## Anderson

 swMATH ID: 15192 Software Authors: Homer F. Walker; Peng Ni Description: Anderson Acceleration for Fixed-Point Iterations. This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [J. Assoc. Comput. Mach., 12 (1965), pp. 547–560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anderson mixing; however, it seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities, this method has received relatively little attention from these communities over the years. A recent paper by H. Fang and Y. Saad [Numer. Linear Algebra Appl., 16 (2009), pp. 197–221] has clarified a remarkable relationship of Anderson acceleration to quasi-Newton (secant updating) methods and extended it to define a broader Anderson family of acceleration methods. In this paper, our goals are to shed additional light on Anderson acceleration and to draw further attention to its usefulness as a general tool. We first show that, on linear problems, Anderson acceleration without truncation is “essentially equivalent” in a certain sense to the generalized minimal residual (GMRES) method. We also show that the Type 1 variant in the Fang–Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method. We then discuss practical considerations for implementing Anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications. Homepage: http://epubs.siam.org/doi/abs/10.1137/10078356X Related Software: NKA; UMFPACK; PETSc; GitHub; SCS; Poblano; TensorToolbox; RODAS; Matlab; MooNMD; EPSfun; SuperMann; NOX Solver AndersonAcceleration; CVX; SHASTA; deal.ii; PRMLT; NSOIA; mftoolbox; SQUAREM Referenced in: 108 Publications
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### Referenced by 250 Authors

 5 De Sterck, Hans 5 John, Volker 5 Kelley, Carl T. 5 Pollock, Sara 5 Rebholz, Leo G. 4 Kuzmin, Dmitri 4 Xiao, Mengying 3 Brezinski, Claude 3 Chacón, Luis 3 Hauck, Cory D. 3 Radu, Florin Adrian 3 Redivo-Zaglia, Michela 3 Taitano, William T. 3 Tchelepi, Hamdi A. 3 Toth, Alex 3 Walker, Homer F. 2 An, Hengbin 2 Barrenechea, Gabriel R. 2 Both, Jakub Wiktor 2 Boyd, Stephen Poythress 2 Ciaramella, Gabriele 2 Hamilton, Steven P. 2 He, Yunhui 2 Huang, Ting-Zhu 2 Jiang, Jiamin 2 Knobloch, Petr 2 Knoll, Dana A. 2 Levitt, Antoine 2 Li, Zhizhi 2 Martínez, José Mario 2 Nordbotten, Jan Martin 2 Pallavicini, Andrea 2 Pawlowski, Roger P. 2 Scheufele, Klaudius 2 Schulte, Miriam 2 Tomin, Pavel 2 Tyrtyshnikov, Evgeniĭ Evgen’evich 2 Zhang, Huai 2 Zhang, Junzi 1 Akian, Marianne 1 Al Nazer, Safaa 1 Almeida, Regina C. 1 Alpak, Faruk O. 1 Anderson, Donald G. M. 1 Anderson, Sarah E. 1 Appelö, Daniel 1 Araya-Polo, Mauricio 1 Augustin, Matthias Albert 1 Bai, Zhaojun 1 Bakurskiy, S. V. 1 Barnafi Wittwer, Nicolás Alejandro 1 Barnafi, Nicolás A. 1 Basting, Christopher 1 Basting, Steffen 1 Beik, Fatemeh Panjeh Ali 1 Benzi, Michele 1 Bermejo, Rodolfo 1 Berndt, Markus 1 Berrill, Mark A. 1 Bindel, David S. 1 Birgin, Ernesto G. 1 Blom, David S. 1 Bochev, M. A. 1 Böhlke, Thomas 1 Bormetti, Giacomo 1 Brune, Peter R. 1 Bruss, Don E. 1 Bungartz, Hans-Joachim 1 Cai, Yunfeng 1 Caiazzo, Alfonso 1 Callegaro, Giorgia 1 Carlson, Neil N. 1 Carpentieri, Bruno 1 Carpio, Jaime 1 Catabriga, Lucia 1 Chehab, Jean-Paul 1 Chen, Guangye 1 Chen, Zheng 1 Cheng, Jian 1 Chu, Risheng 1 Chupin, Maxime 1 Clarno, Kevin T. 1 Colera, Manuel 1 Coutinho, Alvaro L. G. A. 1 DeCaria, Victor P. 1 Deckelnick, Klaus 1 Dede’, Luca 1 Di Gregorio, Simone 1 Dolejší, Vít 1 Dupuy, Mi-Song 1 Ellis, J. Austin 1 Ernesti, Felix 1 Evans, Claire 1 Evans, Tom 1 Fabrini, Giulia 1 Fiebach, André 1 Forbes, Dominique 1 Frank, Florian 1 Frank, Jason 1 Fu, Anqi ...and 150 more Authors
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### Referenced in 45 Serials

 16 Journal of Computational Physics 11 SIAM Journal on Scientific Computing 8 Numerical Algorithms 7 Computer Methods in Applied Mechanics and Engineering 6 Computers & Mathematics with Applications 6 Journal of Computational and Applied Mathematics 5 SIAM Journal on Numerical Analysis 3 SIAM Review 3 Computational Geosciences 2 Computers and Fluids 2 Applied Mathematics and Computation 2 SIAM Journal on Matrix Analysis and Applications 2 SIAM Journal on Optimization 2 Numerical Linear Algebra with Applications 1 Acta Mechanica 1 Archive for Rational Mechanics and Analysis 1 Inverse Problems 1 Mathematics of Computation 1 Applied Mathematics and Optimization 1 Journal of Scientific Computing 1 Journal of Integral Equations and Applications 1 European Journal of Applied Mathematics 1 Applied Mathematical Modelling 1 Linear Algebra and its Applications 1 SIAM Journal on Applied Mathematics 1 Computational Optimization and Applications 1 Journal of Inverse and Ill-Posed Problems 1 ETNA. Electronic Transactions on Numerical Analysis 1 Advances in Computational Mathematics 1 Mathematical and Computer Modelling of Dynamical Systems 1 International Journal of Theoretical and Applied Finance 1 Differential Equations 1 Quantitative Finance 1 Journal of Systems Science and Complexity 1 Journal of Numerical Mathematics 1 Acta Numerica 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 Communications in Computational Physics 1 SIAM Journal on Imaging Sciences 1 Advances in Applied Mathematics and Mechanics 1 Journal of Computational and Graphical Statistics 1 S$$\vec{\text{e}}$$MA Journal 1 Journal of Applied Analysis and Computation 1 SMAI Journal of Computational Mathematics 1 Results in Applied Mathematics
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### Referenced in 26 Fields

 85 Numerical analysis (65-XX) 21 Fluid mechanics (76-XX) 19 Partial differential equations (35-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 10 Mechanics of deformable solids (74-XX) 9 Operations research, mathematical programming (90-XX) 7 Linear and multilinear algebra; matrix theory (15-XX) 6 Computer science (68-XX) 6 Statistical mechanics, structure of matter (82-XX) 4 Statistics (62-XX) 3 Quantum theory (81-XX) 3 Geophysics (86-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Biology and other natural sciences (92-XX) 2 Operator theory (47-XX) 2 Systems theory; control (93-XX) 1 Combinatorics (05-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Difference and functional equations (39-XX) 1 Approximations and expansions (41-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Differential geometry (53-XX) 1 Probability theory and stochastic processes (60-XX) 1 Optics, electromagnetic theory (78-XX) 1 Mathematics education (97-XX)