv: Analysis of PDEs, 2014Co-Authors: Rob De Staelen, Marián SlodičkaAbstract:A semilinear Parabolic Problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe's method, derive a priori estimates and prove convergence of iterates towards the exact solution.
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determination of an unknown diffusion coefficient in a semilinear Parabolic Problem
Journal of Computational and Applied Mathematics, 2013Co-Authors: K Van Bockstal, Marián SlodičkaAbstract:A semilinear Parabolic Problem of second order with an unknown diffusion coefficient in a subregion is considered. The missing data are compensated by a total flux condition through a given surface. The solvability of this Problem is proved. A numerical algorithm based on Rothe's method is designed and the convergence of approximations towards the solution is shown. The results of numerical experiments are discussed.
Vladimir G Kamburg - One of the best experts on this subject based on the ideXlab platform.
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globally strictly convex cost functional for an inverse Parabolic Problem
Mathematical Methods in The Applied Sciences, 2016Co-Authors: Michael V Klibanov, Vladimir G KamburgAbstract:A coefficient inverse Problem for a Parabolic equation is considered. Using a Carleman weight function, a globally strictly convex cost functional is constructed for this Problem. Copyright © 2015 John Wiley & Sons, Ltd.
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globally strictly convex cost functional for an inverse Parabolic Problem
arXiv: Mathematical Physics, 2015Co-Authors: Michael V Klibanov, Vladimir G KamburgAbstract:A coefficient inverse Problem for a Parabolic equation is considered. Using a Carleman Weight Function, a globally strictly convex cost functional is constructed for this Problem.
