An Algorithm for Least-Squares Estimation of Nonlinear Parameters (1963)

Previous article Next article An Algorithm for Least-Squares Estimation of Nonlinear ParametersDonald W. MarquardtDonald W. Marquardthttps://doi.org/10.1137/0111030PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. W. Booth, , G. E. P. Box, , M. E. Muller and , T. I. Peterson, Forecasting by Generalized Regression Methods, Nonlinear Estimation (Princeton—IBM), Mimeo. (IBM Share Program No. 687 WL NL1), International Business Machines Corp., 1959 Google Scholar[2] G. E. P. Box and , G. A. Coutie, Application of digital computers in the exploration of functional relationships, Proc. Inst. Elec. Engrs. B, 103 (1957), 100–107 MR0094906 Google Scholar[3] Haskell B. Curry, The method of steepest descent for non-linear minimization problems, Quart. Appl. Math., 2 (1944), 258–261 MR0010667 0061.26801 CrossrefGoogle Scholar[4] H. O. Hartley, The modified Gauss-Newton method for the fitting of non-linear regression functions by least squares, Technometrics, 3 (1961), 269–280 MR0124117 0096.34603 CrossrefISIGoogle Scholar[5] D. W. Marquardt, , Chem. Eng. Progr., 55 (1959), 65–70 Google Scholar[6] D. W. Marquardt, , R. G. Bennett and , E. J. Burrell, , Jour. Molec. Spectroscopy, 7 (1961), 269–279 10.1016/0022-2852(61)90360-5 CrossrefISIGoogle Scholar[7] D. D. Morrison, Methods for nonlinear least squares problems and convergence proofs, Tracking Programs and Orbit Determination, Proc. Jet Propulsion Laboratory Seminar, 1960, 1–9 Google Scholar[8] Kenneth Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math., 2 (1944), 164–168 MR0010666 0063.03501 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Shadowing-Based Data Assimilation Method for Partially Observed ModelsBart M. de Leeuw and Svetlana DubinkinaSIAM Journal on Applied Dynamical Systems, Vol. 21, No. 2 | 11 April 2022AbstractPDF (2268 KB)Localized Sensitivity Analysis at High-Curvature Boundary Points of Reconstructing Inclusions in Transmission ProblemsHabib Ammari, Yat Tin Chow, and Hongyu LiuSIAM Journal on Mathematical Analysis, Vol. 54, No. 2 | 9 March 2022AbstractPDF (829 KB)A Stochastic Levenberg--Marquardt Method Using Random Models with Complexity ResultsEl Houcine Bergou, Youssef Diouane, Vyacheslav Kungurtsev, and Clément W. RoyerSIAM/ASA Journal on Uncertainty Quantification, Vol. 10, No. 1 | 29 March 2022AbstractPDF (616 KB)Derivative-Free Bayesian Inversion Using Multiscale DynamicsG. A. Pavliotis, A. M. Stuart, and U. VaesSIAM Journal on Applied Dynamical Systems, Vol. 21, No. 1 | 24 January 2022AbstractPDF (2356 KB)Convergence of Newton-MR under Inexact Hessian InformationYang Liu and Fred RoostaSIAM Journal on Optimization, Vol. 31, No. 1 | 7 January 2021AbstractPDF (3140 KB)A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares ProblemsEl Houcine Bergou, Youssef Diouane, Vyacheslav Kungurtsev, and Clément W. RoyerSIAM Journal on Scientific Computing, Vol. 43, No. 5 | 13 September 2021AbstractPDF (951 KB)Random Sampling from Joint Probability Distributions Defined in a Bayesian FrameworkThierry A. Mara, Marwan Fahs, Qian Shao, and Anis YounesSIAM Journal on Scientific Computing, Vol. 41, No. 1 | 15 January 2019AbstractPDF (5620 KB)On the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound ConditionMan-Chung Yue, Zirui Zhou, and Anthony Man-Cho SoSIAM Journal on Optimization, Vol. 29, No. 1 | 28 March 2019AbstractPDF (621 KB)The Conjugate Residual Method in Linesearch and Trust-Region MethodsMarie-Ange Dahito and Dominique OrbanSIAM Journal on Optimization, Vol. 29, No. 3 | 25 July 2019AbstractPDF (977 KB)Variable Projection Methods for an Optimized Dynamic Mode DecompositionTravis Askham and J. Nathan KutzSIAM Journal on Applied Dynamical Systems, Vol. 17, No. 1 | 1 February 2018AbstractPDF (1906 KB)A Duality-Based Optimization Approach for Model Adaptivity in Heterogeneous Multiscale