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An optimal control approach for determining the source term in fractional diffusion equation by different cost functionals

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APPLIED NUMERICAL MATHEMATICS 2022年 181卷 647-664页

作者： Oulmelk, A. Afraites, L. Hadri, A. Nachaoui, M. Sultan Moulay Slimane Univ EMI FST Beni Mellal Beni Mellal Morocco Ibn Zohr Univ Lab SIV Agadir Morocco

This work is devoted to the mathematical analysis of an inverse source problem governed by a time-fractional diffusion equation. The aims of this paper are to identify the source function from additional data based on a regularized optimal control approach, and to determine the regularization parameters using bi-level optimization. To do this, firstly we formulate our inverse problem to an optimal control one with two types of fidelity terms, which are the Least-Squares fitting and L-1 norm. Secondly, we establish the existence of the minimizer for the optimal control problem corresponding to the L-1 cost function, since it is the general case. Thirdly, we present some numerical results for the different cases of the cost function. These results are based on solving the optimal control model by gradient method for the L2 case and the primal-dual method for non-smooth L1 norm. Finally, we propose an adaptive selection strategy for the regularization parameter based on the bi-level optimization method, and we give some numerical results to show the effectiveness of the proposed approach.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Time-fractional diffusion equation Inverse source problem primal-dual algorithm Nonsmooth optimization Optimal control Bi-level optimization

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Convergent Non-overlapping Domain Decomposition Methods for Variational Image Segmentation

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第2期69卷 532-555页

作者： Duan, Yuping Chang, Huibin Tai, Xue-Cheng Tianjin Univ Ctr Appl Math Tianjin Peoples R China Tianjin Normal Univ Sch Math Sci Tianjin Peoples R China Univ Bergen Dept Math Mons Belgium

This paper concerns with the non-overlapping domain decomposition methods (DDMs) for the Chan-Vese model in variational image segmentation. We work with a saddle point formulation for the non-overlapping decompositions, which leads to independent subproblems decoupled by the primal-dual algorithm. With the non-overlapping DDMs, only the interfaces of the adjacent subdomains are coupled, which means the information transmission takes place on such interfaces and therefore makes the proposed DDMs flexible and efficient. Moreover, we consider both the stripe-type and checkerboard-type decomposition methods and provide the rigorous proof of the convergence. Our numerical experiments demonstrate that the proposed DDMs are convergent, efficient, and quite robust with respect to the model parameters and image resolutions.

关键词： Chan-Vese model Non-overlapping domain decomposition methods Image segmentation primal-dual algorithm

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OPTIMAL MATCHING FORESTS AND VALUATED DELTA-MATROIDS

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2014年第1期28卷 445-467页

作者： Takazawa, Kenjiro Kyoto Univ Math Sci Res Inst Kyoto 6068502 Japan UJF CNRS Grenoble INP Lab G SCOP F-38031 Grenoble France

The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(n(2)m)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system. In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid, and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel's valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles' algorithm, we design a simpler O(n(2)m)-time algorithm for the weighted matching forest problem. By incorporating Gabow's method for the weighted matching problem into Giles' algorithm, we also present a faster algorithm for the weighted matching forest problem running in O(n(3))-time, which improves upon the previous best complexity of O(n(2)m).

关键词： matching branching matching forest delta-matroid valuated delta-matroid primal-dual algorithm

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Image reconstruction from data over two orthogonal arcs of limited-angular ranges

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MEDICAL PHYSICS 2022年第3期49卷 1468-1480页

作者： Zhang, Zheng Chen, Buxin Xia, Dan Sidky, Emil Y. Pan, Xiaochuan Univ Chicago Dept Radiol Chicago IL 60637 USA Univ Chicago Dept Radiat & Cellular Oncol Chicago IL 60637 USA

Purpose Computed tomography (CT) scanning over limited-angular ranges (LARs) is of practical interest in possible reduction of imaging dose and time and in design of nonstandard scans. This work aims to investigate image reconstruction for two nonoverlapping arcs of LARs, and to demonstrate that they may allow more accurate image reconstruction than may a single arc of LAR. Methods We consider a configuration with two nonoverlapping arcs of LARs alpha 1$\alpha _1$ and alpha 2$\alpha _2$, whose centers are separated by 90 circle$90<^>\circ$, and refer to it as a two-orthogonal-arc configuration. Data are generated from a chest phantom with two-orthogonal-arc configurations over total angular coverage alpha tau=alpha 1+alpha 2$\alpha _\tau =\alpha _1+\alpha _2$ ranging from 18 circle$18<^>\circ$ to 180 circle$180<^>\circ$, and images are reconstructed subsequently by use of the directional-total-variation (DTV) algorithm. For comparison, we also consider image reconstruction for a single-arc configuration of angular range alpha tau$\alpha _\tau$. Quantitative metrics such as the normalized root-mean-square-error (nRMSE) are used for evaluation of image reconstruction accuracy. Results Visual inspection and quantitative analysis of images reconstructed reveal that a two-orthogonal-arc configuration generally yields more accurate image reconstruction than does its single-arc counterpart. As total angular range alpha tau$\alpha _\tau$ increases, the DTV algorithm yields image reconstruction with enhanced accuracy, as expected. Also, if alpha tau$\alpha _\tau$ remains constant, the two-orthogonal-arc configuration with alpha 1=alpha 2$\alpha _1 = \alpha _2$ generally leads to image reconstruction more accurate than those of two-orthogonal-arc configurations with alpha 1 not equal alpha 2$\alpha _1 \ne \alpha _2$, as the nRMSE of the former can be lower than that of the latter for up to more than one order of magnitude. Conclusions Appropriately designed two-orthogonal-arc

关键词： computed tomography directional total variation (DTV) limited-angular range (LAR) optimization-based reconstruction orthogonal-arc configuration primal-dual algorithm

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algorithm 1001: IPscatt-A MATLAB Toolbox for the Inverse Medium Problem in Scattering

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2019年第4期45卷 1–20页

作者： Buergel, Florian Kazimierski, Kamil S. Lechleiter, Armin Univ Bremen Ctr Ind Math Bibliothekstr 5 D-28359 Bremen Germany Karl Franzens Univ Graz Inst Math & Sci Comp Heinrichstr 36-3 A-8010 Graz Austria

IPscatt is a free, open-source MATLAB toolbox facilitating the solution for time-independent scattering (also known as time-harmonic scattering) in two- and three-dimensional settings. The toolbox has three main application cases: simulation of the scattered field for a given transmitter-receiver geometry;the generation of simulated data as well as the handling of the real-world data from Institute Fresnel;and the reconstruction of the contrast from several measured, scattered fields. In each case, a variety of options tailored to the needs of practitioners is provided. For example, the toolbox allows the simulation of the scattered near field as well as of the far field. Also, it provides methods for the modeling of the incident field as point sources as well as plane waves. Finally, many common geometries of transmitters and receivers are included out of the box. Regarding the reconstruction, the provided functions implement the regularization scheme that relies on a primal-dual algorithm and was introduced by F. Burgel, K. S. Kazimierski, and A. Lechleiter [Journal of Computational Physics 339 (2017),1-30]. This article provides a survey of the mathematical concepts in scattering, connects them with the provided implementation, gives an overview of the software framework as well as its application areas, and compares it with existing software packages solving the same problem.

关键词： Inverse scattering problem parameter identification Helmholtz equation MATLAB toolbox denoising sparsity regularization total variation regularization primal-dual algorithm

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Internet primal-dual congestion control: Stability and applications

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CONTROL ENGINEERING PRACTICE 2013年第1期21卷 87-95页

作者： Zhang, Wei Tan, Liansheng Yuan, Cao Chen, Guanrong Ge, Fei Cent China Normal Univ Dept Comp Sci Wuhan 430079 Peoples R China Wuhan Polytech Univ Dept Comp & Informat Engn Wuhan 430023 Peoples R China City Univ Hong Kong Dept Elect Engn Hong Kong Hong Kong Peoples R China

This paper studies the stability of the primal-dual algorithm for Internet congestion control with round-trip communication delays. The model is formulated as a time-delayed multivariable control system for a general network, and then is studied by using time-delayed control theory. A necessary and sufficient condition is established for a network with a general topology, and then is further discussed for the representative dumbbell network The stability condition is applied to the design of FAST TCP and adaptive virtual queue (AVQ) scheme as guideline, which is validated by simulation results. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Internet primal-dual algorithm Stability FAST TCP Time-delayed control Congestion control

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A Variable Metric Extension of the Forward-Backward-Forward algorithm for Monotone Operators

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2013年第9期34卷 1050-1065页

作者： Vu, Bang Cong Univ Paris 06 Lab Jacques Louis Lions CNRS UMR 7598 F-75005 Paris France

We propose a variable metric extension of the forward-backward-forward algorithm for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. In turn, this framework provides a variable metric splitting algorithm for solving monotone inclusions involving sums of composite operators. Monotone operator splitting methods recently proposed in the literature are recovered as special cases.

关键词： Composite operator duality Monotone inclusion Monotone operator Operator splitting primal-dual algorithm Variable metric 47H05 49M29 49M27 90C25

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Continuous-Time algorithm Based on Finite-Time Consensus for Distributed Constrained Convex Optimization

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2022年第5期67卷 2552-2559页

作者： Liu, Hongzhe Zheng, Wei Xing Yu, Wenwu Western Sydney Univ Sch Comp Data & Math Sci Sydney NSW 2751 Australia Southeast Univ Sch Math Nanjing 210096 Peoples R China Southeast Univ Sch Automat Nanjing 210096 Peoples R China

This article studies the convex optimization problem with general constraints, where its global objective function is composed of the sum of local objective functions. The objective is to design a distributed algorithm to cooperatively resolve the optimization problem under the condition that only the information of each node's own local cost function and its neighbors' states can be obtained. To this end, the optimality condition of the researched optimization problem is developed in terms of the saddle point theory. On this basis, the corresponding continuous-time primal-dual algorithm is constructed for the considered constrained convex optimization problem under time-varying undirected and connected graphs. In the case that the parameters involved in the proposed algorithm satisfy certain inequality, the states of all nodes will reach consensus in finite time. Meanwhile, the average state is globally convergent to the optimal solution of the considered optimization problem under some mild and standard assumptions.

关键词： Optimization Convex functions Distributed algorithms Convergence Linear programming Multi-agent systems Graph theory Distributed optimization finite-time consensus general constraints primal-dual algorithm

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Interior-point algorithm for linear programming based on a new descent direction

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RAIRO-OPERATIONS RESEARCH 2023年第5期57卷 2473-2491页

作者： Billel, Zaoui Djamel, Benterki Aicha, Kraria Hadjer, Raouache Setif 1 Ferhat Abbas Univ Fac Sci Dept Math Lab Fundamental & Numer Math Setif 19000 Algeria Setif 1 Ferhat Abbas Univ Fac Sci Dept Math Setif 19000 Algeria

We present a full-Newton step feasible interior-point algorithm for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on a new type of transformation on the centering equations of the system which characterizes the central path. For this, we consider a new function psi(t) = t7\4 psi(t) =t74$ \psi (t)\enspace ={t}<^>{\frac{7}{4}}$. Furthermore, we show that the algorithm finds the epsilon-optimal solution of the underlying problem in polynomial time, namely O(root nlog(n+3\7 root 2))On logn+327 epsilon$ O\left(\sqrt{n}\enspace \mathrm{log}\frac{\left(n+\fracalgorithm{\sqrt[7]algorithm}\right)}{\epsilon }\right)$ iterations. Finally, a comparative numerical study is reported in order to analyze the efficiency of the proposed algorithm.

关键词： Linear programming interior-point methods descent direction primal-dual algorithm

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A Reflected Forward-Backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators

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SET-VALUED AND VARIATIONAL ANALYSIS 2021年第1期29卷 163-174页

作者： Cevher, Volkan Bang Cong Vu Ecole Polytech Fed Lausanne Lab Informat & Inference Syst Lausanne Switzerland

In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing works are clearly stated. We also provide an application of the proposed method to the image denoising by the total variation.

关键词： Monotone inclusion Monotone operator Operator splitting Cocoercive Forward-backward-forward method Forward-backward algorithm Composite operator duality primal-dual algorithm

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