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20 pages, 2155 KB

Open AccessArticle

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

by Ruixue Gu, Hongsun Fu and Zhuoyue Wang

Mathematics 2023, 11(7), 1706; https://doi.org/10.3390/math11071706 - 3 Apr 2023

Viewed by 1529

In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including

L^{1}

-like and total [...] Read more.

L^{1}

-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on

ε

-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems. Full article

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20 pages, 343 KB

Open AccessArticle

An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

by Jamilu Sabi’u, Kanikar Muangchoo, Abdullah Shah, Auwal Bala Abubakar and Kazeem Olalekan Aremu

Symmetry 2021, 13(10), 1829; https://doi.org/10.3390/sym13101829 - 1 Oct 2021

Cited by 8 | Viewed by 2429

Abstract

This article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai–Liao (DL) and the extended three-term Polak–Ribiére–Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers. Full article

(This article belongs to the Special Issue Symmetry in Abstract Differential Equations)

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16 pages, 397 KB

Open AccessArticle

Progressive Iterative Approximation with Preconditioners

by Chengzhi Liu and Zhongyun Liu

Mathematics 2020, 8(9), 1503; https://doi.org/10.3390/math8091503 - 4 Sep 2020

Cited by 15 | Viewed by 2839

Abstract

The progressive iterative approximation (PIA) plays an important role in curve and surface fitting. By using the diagonally compensated reduction of the collocation matrix, we propose the preconditioned progressive iterative approximation (PPIA) to improve the convergence rate of PIA. For most of the normalized totally positive bases, we show that the presented PPIA can accelerate the convergence rate significantly in comparison with the weighted progressive iteration approximation (WPIA) and the progressive iterative approximation with different weights (DWPIA). Furthermore, we propose an inexact variant of the PPIA (IPPIA) to reduce the computational complexity of the PPIA. We introduce the inexact solver of the preconditioning system by employing some state-of-the-art iterative methods. Numerical results show that both the PPIA and the IPPIA converge faster than the WPIA and DWPIA, while the elapsed CPU times of the PPIA and IPPIA are less than those of the WPIA and DWPIA. Full article

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19 pages, 570 KB

Open AccessEditor’s ChoiceArticle

A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

by Luca Bergamaschi

Algorithms 2020, 13(4), 100; https://doi.org/10.3390/a13040100 - 21 Apr 2020

Cited by 16 | Viewed by 4904

Abstract

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems

A_{k} x_{k} = b_{k}, k = 1, \dots

arising in many scientific applications, such [...] Read more.

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems

A_{k} x_{k} = b_{k}, k = 1, \dots

arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates. Full article

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17 pages, 1342 KB

Open AccessArticle

Distributing Load Flow Computations Across System Operators Boundaries Using the Newton–Krylov–Schwarz Algorithm Implemented in PETSc

by Stefano Guido Rinaldo, Andrea Ceresoli, Domenico J. P. Lahaye, Marco Merlo, Miloš Cvetković, Silvia Vitiello and Gianluca Fulli

Energies 2018, 11(11), 2910; https://doi.org/10.3390/en11112910 - 25 Oct 2018

Cited by 3 | Viewed by 3275

Abstract

The upward trends in renewable energy penetration, cross-border flow volatility and electricity actors’ proliferation pose new challenges in the power system management. Electricity and market operators need to increase collaboration, also in terms of more frequent and detailed system analyses, so as to ensure adequate levels of quality and security of supply. This work proposes a novel distributed load flow solver enabling for better cross border flow analysis and fulfilling possible data ownership and confidentiality arrangements in place among the actors. The model exploits an Inexact Newton Method, the Newton–Krylov–Schwarz method, available in the portable, extensible toolkit for scientific computation (PETSc) libraries. A case-study illustrates a real application of the model for the TSO–TSO (transmission system operator) cross-border operation, analyzing the specific policy context and proposing a test case for a coordinated power flow simulation. The results show the feasibility of performing the distributed calculation remotely, keeping the overall simulation times only a few times slower than locally. Full article

(This article belongs to the Section F: Electrical Engineering)

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