V. Shutyaev

On Analysis Error Covariances in Variational Data Assimilation

The 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 equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented.

On optimal solution error covariances in variational data assimilation problems

Abstract The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection–diffusion model.

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

Abstract The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.

On Newton methods in data assimilation

In this paper we consider the Newton method both for solving the optimality system and for minimising the misfit functional in a data assimilation problem. The convergence of the algorithms is studied. 1. STATEMENT OF THE PROBLEM Consider the evolution problem , *€ (0 ,T) (11) .-o = « where φ = <p(t) is the unknown function belonging to a Hubert space X for any t, u e X, F is a nonlinear operator mapping X into X. Let Υ L2(Q,T;X), (·,·)^(ΟΓ·Λ) = (·,·), 1 1 1 1 =(v)· Let us introduce the functional 1 T S(u) = |||t, Udbsx + -J 0φ φ*Βχύί (1.2) where α = const > 0, u^ G Χ, φ&>5 £ Y<*>s are prescribed functions (observational data), YObs is a subspace of y, and C : y — > y0ts is a linear operator. Consider the following data assimilation problem with the aim to identify the initial condition: find u and φ such that , *e(0,T) Vlt-o = u 0-3) 5(«) = inf 5(v). The necessary optimality condition reduces the problem (1.3) to the system <6(0 ,Γ) = u e (Ο,τ) a(«-uebi)-^| t.0=0 (1.6) with the unknowns φ, φ, and u, where (Ρ(φ))* is the adjoint to the Frechet derivative of F, and C* is the adjoint to C. •LMC-IMAG Joseph Fourier University, BP53, 38041 Grenoble cedex 9, France Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 117951, GSP-1, Russia 420 F.-X. Le Dimet and V.P. Shutyaev The existence and uniqueness of the solution to systems of the form (1.4)-(1.6) have been studied by many authors (see, for example, [1, 3, 8, 18]). Having supposed that the solution of the problem (1.4)-(1.6) exists, we will consider the Newton method to solve it. 2. THE NEWTON METHOD The system (1.4)-(1.6) with the three unknowns φ, φ*, u may be treated as an operator equation of the form F(U) = 0 (2.1) where U = (φ, φ*, u). To implement the Newton method it is necessary to calculate F(U). We suppose that the original operator F is twice continuously Frechet differentiable. Then the Newton algorithm i/n + l = £/n (F(Un)}-f(Unl Un = (?»,¥>>η) consists of the following steps. 1. Find Vn = [^(i/n)]^!/») as the solution to the problem F(Un)Vn = With Vn = (^n,#>n): (23) V>n|i=0 = ^n + ^n| i=0~n v^nUo(·) 2. Set i/n+i = i/n l/n, that is, ψη + l = ψη~ -0η, ^ή + 1 = Ψη ~ ^ή, n^l = tin ~ t>n· (2.6) Since t/n+1 = Un Vn, the two steps (2.3)-(2.5) may be formulated as follows: for given φη,φ*η,υ,η find <J0n+1,y?;+1,un+i such that

On error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The equation for the error of the optimal solution (analysis) is derived through the statistical errors of the input data (background and observation errors). The numerical algorithm is developed to construct the covariance operator of the analysis error using the covariance operators of the input errors. Numerical examples are presented.

A numerical algorithm of variational data assimilation for reconstruction of salinity fluxes on the ocean surface

Abstract A problem of variational data assimilation concerning salinity on the ocean surface is formulated and studied. A solution algorithm for this problem is developed. The results of numerical experiments are presented.

On error analysis in data assimilation problems

Abstract - We consider the data assimilation problem for a nonlinear evolution model to identify the initial condition. We derive an equation for the error of the optimal solution through the errors of the input data, which is based on the Hessian of a misfit functional, and study the solvability of the error equation. Fundamental control functions are used for error analysis. We obtain error sensitivity coefficients, using the singular vectors of the specific response operators in the error equation. We show the application of the data assimilation problem in hydrology and give numerical results.

On data assimilation for quasilinear parabolic problems

Abstract We consider the problem of data assimilation for a quasilinear parabolic equation to identify the initial condition. We study the properties of the nonlinear operator of the problem and prove the solvability of the data assimilation problem in a class of functional spaces. The Newton algorithm for the optimality system is given.

On Computation of the Design Function Gradient for the Sensor-Location Problem in Variational Data Assimilation

The optimal sensor-location problem is considered in the framework of variational data assimilation for a large-scale dynamical model governed by partial differential equations. This problem is formulated as an optimization problem for the design function defined on the limited-memory approximation of the inverse Hessian of the data assimilation cost function. The expression for the gradient of the design function with respect to the sensor-location coordinates is derived via the adjoint to the Hessian derivative. An efficient algorithm for the gradient evaluation suitable for large-scale applications is suggested. This algorithm exploits the special structure of the limited-memory inverse Hessian defined by a small number of Ritz pairs obtained by the Lanczos method. If additional memory is allocated and certain data are stored during the computation of the Ritz pairs, no additional runs of the tangent linear model are required to evaluate the gradient. The accuracy of the gradients is checked in the numerical experiments. These gradients can be used for the gradient-based optimization of the design function within the chosen global optimization procedure.

Adjoint to the Hessian derivative and error covariances in variational data assimilation

Abstract The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.