ProblemsMatthias Maier and Rolf RannacherMultiscale Modeling & Simulation, Vol. 16, No. 1 | 1 March 2018AbstractPDF (4572 KB)Iterative Importance Sampling Algorithms for Parameter EstimationMatthias Morzfeld, Marcus S. Day, Ray W. Grout, George Shu Heng Pau, Stefan A. Finsterle, and John B. BellSIAM Journal on Scientific Computing, Vol. 40, No. 2 | 1 March 2018AbstractPDF (1644 KB)Newton Correction Methods for Computing Real Eigenpairs of Symmetric TensorsAriel Jaffe, Roi Weiss, and Boaz NadlerSIAM Journal on Matrix Analysis and Applications, Vol. 39, No. 3 | 3 July 2018AbstractPDF (1076 KB)Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and OptimizationBenjamin Peherstorfer, Karen Willcox, and Max GunzburgerSIAM Review, Vol. 60, No. 3 | 8 August 2018AbstractPDF (825 KB)The Use of Quadratic Regularization with a Cubic Descent Condition for Unconstrained OptimizationE. G. Birgin and J. M. MartínezSIAM Journal on Optimization, Vol. 27, No. 2 | 6 June 2017AbstractPDF (586 KB)An ODE-Based Method for Computing the Distance of Coprime Polynomials to Common DivisibilityNicola Guglielmi and Ivan MarkovskySIAM Journal on Numerical Analysis, Vol. 55, No. 3 | 22 June 2017AbstractPDF (593 KB)Levenberg--Marquardt Methods Based on Probabilistic Gradient Models and Inexact Subproblem Solution, with Application to Data AssimilationSIAM/ASA Journal on Uncertainty Quantification, Vol. 4, No. 1 | 16 August 2016AbstractPDF (558 KB)Synchronization of Independently Moving Cameras via Motion RecoverySIAM Journal on Imaging Sciences, Vol. 9, No. 3 | 7 July 2016AbstractPDF (4822 KB)An Efficient Gauss--Newton Algorithm for Symmetric Low-Rank Product Matrix ApproximationsSIAM Journal on Optimization, Vol. 25, No. 3 | 6 August 2015AbstractPDF (521 KB)An Algebraic Solution for the Candecomp/PARAFAC Decomposition with Circulant FactorsSIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 4 | 11 December 2014AbstractPDF (354 KB)Factorization Approach to Structured Low-Rank Approximation with ApplicationsSIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 3 | 11 September 2014AbstractPDF (401 KB)Structured Low-Rank Approximation with Missing DataSIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 2 | 27 June 2013AbstractPDF (255 KB)Rank-Deficient Nonlinear Least Squares Problems and Subset SelectionSIAM Journal on Numerical Analysis, Vol. 49, No. 3 | 23 June 2011AbstractPDF (267 KB)Computing Points that Satisfy Second Order Necessary Optimality Conditions for Unconstrained MinimizationSIAM Journal on Optimization, Vol. 20, No. 4 | 19 February 2010AbstractPDF (214 KB)On the Convergence of Recursive Trust-Region Methods for Multiscale Nonlinear Optimization and Applications to Nonlinear MechanicsSIAM Journal on Numerical Analysis, Vol. 47, No. 4 | 4 September 2009AbstractPDF (1673 KB)Relative Periodic Solutions of the Complex Ginzburg--Landau EquationSIAM Journal on Applied Dynamical Systems, Vol. 4, No. 4 | 7 August 2006AbstractPDF (871 KB)Linear versus Nonlinear Dimensionality Reduction of High-Dimensional Dynamical SystemsSIAM Journal on Scientific Computing, Vol. 25, No. 6 | 25 July 2006AbstractPDF (814 KB)Analyzing the Dynamics of Cellular Flames Using Karhunen--Loève Decomposition and Autoassociative Neural NetworksSIAM Journal on Scientific Computing, Vol. 24, No. 5 | 25 July 2006AbstractPDF (932 KB)An Elementary Model for Control of a Semiconductor Etching ProcessSIAM Review, Vol. 44, No. 4 | 4 August 2006AbstractPDF (1001 KB)A Global Convergence Theory for Dennis, El-Alem, and Maciel's Class of Trust-Region Algorithms for Constrained Optimization without Assuming RegularitySIAM Journal on Optimization, Vol. 9, No. 4 | 31 July 2006AbstractPDF (426 KB)A Trust-Region Approach to Nonlinear Systems of Equalities and InequalitiesSIAM Journal on Optimization, Vol. 9, No. 2 | 31 July 2006AbstractPDF (436 KB)Convergence Properties of Minimization