| 1. |
- Aspri, A., et al.
(author)
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A Data-Driven Iteratively Regularized Landweber Iteration
- 2020
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In: Numerical Functional Analysis and Optimization. - : Taylor and Francis Inc.. - 0163-0563 .- 1532-2467.
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Journal article (peer-reviewed)abstract
- We derive and analyze a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by some numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.
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| 2. |
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| 3. |
- Baravdish, George, et al.
(author)
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On Backward p(x)-Parabolic Equations for Image Enhancement
- 2015
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In: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 36:2, s. 147-168
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Journal article (peer-reviewed)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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| 4. |
- Berntsson, Fredrik, et al.
(author)
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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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In: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 38:3, s. 293-305
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Journal article (peer-reviewed)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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| 5. |
- Burdakov, Oleg, 1953-
(author)
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Methods of the secant type for systems of equations with symmetric Jacobian matrix
- 1983
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In: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 6:2, s. 183-195
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Journal article (peer-reviewed)abstract
- Symmetric methods (SS methods) of the secant type are proposed for systems of equations with symmetric Jacobian matrix. The SSI and SS2 methods generate sequences of symmetric matrices J and H which approximate the Jacobian matrix and inverse one, respectively. Rank-two quasi-Newton formulas for updating J and H are derived. The structure of the approximations J and H is better than the structure of the corresponding approximations in the traditional secant method because the SS methods take into account symmetry of the Jacobian matrix. Furthermore, the new methods retain the main properties of the traditional secant method, namely, J and H-1are consistent approximations to the Jacobian matrix; the SS methods converge superlinearly; the sequential (n + 1)-point SS methods have the R-order at least equal to the positive root of tn+1-tn-1=0.
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| 6. |
- Casazza, P. G., et al.
(author)
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Preface
- 2012
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In: Numerical Functional Analysis and Optimization. - : Informa UK Limited. - 0163-0563 .- 1532-2467. ; 33:7-9, s. 705-707
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Journal article (other academic/artistic)
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| 7. |
- Lind, Martin, 1985-, et al.
(author)
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A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems
- 2018
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In: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 39:4, s. 413-437
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Journal article (peer-reviewed)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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| 8. |
- Mirzapour, Mahdi, et al.
(author)
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Convergence and Semi-Convergence of a Class of Constrained Block Iterative Methods
- 2021
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In: Numerical Functional Analysis and Optimization. - : TAYLOR & FRANCIS INC. - 0163-0563 .- 1532-2467. ; 42:14, s. 1718-1746
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Journal article (peer-reviewed)abstract
- In this paper, we analyze the convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the kth iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.
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| 9. |
- Persson, Tomas, et al.
(author)
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Commuting Operators for Representations of Commutation Relations Defined by Dynamical Systems
- 2012
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In: Numerical Functional Analysis and Optimization. - : Informa UK Limited. - 1532-2467 .- 0163-0563. ; 33:7-9, s. 1126-1165
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Journal article (peer-reviewed)abstract
- In this article, using orbits of the dynamical system generated by the function F, operator representations of commutation relations XX* = F (X* X) and AB = BF (A) are studied and used to investigate commuting operators expressed using polynomials in A and B. Various conditions on the function F, defining the commutation relations, are derived for monomials and polynomials in operators A and B to commute. These conditions are further studied for dynamical systems generated by affine and q-deformed power functions, and for the beta-shift dynamical system.
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