36,459 research outputs found
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
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
Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials
We study the seismic inverse problem for the recovery of subsurface
properties in acoustic media. In order to reduce the ill-posedness of the
problem, the heterogeneous wave speed parameter to be recovered is represented
using a limited number of coefficients associated with a basis of eigenvectors
of a diffusion equation, following the regularization by discretization
approach. We compare several choices for the diffusion coefficient in the
partial differential equations, which are extracted from the field of image
processing. We first investigate their efficiency for image decomposition
(accuracy of the representation with respect to the number of variables and
denoising). Next, we implement the method in the quantitative reconstruction
procedure for seismic imaging, following the Full Waveform Inversion method,
where the difficulty resides in that the basis is defined from an initial model
where none of the actual structures is known. In particular, we demonstrate
that the method is efficient for the challenging reconstruction of media with
salt-domes. We employ the method in two and three-dimensional experiments and
show that the eigenvector representation compensates for the lack of low
frequency information, it eventually serves us to extract guidelines for the
implementation of the method.Comment: 45 pages, 37 figure
Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales
We consider the reconstruction of a diffusion coefficient in a quasilinear
elliptic problem from a single measurement of overspecified Neumann and
Dirichlet data. The uniqueness for this parameter identification problem has
been established by Cannon and we therefore focus on the stable solution in the
presence of data noise. For this, we utilize a reformulation of the inverse
problem as a linear ill-posed operator equation with perturbed data and
operators. We are able to explicitly characterize the mapping properties of the
corresponding operators which allow us to apply regularization in Hilbert
scales. We can then prove convergence and convergence rates of the regularized
reconstructions under very mild assumptions on the exact parameter. These are,
in fact, already needed for the analysis of the forward problem and no
additional source conditions are required. Numerical tests are presented to
illustrate the theoretical statements.Comment: 17 pages, 2 figure
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