Approximate Gauss–Newton Methods for Nonlinear Least Squares Problems

Gratton, Serge; Lawless, Amos S.; Nichols, Nancy K.

doi:10.1137/050624935

Cited by 175 publications

(130 citation statements)

References 19 publications

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“…2.3.2). In Appendix A, we show (similar to Lawless et al, 2005;Gratton et al, 2007;Tshimanga et al, 2008) that incremental 4D-Var is equivalent to a GaussNewton (GN) optimization, where a cost function,…”

Section: Prior Emission Inventoriesmentioning

confidence: 99%

Four-dimensional variational inversion of black carbon emissions during ARCTAS-CARB with WRFDA-Chem

Guerrette

Henze

2017

Atmos. Chem. Phys.

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Abstract. Biomass burning emissions of atmospheric aerosols, including black carbon, are growing due to increased global drought, and comprise a large source of uncertainty in regional climate and air quality studies. We develop and apply new incremental four-dimensional variational (4D-Var) capabilities in WRFDA-Chem to find optimal spatially and temporally distributed biomass burning (BB) and anthropogenic black carbon (BC) aerosol emissions. The constraints are provided by aircraft BC concentrations from the Arctic Research of the Composition of the Troposphere from Aircraft and Satellites in collaboration with the California Air Resources Board (ARCTAS-CARB) field campaign and surface BC concentrations from the Interagency Monitoring of PROtected Visual Environment (IMPROVE) network on 22, 23, and 24 June 2008. We consider three BB inventories, including Fire INventory from NCAR (FINN) v1.0 and v1.5 and Quick Fire Emissions Database (QFED) v2.4r8. On 22 June, aircraft observations are able to reduce the spread between a customized QFED inventory and FINNv1.0 from a factor of 3.5 (×3.5) to only ×2.1. On 23 and 24 June, the spread is reduced from ×3.4 to ×1.4. The posterior corrections to emissions are heterogeneous in time and space, and exhibit similar spatial patterns of sign for both inventories. The posterior diurnal BB patterns indicate that multiple daily emission peaks might be warranted in specific regions of California. The US EPA's 2005 National Emissions Inventory (NEI05) is used as the anthropogenic prior. On 23 and 24 June, the coastal California posterior is reduced by ×2, where highway sources dominate, while inland sources are increased near Barstow by ×5. Relative BB emission variances are reduced from the prior by up to 35 % in grid cells close to aircraft flight paths and by up to 60 % for fires near surface measurements. Anthropogenic variance reduction is as high as 40 % and is similarly limited to sources close to observations. We find that the 22 June aircraft observations are able to constrain approximately 14 degrees of freedom of signal (DOF), while surface and aircraft observations together on 23/24 June constrain 23 DOF. Improving hourly-to daily-scale concentration predictions of BC and other aerosols during BB events will require more comprehensive and/or targeted measurements and a more complete accounting of sources of error besides the emissions.

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Section: Prior Emission Inventoriesmentioning

confidence: 99%

Four-dimensional variational inversion of black carbon emissions during ARCTAS-CARB with WRFDA-Chem

Guerrette

Henze

2017

Atmos. Chem. Phys.

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“…One important area, in which non-linear least squares problems arise, is in data fitting [16]. The Gauss-Newton method for the problem above: A start with an initial guess x (0) for the minimum, the method proceeds by the iterations [16,17].…”

Section: Levenberg-marquardt and Gauss-newton Algorithmsmentioning

confidence: 99%

Motion of a Point Mass in a Rotating Disc: A Quantitative Analysis of the Coriolis and Centrifugal Force

Haddout¹

2016

Journal of Theoretical and Applied Mechanics

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Abstract. In Newtonian mechanics, the non-inertial reference frames is a generalization of Newton's laws to any reference frames. While this approach simplifies some problems, there is often little physical insight into the motion, in particular into the effects of the Coriolis force. The fictitious Coriolis force can be used by anyone in that frame of reference to explain why objects follow curved paths. In this paper, a mathematical solution based on differential equations in non-inertial reference is used to study different types of motion in rotating system. In addition, the experimental data measured on a turntable device, using a video camera in a mechanics laboratory was conducted to compare with mathematical solution in case of parabolically curved, solving non-linear least-squares problems, based on Levenberg-Marquardt's and Gauss-Newton algorithms.

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“…Iterative techniques must be used to identify an approximate minimum of the cost function when n is large. In meteorological applications, variational assimilation is implemented using an iterative technique based on the incremental approach (Courtier et al, 1994), which in optimization theory is known as a Truncated Gauss-Newton (TGN) method (Lawless et al, 2005;Gratton et al, 2007). This approach is also widely used in oceanographic applications (Weaver et al, 2003;Moore et al, 2011a).…”

Section: Introductionmentioning

confidence: 99%

B‐preconditioned minimization algorithms for variational data assimilation with the dual formulation

Gürol

Weaver

Moore

et al. 2013

Quart J Royal Meteoro Soc

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Variational data assimilation problems arising in meteorology and oceanography require the solution of a regularized nonlinear least-squares problem. Practical solution algorithms are based on the incremental (Truncated Gauss-Newton) approach, which involves the iterative solution of a sequence of linear least-squares (quadratic minimization) sub-problems. Each sub-problem can be solved using a primal approach, where the minimization is performed in a space spanned by vectors of the size of the model control vector, or a dual approach, where the minimization is performed in a space spanned by vectors of the size of the observation vector. The dual formulation can be advantageous for two reasons. First, the dimension of the minimization problem with the dual formulation does not increase when additional control variables, such as those accounting for model error in a weak-constraint formulation, are considered. Second, whenever the dimension of observation space is significantly smaller than that of the model control space, the dual formulation can reduce both memory usage and computational cost.In this paper, a new dual-based algorithm called Restricted B-preconditioned Lanczos (RBLanczos) is introduced, where B denotes the background-error covariance matrix. RBLanczos is the Lanczos formulation of the Restricted B-preconditioned Conjugate Gradient (RBCG) method. RBLanczos generates mathematically equivalent iterates to those of RBCG and the corresponding B-preconditioned Conjugate Gradient and Lanczos algorithms used in the primal approach. All these algorithms can be implemented without the need for a square-root factorization of B. RBCG and RBLanczos, as well as the corresponding primal algorithms, are implemented in two operational ocean data assimilation systems and numerical results are presented. Practical diagnostic formulae for monitoring the convergence properties of the minimization are also presented.

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Approximate Gauss–Newton Methods for Nonlinear Least Squares Problems

Cited by 175 publications

References 19 publications

Four-dimensional variational inversion of black carbon emissions during ARCTAS-CARB with WRFDA-Chem

Four-dimensional variational inversion of black carbon emissions during ARCTAS-CARB with WRFDA-Chem

Motion of a Point Mass in a Rotating Disc: A Quantitative Analysis of the Coriolis and Centrifugal Force

B‐preconditioned minimization algorithms for variational data assimilation with the dual formulation

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