Adjoint equations in variational data assimilation problems

2018 ◽  
Vol 33 (2) ◽  
pp. 137-147 ◽  
Author(s):  
Victor P. Shutyaev
Keyword(s):  
Data Assimilation ◽  
Iterative Algorithms ◽  
Variational Data Assimilation ◽  
Evolution Model ◽  
Adjoint Equations ◽  
Optimality System ◽  
Initial Condition

Abstract The problem of variational data assimilation for an evolution model is considered with the aim to identify the initial condition. The solvability of the optimality system is studied. Based on the adjoint equations, iterative algorithms for solving the problem are developed and justified.

scholarly journals Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation

2018 ◽  
Vol 25 (2) ◽  
pp. 429-439 ◽  
Author(s):  
Victor Shutyaev ◽  
Francois-Xavier Le Dimet ◽  
Eugene Parmuzin
Keyword(s):  
Data Assimilation ◽  
Response Function ◽  
Optimal Solution ◽  
Coupled System ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Observation Data ◽  
Unknown Parameters ◽  
The Baltic ◽  
Nonstandard Problem

Abstract. The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a nonstandard problem involving the coupled system of direct and adjoint equations. The nonstandard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is presented. A numerical example is given for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model.

Sensitivity with respect to observations in variational data assimilation

2017 ◽  
Vol 32 (1) ◽  
Author(s):  
Victor Shutyaev ◽  
Francois-Xavier Le Dimet ◽  
Elena Shubina
Keyword(s):  
Data Assimilation ◽  
Response Function ◽  
Nonlinear Evolution ◽  
Function Analysis ◽  
Optimal Solution ◽  
Coupled System ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Observation Data ◽  
Standard Problem

AbstractThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. The solvability of the non-standard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is developed and justified.

scholarly journals Numerical Modeling of Marine Circulation with 4D Variational Data Assimilation

2020 ◽  
Vol 8 (7) ◽  
pp. 503
Author(s):  
Vladimir Zalesny ◽  
Valeriy Agoshkov ◽  
Victor Shutyaev ◽  
Eugene Parmuzin ◽  
Natalia Zakharova
Keyword(s):  
Data Assimilation ◽  
Finite Difference Methods ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Special Choice ◽  
Modeling And Prediction ◽  
Solving Equations ◽  
The Baltic ◽  
Academy Of Sciences ◽  
Data Assimilation Technique

The technology is presented for modeling and prediction of marine hydrophysical fields based on the 4D variational data assimilation technique developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS). The technology is based on solving equations of marine hydrodynamics using multicomponent splitting, thereby solving an optimality system that includes adjoint equations and covariance matrices of observation errors. The hydrodynamic model is described by primitive equations in the sigma-coordinate system, which is solved by finite-difference methods. The technology includes original algorithms for solving the problems of variational data assimilation using modern iterative processes with a special choice of iterative parameters. The methods and technology are illustrated by the example of solving the problem of circulation of the Baltic Sea with 4D variational data assimilation of sea surface temperature information.

scholarly journals Sensitivity of functionals of variational data assimilation problems

2019 ◽  
Vol 486 (4) ◽  
pp. 421-425
Author(s):  
V. P. Shutyaev ◽  
F.-X. Le Dimet
Keyword(s):  
Data Assimilation ◽  
Observational Data ◽  
Response Function ◽  
Optimal Solution ◽  
Evolutionary Model ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Unknown Parameters ◽  
Initial State ◽  
Nonstandard Problem

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. On the basis of the Hessian of the original cost function, the solvability of the nonstandard problem is studied. An algorithm for calculating the gradient of the response function with respect to observational data is formulated and justified.

scholarly journals Domain Decomposition Method for the Variational Assimilation of the Sea Level in a Model of Open Water Areas Hydrodynamics

2019 ◽  
Vol 7 (6) ◽  
pp. 195 ◽  
Author(s):  
Valery Agoshkov ◽  
Natalia Lezina ◽  
Tatiana Sheloput
Keyword(s):  
Data Assimilation ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Open Water ◽  
Coastal Ocean ◽  
Variational Data Assimilation ◽  
Adjoint Equations ◽  
Open Boundary ◽  
Open Boundary Conditions

One of the modern fields in mathematical modelling of water areas is developing hybrid coastal ocean models based on domain decomposition. In coastal ocean modelling a problem to be solved is setting open boundary conditions. One of the methods dealing with open boundaries is variational data assimilation. The purpose of this work is to apply the domain decomposition method to the variational data assimilation problem. The method to solve the problem of restoring boundary functions at the liquid boundaries for a system of linearized shallow water equations is studied. The problem of determining additional unknowns is considered as an inverse problem and solved using well-known approaches. The methodology based on the theory of optimal control and adjoint equations is used. In the paper the theoretical study of the problem is carried out, unique and dense solvability of the problem is proved, an iterative algorithm is proposed and its convergence is studied. The results of the numerical experiments are presented and discussed.

scholarly journals Methods for assimilation of observational data in problems of the physics of the atmosphere and the ocean

2019 ◽  
Vol 55 (1) ◽  
pp. 17-34
Author(s):  
V. P. Shutyaev
Keyword(s):  
Data Assimilation ◽  
Variational Methods ◽  
Observational Data ◽  
Optimal Solution ◽  
Variational Data Assimilation ◽  
Optimality System ◽  
Weak Formulation ◽  
Variational Assimilation ◽  
Geophysical Hydrodynamics

In this paper we review and analyze approaches to data assimilation in geophysical hydrodynamics problems, starting with the simplest successive schemes of assimilation and ending with modern variational methods. Special attention is paid to the the study of the problem of variational assimilation in the weak formulation and construction of covariance error matrices of the optimal solution. This is a new direction, to which the author made a contribution: an optimality system is formulated for the problem of variational data assimilation in a weak formulation and algorithms for deriving the covariance error matrices of the optimal solution are developed.

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