| 1. |
- Baravdish, George, et al.
(författare)
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On Backward p(x)-Parabolic Equations for Image Enhancement
- 2015
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Ingår i: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 36:2, s. 147-168
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Tidskriftsartikel (refereegranskat)abstract
- In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems we obtain a smooth image which is denoised. Second, by choosing the initial data properly we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.
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| 2. |
- Berntsson, Fredrik, et al.
(författare)
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Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots
- 2017
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Ingår i: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 38:3, s. 293-305
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Tidskriftsartikel (refereegranskat)abstract
- The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.
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| 3. |
- Lind, Martin, 1985-, et al.
(författare)
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A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems
- 2018
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Ingår i: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 39:4, s. 413-437
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Tidskriftsartikel (refereegranskat)abstract
- We study the approximation of a multiscale reaction–diusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done through micro– macro ux conditions. Our target system has a typical structure for reaction–diusion ow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semidiscrete Galerkin approximations, we provide a set of a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally ecient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a ne bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions. A
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