The role of the Hessian matrix in fitting models to measurements

Thacker, W. C.

doi:10.1029/jc094ic05p06177

Cited by 203 publications

(179 citation statements)

References 41 publications

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“…There are several papers presenting essentially the same result for dynamical models [26,32]. However, there has also been some confusion.…”

Section: Clarification Of the Existing Theorymentioning

confidence: 69%

“…The importance of the Hessian matrix and its inverse in variational DA for geophysical applications is underlined in [32], although this has been a well-known fact in statistics for decades (see, for example, [3]). Section 2.1 illustrates that for linear and moderately non-linear DA/estimation problems H −1 (⋅) can serve as an approximation of the estimation (analysis) error covariance matrix.…”

Section: On the Role Of The Hessian And Its Inversementioning

confidence: 99%

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Hessian-based covariance approximations in variational data assimilation

Gejadze

Shutyaev

Dimet

2018

Russian Journal of Numerical Analysis and Mathematical Modelling

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Abstract:The problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed. Keywords:Variational data assimilation, optimality system, Hessian, estimation error covariance. MSC 2010: 65K10, 86A22, 93B40Data assimilation (state or parameter estimation) for high-dimensional distributed parameter dynamical systems has become a powerful analysis tool in recent decades. It is a fusion process of incomplete and, possibly, indirect observations of state variables with a mathematical model governed by partial differential equations, complemented by a priori information. The applications include model initialization in meteorology and oceanography, air and water quality monitoring, 'calibration' of groundwater and reservoir models, discharge estimation and forecasting in river hydraulics and hydrology, flow estimation and control in aerospace engineering, process control in chemical and nuclear engineering, etc. In different applications these estimation problems are also referred as 'inverse problems', 'data assimilation' (DA), and 'calibration'. Variational methods were introduced in meteorology by Sasaki [27]. These methods consider the equations governing the flow as constraints and the problem is closed by using a variational principle, e.g., the minimization of the discrepancy between the model prediction and the observations. Optimal Control Approach ( [30]. Therefore, variational estimation/DA is a method based on the optimal control theory, which can also be understood as a special case of the maximum a-posteriory probability (MAP) estimator [8]. This method is preferred for weather and ocean forecasting in major operational centers around the globe, particularly in the form of the incremental 4D-Var [7], and in the form of the ensemble 4D-Var [6]. Variational estimation is widely used in other scientific and engineering applications, such as aerospace engineering [2] and astrophysics [4], to mention a few.Uncertainty quantification is an important topic closely associated with estimation and data assimilation. An overview of the original works by the authors on the Hessian-based advanced methodology for computing the estimation error covariances in the framework of variational estimation is presented in the coming sections. The main results are given in a general operator formulation. The accent is made on feasibility of the suggested methods for the uncertainty quantification in high-dimensions, where the statistical methods (Monte Carlo involving associated tricks, e.g., localization, importance sampling, etc.) may not produce a sensible outcome due to a very small sample being available. Some of the approaches could equally be useful for improving (in terms of acceleration, memory savings and robustness) the estimation/DA technique itself.

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“…There are several papers presenting essentially the same result for dynamical models [26,32]. However, there has also been some confusion.…”

Section: Clarification Of the Existing Theorymentioning

confidence: 69%

Section: On the Role Of The Hessian And Its Inversementioning

confidence: 99%

Hessian-based covariance approximations in variational data assimilation

Gejadze

Shutyaev

Dimet

2018

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…Generally, difficulties might be associated with the formulation of the inverse problem that is to be solved and with the numerical approach to its solu-Ž . tion Thacker, 1989 . Difficulties might stem from the model formulation itself.…”

Section: Discussion Of the Sensitivity Analysismentioning

confidence: 99%

“…Near the global minimum, the inverse of the Hessian matrix provides a good approximation of the covariance matrix for the inde-Ž . pendent model parameters Thacker, 1989 . The condition number of the Hessian, which is defined as the ratio of its largest to its smallest eigenvalue, determines the rate of convergence of the minimization algorithm and indicates how singular the problem is.…”

Section: The Hessian Matrixmentioning

confidence: 99%

Testing a marine ecosystem model: sensitivity analysis and parameter optimization

Fennel

Lösch

Schröter

et al. 2001

Journal of Marine Systems

166

119

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A data assimilation technique is used with a simple but widely used marine ecosystem model to optimize poorly known model parameters. A thorough analysis of the a posteriori errors to be expected for the estimated parameters was carried out. The errors have been estimated by calculating the Hessian matrices for different problem formulations based on identical twin experiments. The error analysis revealed inadequacies in the formulation of the optimization problem and insufficiencies of the applied data set. Modifications of the actual problem formulation, which improved the accuracy of the estimated parameters considerably, are discussed.The optimization procedure was applied to real measurements of nitrate and chlorophyll at the Atlantic Bermuda site. The parameter optimization gave poor results. We suggest this to be due to features of the ecosystem that are unresolved by the present model formulation. Our results emphasize the necessity of an error analysis to accompany any parameter optimization study. q

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“…That increases computational efficiency but expands the space of control variables which has to include density values in all nodal points as the parameters to be varied in search for a dynamically constrained minimum of the cost function J. Therefore full list of the varied parameters (control variables) includes (i) density anomaly in all nodal points, (ii) wind stress values in all nodal points on C 1 , The cost function J which basically measures the magnitude of the model/data misfit can be treated as the argument of the Gaussian probability density function on the space of model solutions (Thacker, 1989). We assume that independent types of data are d-correlated, so that J has the following structure: The cost function contains three basic groups of terms.…”

Section: Optimizationmentioning

confidence: 99%

A diagnostic stabilized finite-element ocean circulation model

2003

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A stabilized finite-element (FE) algorithm for the solution of oceanic large scale circulation equations and optimization of the solutions is presented. Pseudo-residual-free bubble function (RFBF) stabilization technique is utilized to enforce robustness of the numerics and override limitations imposed by the Babus ska-Brezzi condition on the choice of functional spaces. The numerical scheme is formulated on an unstructured tetrahedral 3d grid in velocity-pressure variables defined as piecewise linear continuous functions. The model is equipped with a standard variational data assimilation scheme, capable to perform optimization of the solutions with respect to open lateral boundary conditions and external forcing imposed at the ocean surface. We demonstrate the model performance in applications to idealized and realistic basin-scale flows. Using the adjoint method, the code is tested against a synthetic climatological data set for the South Atlantic ocean which includes hydrology, fluxes at the ocean surface and satellite altimetry. The optimized solution proves to be consistent with all these data sets, fitting them within the error bars.The presented diagnostic tool retains the advantages of existing FE ocean circulation models and in addition (1) improves resolution of the bottom boundary layer due to employment of the 3d tetrahedral elements; (2) enforces numerical robustness through utilization of the RFBF stabilization, and (3) provides an opportunity to optimize the solutions by means of 3d variational data assimilation. Numerical efficiency of the code makes this a desirable tool for dynamically constrained analyses of large datasets. Ó 2002 Elsevier Science Ltd. All rights reserved.

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The role of the Hessian matrix in fitting models to measurements

Cited by 203 publications

References 41 publications

Hessian-based covariance approximations in variational data assimilation

Hessian-based covariance approximations in variational data assimilation

Testing a marine ecosystem model: sensitivity analysis and parameter optimization

A diagnostic stabilized finite-element ocean circulation model

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