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Global linear convergence in operator splitting methods

Goran Banjac, Paul J. Goulart
2016 2016 IEEE 55th Conference on Decision and Control (CDC)  
We establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged  ...  Moreover, we propose a novel linearly convergent splitting method for linear programming.  ...  splitting (PRS), Douglas-Rachford splitting (DRS), and the alternating direction method of multipliers (ADMM) [6] , [7] .  ... 
doi:10.1109/cdc.2016.7798275 dblp:conf/cdc/BanjacG16 fatcat:244sj2o7t5aptkvo2q42gk5ry4

Estimating numerical errors due to operator splitting in global atmospheric chemistry models: Transport and chemistry

Mauricio Santillana, Lin Zhang, Robert Yantosca
2016 Journal of Computational Physics  
We present upper bounds for the numerical errors introduced when using operator splitting methods to integrate transport and non-linear chemistry processes in global chemical transport models (CTM).  ...  We find similar upper bounds in operator splitting numerical errors in global CTM simulations.  ...  MS would like to thank Jonathan Pines for his involvement in the exploratory phases of this project.  ... 
doi:10.1016/j.jcp.2015.10.052 fatcat:d5fnkswrpzhu3bnc4j6kcp7lku

Nonlinear Analysis: Algorithm, Convergence, and Applications 2014

Yisheng Song, Rudong Chen, Guoyin Li, Changsen Yang, Gaohang Yu
2014 Abstract and Applied Analysis  
are two bounded linear operators.  ...  Liu showed the strong convergence theorems of the CQ algorithm for H-monotone operators in Hilbert spaces by hybrid method in the mathematical programming.  ...  Liu showed the strong convergence theorems of the CQ algorithm for H-monotone operators in Hilbert spaces by hybrid method in the mathematical programming.  ... 
doi:10.1155/2014/912910 fatcat:oy6y6a3eorazxi34ckznkxght4

Page 5045 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
For non-interior continuation/smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence  ...  In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous.  ... 

Page 8469 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
In the scalar case, the author proves that the operator splitting scheme converges to the unique entropy solution in the sense of Kruzhkov.  ...  The main result of the paper is the proof of the rate of convergence for a semi-discrete operator splitting method applied to evolutive For the web version of Mathematical Reviews, see http: //www.ams.org  ... 

An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime

Stéphane Descombes, Mechthild Thalhammer
2010 BIT Numerical Mathematics  
From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrödinger equations in the semi-classical regime  ...  In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters.  ...  convergence orders of the time-splitting methods. critical parameter global error (p = 1) ratio (p = 1)  ... 
doi:10.1007/s10543-010-0282-4 fatcat:bag4szpueneh3djfh67rhszvpm

The error structure of the Douglas–Rachford splitting method for stiff linear problems

Eskil Hansen, Alexander Ostermann, Katharina Schratz
2016 Journal of Computational and Applied Mathematics  
In this paper we derive a rigorous error analysis in the setting of linear dissipative operators and inhomogeneous evolution equations.  ...  It is unconditionally stable and is considered to be a robust choice of method in most settings. However, it possesses a rather unfavorable local error structure.  ...  Hence, the previous lack of convergence for the Lie splitting is no longer present for the DR and PR methods.  ... 
doi:10.1016/j.cam.2016.02.037 fatcat:3jzr4cenybew5a7vjwqza75w7u

Convergence rates of Forward--Douglas--Rachford splitting method [article]

Cesare Molinari, Jingwei Liang, Jalal Fadili
2018 arXiv   pre-print
In this paper, we consider the Forward--Douglas--Rachford splitting method (FDR) [10,40], and study both global and local convergence rates of this method.  ...  Over the past years, operator splitting methods have become ubiquitous for non-smooth optimization owing to their simplicity and efficiency.  ...  and Leverhulme Trust project "Breaking the non-convexity barrier", the EPSRC grant "EP/M00483X/1", EPSRC centre "EP/N014588/1", the Cantab Capital Institute for the Mathematics of Information, and the Global  ... 
arXiv:1801.01088v1 fatcat:ctue4jwj7rcuxmq3ulb5syl4dy

Page 3350 of Mathematical Reviews Vol. , Issue 92f [page]

1992 Mathematical Reviews  
Suppose A,B are linear selfadjoint operators in a Hilbert space H with domains D(A) = D(B) dense in H.  ...  For the iterative solution of a linear system Au = f, a splitting A=M —N is typically considered such that the iteration operator M~—!N has spectral radius less than one.  ... 

On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential

Christof Neuhauser, Mechthild Thalhammer
2009 BIT Numerical Mathematics  
In this paper, we study the convergence behaviour of high-order exponential operator splitting methods for the time integration of linear Schrödinger equations involving an unbounded potential; in particular  ...  We deduce a global error estimate which implies that any time-splitting method retains its classical convergence order for linear Schrödinger equations, provided that the exact solution fulfills suitable  ...  also Convergence analysis In the following, we deduce a global error estimate for exponential operator splitting methods of the form (2.2) when applied to linear evolution equations (1.2).  ... 
doi:10.1007/s10543-009-0215-2 fatcat:szdzlnrsdzb3beype7hmolfyvy

Additive domain decomposition operator splittings—convergence analyses in a dissipative framework

Eskil Hansen, Erik Henningsson
2016 IMA Journal of Numerical Analysis  
The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field.  ...  Namely, alternating direction implicit schemes and additive splitting schemes of first and second order.  ...  Acknowledgments The authors thank Philipp Birken for helpful input on domain decomposition methods during the initial preparation of the paper.  ... 
doi:10.1093/imanum/drw043 fatcat:oovhoah4bbeujh2prhe67jtt5i

On the convergence rates of proximal splitting algorithms

Jingwei Liang, Jalal M. Fadili, Gabriel Peyre
2014 2014 IEEE International Conference on Image Processing (ICIP)  
Moreover, under an appropriate regularity assumption on the fixed point operator, local linear convergence rate is also established.  ...  These results are then applied to analyze the convergence rate of various proximal splitting methods in the literature, which includes the Forward-Backward, generalized Forward-Backward, Douglas-Rachford  ...  Forward-Backward splitting method (GFB) [3] when L i = Id, or primal-dual splitting methods.  ... 
doi:10.1109/icip.2014.7025842 dblp:conf/icip/LiangFP14 fatcat:ym6zh5tbzzeqlc24fsjtljxmrm

Page 5659 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
On improving the convergence of the method of successive approximations in the solution of a linear operator equation. (Russian. English and Azerbaijani summaries) Izv. Akad. Nauk Azerb. Ser. Fiz.  ...  Summary: “In this paper, we consider robust inversion of linear operators with convex constraints.  ... 

Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type

Philippe Chartier, Florian Méhats, Mechthild Thalhammer, Yong Zhang
2017 Mathematical Modelling and Numerical Analysis  
The convergence behaviour of multi-revolution composition methods combined with timesplitting methods is analysed for highly oscillatory linear differential equations of Schrödinger type.  ...  Numerical experiments In this section, we illustrate the convergence behaviour of multi-revolution composition methods combined with time-splitting methods for linear and nonlinear Schrödinger equations  ...  Numerical results In Figure 3 , the global errors of the second-order and fourth-order multi-revolution composition methods combined with the fourth-order splitting method, obtained for the linear and  ... 
doi:10.1051/m2an/2017010 fatcat:7copeuepz5fxxbkwwqzz6vnpm4

Page 2219 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
In particular, the convergence proofs do not require the affine operator to be symmetric.  ...  We specialize our matrix-splitting-like method to discrete-time op- timal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets.  ... 
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