scholarly journals Quasi static ensemble variational data assimilation

2017 ◽  
Author(s):  
Anthony Fillion ◽  
Marc Bocquet ◽  
Serge Gratton
Keyword(s):  
Data Assimilation ◽  
Cost Function ◽  
Global Minimum ◽  
Positive Impact ◽  
Computational Cost ◽  
Variational Data Assimilation ◽  
Local Extrema ◽  
Starting Point ◽  
The Cost ◽  
Temporal Extent

Abstract. The analysis in nonlinear variational data assimilation is the solution of a non-quadratic minimization. Thus, the analysis efficiency relies on its ability to locate a global minimum of the cost function. If this minimization uses a Gauss-Newton (GN) method, it is critical for the starting point to be in the attraction basin of a global minimum. Otherwise the method may converge to a local extremum, which degrades the analysis. With chaotic models, the number of local extrema often increases with the temporal extent of the data assimilation window, making the former condition harder to satisfy. This is unfortunate because the assimilation performance also increases with this temporal extent. However, a quasi-static (QS) minimization may overcome these local extrema. It consists in gradually injecting the observations in the cost function. This method was introduced by Pires et al. (1996) in a 4D-Var context. We generalize this approach to four-dimensional nonlinear EnVar methods, which are based on both a nonlinear variational analysis and the propagation of dynamical error statistics via an ensemble. This forces to consider the cost function minimizations in the broader context of cycled data assimilation algorithms. We adapt this QS approach to the iterative ensemble Kalman smoother (IEnKS), an exemplar of nonlinear deterministic 4D EnVar methods. Using low-order models, we quantify the positive impact of the QS approach on the IEnKS, especially for long data assimilation windows. We also examine the computational cost of QS implementations and suggest cheaper algorithms.

scholarly journals Quasi-static ensemble variational data assimilation: a theoretical and numerical study with the iterative ensemble Kalman smoother

2018 ◽  
Vol 25 (2) ◽  
pp. 315-334 ◽  
Author(s):  
Anthony Fillion ◽  
Marc Bocquet ◽  
Serge Gratton
Keyword(s):  
Data Assimilation ◽  
Cost Function ◽  
Global Minimum ◽  
Numerical Study ◽  
Variational Data Assimilation ◽  
Kalman Smoother ◽  
Local Extrema ◽  
Starting Point ◽  
The Cost ◽  
Temporal Extent

Abstract. The analysis in nonlinear variational data assimilation is the solution of a non-quadratic minimization. Thus, the analysis efficiency relies on its ability to locate a global minimum of the cost function. If this minimization uses a Gauss–Newton (GN) method, it is critical for the starting point to be in the attraction basin of a global minimum. Otherwise the method may converge to a local extremum, which degrades the analysis. With chaotic models, the number of local extrema often increases with the temporal extent of the data assimilation window, making the former condition harder to satisfy. This is unfortunate because the assimilation performance also increases with this temporal extent. However, a quasi-static (QS) minimization may overcome these local extrema. It accomplishes this by gradually injecting the observations in the cost function. This method was introduced by Pires et al. (1996) in a 4D-Var context. We generalize this approach to four-dimensional strong-constraint nonlinear ensemble variational (EnVar) methods, which are based on both a nonlinear variational analysis and the propagation of dynamical error statistics via an ensemble. This forces one to consider the cost function minimizations in the broader context of cycled data assimilation algorithms. We adapt this QS approach to the iterative ensemble Kalman smoother (IEnKS), an exemplar of nonlinear deterministic four-dimensional EnVar methods. Using low-order models, we quantify the positive impact of the QS approach on the IEnKS, especially for long data assimilation windows. We also examine the computational cost of QS implementations and suggest cheaper algorithms.

scholarly journals An Overlooked Issue of Variational Data Assimilation

2015 ◽  
Vol 143 (10) ◽  
pp. 3925-3930 ◽  
Author(s):  
Benjamin Ménétrier ◽  
Thomas Auligné
Keyword(s):  
Data Assimilation ◽  
Cost Function ◽  
Control Variable ◽  
Variational Data Assimilation ◽  
Inner Product ◽  
Background Error ◽  
Error Covariance ◽  
Definition Of ◽  
Linear Unbiased Estimate ◽  
The Cost