Algorithms for Convex Constraints Using a Structured Trust RegionSIAM Journal on Optimization, Vol. 6, No. 4 | 31 July 2006AbstractPDF (3077 KB)Restricted Step and Levenberg–Marquardt Techniques in Proximal Bundle Methods for Nonconvex Nondifferentiable OptimizationSIAM Journal on Optimization, Vol. 6, No. 1 | 31 July 2006AbstractPDF (2829 KB)A Parallel Nonlinear Least-Squares Solver: Theoretical Analysis and Numerical ResultsSIAM Journal on Scientific and Statistical Computing, Vol. 13, No. 3 | 13 July 2006AbstractPDF (2479 KB)The Early Use of Matrix Diagonal Increments in Statistical ProblemsSIAM Review, Vol. 31, No. 3 | 18 July 2006AbstractPDF (937 KB)Nonstandard Scaling Matrices for Trust Region Gauss–Newton MethodsSIAM Journal on Scientific and Statistical Computing, Vol. 10, No. 4 | 13 July 2006AbstractPDF (1559 KB)A Continuation Approach for Solving Large-Residual Nonlinear Least Squares ProblemsSIAM Journal on Scientific and Statistical Computing, Vol. 8, No. 4 | 14 July 2006AbstractPDF (1744 KB)The Use of Linear Programming for the Solution of Sparse Sets of Nonlinear EquationsSIAM Journal on Scientific and Statistical Computing, Vol. 8, No. 2 | 14 July 2006AbstractPDF (1172 KB)Newton’s Method with a Model Trust Region ModificationSIAM Journal on Numerical Analysis, Vol. 19, No. 2 | 17 July 2006AbstractPDF (2105 KB)Computing Optimal Locally Constrained StepsSIAM Journal on Scientific and Statistical Computing, Vol. 2, No. 2 | 16 May 2012AbstractPDF (1401 KB)Robust Methods for Solving Systems of Nonlinear EquationsSIAM Journal on Scientific and Statistical Computing, Vol. 1, No. 1 | 16 May 2012AbstractPDF (1004 KB)Some Problems in the Control of Distributed Systems, and Their Numerical SolutionSIAM Journal on Control and Optimization, Vol. 17, No. 1 | 18 July 2006AbstractPDF (1597 KB)Minimum Principles for Ill-Posed ProblemsSIAM Journal on Mathematical Analysis, Vol. 9, No. 4 | 17 February 2012AbstractPDF (937 KB)Approximation to Data by Splines with Free KnotsSIAM Journal on Numerical Analysis, Vol. 15, No. 2 | 14 July 2006AbstractPDF (1543 KB)Dynamic Optimal Control Models in Advertising: a SurveySIAM Review, Vol. 19, No. 4 | 17 February 2012AbstractPDF (4271 KB)Use of Smoothing and Damping Techniques in the Solution of Nonlinear EquationsSIAM Review, Vol. 19, No. 1 | 18 July 2006AbstractPDF (1086 KB)Nonlinear Mean-Square Approximation of Finite SetsSIAM Journal on Numerical Analysis, Vol. 12, No. 1 | 14 July 2006AbstractPDF (522 KB)Complex Linear Least SquaresSIAM Review, Vol. 15, No. 4 | 18 July 2006AbstractPDF (2012 KB)Multivariate Splines with Nondegenerate PartitionsSIAM Journal on Numerical Analysis, Vol. 10, No. 4 | 14 July 2006AbstractPDF (836 KB)Parameter Selection for Modified Newton Methods for Function MinimizationSIAM Journal on Numerical Analysis, Vol. 7, No. 3 | 14 July 2006AbstractPDF (692 KB)Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation ProblemsSIAM Journal on Numerical Analysis, Vol. 7, No. 1 | 14 July 2006AbstractPDF (2230 KB)An Accelerated Gradient Projection Method for Linearly Constrained Nonlinear EstimationSIAM Journal on Applied Mathematics, Vol. 18, No. 2 | 1 August 2006AbstractPDF (1092 KB)A Survey of Numerical Methods for Unconstrained OptimizationSIAM Review, Vol. 12, No. 1 | 18 July 2006AbstractPDF (2399 KB)On a Theorem Used in Nonlinear Least SquaresSIAM Journal on Applied Mathematics, Vol. 14, No. 5 | 13 July 2006AbstractPDF (312 KB)On the Stationary Values of a Second-Degree Polynomial on the Unit SphereJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (1358 KB) Volume 11, Issue 2| 1963Journal of the Society for Industrial and Applied Mathematics205-519 History Submitted:13 June 1962Published online:13 July 2006 InformationCopyright © 1963 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0111030Article page range:pp. 431-441ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics