We present the convergence analysis of convex combination of the alternating projection and Douglas-Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results demonstrate the advantages of the algorithm over the other widely known projection methods in practically relevant simulations.
Douglas-Rachford (DR) algorithm is analyzed for Fourier phase retrieval with a single random phase mask. Local, geometric convergence to a unique fixed point is proved with numerical demonstration of global convergence.
In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge. Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. A…
Enhanced randomized Douglas-Rachford method for linear systems: Improved probabilities and adaptive momentum
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University（教育部重点实验室，北京航空航天大学数学科学学院）
Douglas-Rachford Algorithm for Magnetorelaxometry Imaging using Random and Deterministic Activations
University of Innsbruck（因斯布鲁克大学）；Private University for Health Sciences, Medical Informatics and Technology (UMIT)（健康科学、医学信息学和技术私立大学（UMIT））；Technische Universität Ilmenau（伊尔梅瑙工业大学）
Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsis…
Local Linear Convergence of the ADMM/Douglas--Rachford Algorithms without Strong Convexity and Application to Statistical Imaging
Georg-August Universität Göttingen（哥廷根大学）；Institut für Numerische und Angewandte Mathematik, Universität Göttingen（哥廷根大学数值与应用数学研究所）；Institute for Mathematical Stochastics and Felix Bernstein Institute for Mathematical Statistics in the Biosciences, Georg-August Universität Göttingen（哥廷根大学数学随机学与生物科学费利克斯·伯恩斯坦数学统计研究所）
The last two decades witnessed the increasing of the interests on the absolute value equations (AVE) of finding $x\in\mathbb{R}^n$ such that $Ax-|x|-b=0$, where $A\in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^n$. In this paper, we pay our attention on designing efficient algorithms. To this end, we reformulate AVE to a generalized linear complementarity problem (GLCP), which, among the equivalent forms, is the most economical one in the sense that it does not increase the dimension of the variables. For solving the GLCP, we propose an inexact Douglas-Rachford splitting method which can adop…
The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equ…