Abstract The control variable transform (CVT) is a keystone of variational data assimilation. In publications using such a technique, the background term of the transformed cost function is defined as a canonical inner product of the transformed control variable with itself. However, it is shown in this paper that this practical definition of the cost function is not correct if the CVT uses a square root of the background error covariance matrix that is not square. Fortunately, it is then shown that there is a manifold of the control space for which this flaw has no impact, and that most minimizers used in practice precisely work in this manifold. It is also shown that both correct and practical transformed cost functions have the same minimum. This explains more rigorously why the CVT is working in practice. The case of a singular is finally detailed, showing that the practical cost function still reaches the best linear unbiased estimate (BLUE).

scholarly journals Ensemble data assimilation of total column ozone using a coupled meteorology–chemistry model and its impact on the structure of Typhoon Nabi (2005)

2015 ◽  
Vol 15 (17) ◽  
pp. 10019-10031 ◽  
Author(s):  
S. Lim ◽  
S. K. Park ◽  
M. Zupanski
Keyword(s):  
Data Assimilation ◽  
Cost Function ◽  
Positive Impact ◽  
Eastern China ◽  
Chemical Information ◽  
Elevated Concentration ◽  
Total Column Ozone ◽  
Rms Error ◽  
The Cost ◽  
The Impact

Abstract. Ozone (O3) plays an important role in chemical reactions and is usually incorporated in chemical data assimilation (DA). In tropical cyclones (TCs), O3 usually shows a lower concentration inside the eyewall and an elevated concentration around the eye, impacting meteorological as well as chemical variables. To identify the impact of O3 observations on TC structure, including meteorological and chemical information, we developed a coupled meteorology–chemistry DA system by employing the Weather Research and Forecasting model coupled with Chemistry (WRF-Chem) and an ensemble-based DA algorithm – the maximum likelihood ensemble filter (MLEF). For a TC case that occurred over East Asia, Typhoon Nabi (2005), our results indicate that the ensemble forecast is reasonable, accompanied with larger background state uncertainty over the TC, and also over eastern China. Similarly, the assimilation of O3 observations impacts meteorological and chemical variables near the TC and over eastern China. The strongest impact on air quality in the lower troposphere was over China, likely due to the pollution advection. In the vicinity of the TC, however, the strongest impact on chemical variables adjustment was at higher levels. The impact on meteorological variables was similar in both over China and near the TC. The analysis results are verified using several measures that include the cost function, root mean square (RMS) error with respect to observations, and degrees of freedom for signal (DFS). All measures indicate a positive impact of DA on the analysis – the cost function and RMS error have decreased by 16.9 and 8.87 %, respectively. In particular, the DFS indicates a strong positive impact of observations in the TC area, with a weaker maximum over northeastern China.

scholarly journals Impact of ozone observations on the structure of a tropical cyclone using coupled atmosphere–chemistry data assimilation

2015 ◽  
Vol 15 (8) ◽  
pp. 11573-11597
Author(s):  
S. Lim ◽  
S. K. Park ◽  
M. Zupanski
Keyword(s):  
Air Quality ◽  
Data Assimilation ◽  
Cost Function ◽  
Positive Impact ◽  
Eastern China ◽  
Chemical Information ◽  
Chemistry Data ◽  
Air Quality Forecast ◽  
The Cost ◽  
The Impact

Abstract. Since the air quality forecast is related to both chemistry and meteorology, the coupled atmosphere–chemistry data assimilation (DA) system is essential to air quality forecasting. Ozone (O3) plays an important role in chemical reactions and is usually assimilated in chemical DA. In tropical cyclones (TCs), O3 usually shows a lower concentration inside the eyewall and an elevated concentration around the eye, impacting atmospheric as well as chemical variables. To identify the impact of O3 observations on TC structure, including atmospheric and chemical information, we employed the Weather Research and Forecasting model coupled with Chemistry (WRF-Chem) with an ensemble-based DA algorithm – the maximum likelihood ensemble filter (MLEF). For a TC case that occurred over the East Asia, our results indicate that the ensemble forecast is reasonable, accompanied with larger background state uncertainty over the TC, and also over eastern China. Similarly, the assimilation of O3 observations impacts atmospheric and chemical variables near the TC and over eastern China. The strongest impact on air quality in the lower troposphere was over China, likely due to the pollution advection. In the vicinity of the TC, however, the strongest impact on chemical variables adjustment was at higher levels. The impact on atmospheric variables was similar in both over China and near the TC. The analysis results are validated using several measures that include the cost function, root-mean-squared error with respect to observations, and degrees of freedom for signal (DFS). All measures indicate a positive impact of DA on the analysis – the cost function and root mean square error have decreased by 16.9 and 8.87%, respectively. In particular, the DFS indicates a strong positive impact of observations in the TC area, with a weaker maximum over northeast China.

scholarly journals An Adaptive Optimization Method Based on Learning Rate Schedule for Neural Networks

2021 ◽  
Vol 11 (2) ◽  
pp. 850
Author(s):  
Dokkyun Yi ◽  
Sangmin Ji ◽  
Jieun Park
Keyword(s):  
Artificial Intelligence ◽  
Cost Function ◽  
Numerical Experiments ◽  
Global Minimum ◽  
Optimization Method ◽  
Learning Method ◽  
Adaptive Optimization ◽  
The Cost ◽  
Proof Of Convergence ◽  
Learning Data

Artificial intelligence (AI) is achieved by optimizing the cost function constructed from learning data. Changing the parameters in the cost function is an AI learning process (or AI learning for convenience). If AI learning is well performed, then the value of the cost function is the global minimum. In order to obtain the well-learned AI learning, the parameter should be no change in the value of the cost function at the global minimum. One useful optimization method is the momentum method; however, the momentum method has difficulty stopping the parameter when the value of the cost function satisfies the global minimum (non-stop problem). The proposed method is based on the momentum method. In order to solve the non-stop problem of the momentum method, we use the value of the cost function to our method. Therefore, as the learning method processes, the mechanism in our method reduces the amount of change in the parameter by the effect of the value of the cost function. We verified the method through proof of convergence and numerical experiments with existing methods to ensure that the learning works well.

Maximum Likelihood Ensemble Filter: Theoretical Aspects

2005 ◽  
Vol 133 (6) ◽  
pp. 1710-1726 ◽  
Author(s):  
Milija Zupanski
Keyword(s):  
Maximum Likelihood ◽  
Data Assimilation ◽  
Cost Function ◽  
Ensemble Data Assimilation ◽  
Error Covariance ◽  
Ensemble Data ◽  
Analysis Error ◽  
Nonlinear Observation ◽  
The Cost ◽  
Maximum Likelihood Ensemble Filter

Abstract A new ensemble-based data assimilation method, named the maximum likelihood ensemble filter (MLEF), is presented. The analysis solution maximizes the likelihood of the posterior probability distribution, obtained by minimization of a cost function that depends on a general nonlinear observation operator. The MLEF belongs to the class of deterministic ensemble filters, since no perturbed observations are employed. As in variational and ensemble data assimilation methods, the cost function is derived using a Gaussian probability density function framework. Like other ensemble data assimilation algorithms, the MLEF produces an estimate of the analysis uncertainty (e.g., analysis error covariance). In addition to the common use of ensembles in calculation of the forecast error covariance, the ensembles in MLEF are exploited to efficiently calculate the Hessian preconditioning and the gradient of the cost function. A sufficient number of iterative minimization steps is 2–3, because of superior Hessian preconditioning. The MLEF method is well suited for use with highly nonlinear observation operators, for a small additional computational cost of minimization. The consistent treatment of nonlinear observation operators through optimization is an advantage of the MLEF over other ensemble data assimilation algorithms. The cost of MLEF is comparable to the cost of existing ensemble Kalman filter algorithms. The method is directly applicable to most complex forecast models and observation operators. In this paper, the MLEF method is applied to data assimilation with the one-dimensional Korteweg–de Vries–Burgers equation. The tested observation operator is quadratic, in order to make the assimilation problem more challenging. The results illustrate the stability of the MLEF performance, as well as the benefit of the cost function minimization. The improvement is noted in terms of the rms error, as well as the analysis error covariance. The statistics of innovation vectors (observation minus forecast) also indicate a stable performance of the MLEF algorithm. Additional experiments suggest the amplified benefit of targeted observations in ensemble data assimilation.

scholarly journals Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization

2020 ◽  
Vol 30 (6) ◽  
pp. 1645-1663
Author(s):  
Ömer Deniz Akyildiz ◽  
Dan Crisan ◽  
Joaquín Míguez
Keyword(s):  
Monte Carlo ◽  
Cost Function ◽  
Global Minimum ◽  
Sequential Monte Carlo ◽  
Convergence Rates ◽  
Optimization Problems ◽  
Search Space ◽  
Gradient Based ◽  
Multiple Minima ◽  
The Cost

Abstract We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.

Cycling the Representer Algorithm for Variational Data Assimilation with the Lorenz Attractor

2007 ◽  
Vol 135 (2) ◽  
pp. 373-386 ◽  
Author(s):  
H. E. Ngodock ◽  
S. R. Smith ◽  
G. A. Jacobs
Keyword(s):  
Data Assimilation ◽  
Variational Data Assimilation ◽  
Lorenz Attractor ◽  
Strong Constraint ◽  
Strongly Nonlinear ◽  
Time Period ◽  
Representer Method ◽  
The Cost ◽  
Solution Accuracy ◽  
Perturbation Magnitude

Abstract Realistic dynamic systems are often strongly nonlinear, particularly those for the ocean and atmosphere. Applying variational data assimilation to these systems requires a tangent linearization of the nonlinear dynamics about a background state for the cost function minimization. The tangent linearization may be accurate for limited time scales. Here it is proposed that linearized assimilation systems may be accurate if the assimilation time period is less than the tangent linear accuracy time limit. In this paper, the cycling representer method is used to test this assumption with the Lorenz attractor. The outer loops usually required to accommodate the linear assimilation for a nonlinear problem may be dropped beyond the early cycles once the solution (and forecast used as the background in the tangent linearization) is sufficiently accurate. The combination of cycling the representer method and limiting the number of outer loops significantly lowers the cost of the overall assimilation problem. In addition, this study shows that weak constraint assimilation corrects tangent linear model inaccuracies and allows extension of the limited assimilation period. Hence, the weak constraint outperforms the strong constraint method. Assimilated solution accuracy at the first cycle end is computed as a function of the initial condition error, model parameter perturbation magnitude, and outer loops. Results indicate that at least five outer loops are needed to achieve solution accuracy in the first cycle for the selected error range. In addition, this study clearly shows that one outer loop in the first cycle does not preclude accuracy convergence in future cycles.

scholarly journals Optimal solution error covariance in highly nonlinear problems of variational data assimilation

2012 ◽  
Vol 19 (2) ◽  
pp. 177-184 ◽  
Author(s):  
V. Shutyaev ◽  
I. Gejadze ◽  
G. J. M. Copeland ◽  
F.-X. Le Dimet
Keyword(s):  
Nonlinear Dynamics ◽  
Data Assimilation ◽  
Optimal Solution ◽  
Nonlinear Problems ◽  
Variational Data Assimilation ◽  
Model Parameters ◽  
Efficient Computation ◽  
Viscous Term ◽  
Highly Nonlinear ◽  
The Cost

Abstract. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost function. For problems with strongly nonlinear dynamics, a new statistical method based on the computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. Numerical examples are presented for the model governed by the Burgers equation with a nonlinear viscous term.

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