A stochastic Levenberg-Marquardt method using random models with complexity results and application to data assimilation

Globally convergent variants of the Gauss-Newton algorithm are often the methods of choice to tackle nonlinear least-squares problems. Among such frameworks, Levenberg-Marquardt and trust-region methods are two well-established, similar paradigms. Both schemes have been studied when the Gauss-Newton model is replaced by a random model that is only accurate with a given probability. Trust-region schemes … Read more

A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation

The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed, showing the interplay of the eigenstructures of both the model and weighting matrices. A small example is given illustrating the resulting potential inefficiency of such preconditioners. Consequences of these results in the context of the weakly-constrained 4D-Var data assimilation … Read more

On the use of the saddle formulation in weakly-constrained 4D-VAR data assimilation

This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationaly coherent variants of the algorithm … Read more

Adaptive Observations And Multilevel Optimization In Data Assimilation

We propose to use a decomposition of large-scale incremental four dimensional (4D-Var) data assimilation problems in order to make their numerical solution more efficient. This decomposition is based on exploiting an adaptive hierarchy of the observations. Starting with a low-cardinality set and the solution of its corresponding optimization problem, observations are adaptively added based on … Read more

A framework for automated PDE-constrained optimisation

A generic framework for the solution of PDE-constrained optimisation problems based on the FEniCS system is presented. Its main features are an intuitive mathematical interface, a high degree of automation, and an efficient implementation of the generated adjoint model. The framework is based upon the extension of a domain-specific language for variational problems to cleanly … Read more

A Low-Memory Approach For Best-State Estimation Of Hidden Markov Models With Model Error

We present a low-memory approach for the best-state estimate (data assimilation) of hidden Markov models where model error is considered. In particular, our findings apply for the 4D- Var framework. The novelty of our approach resides in the fact that the storage needed by our estimation framework, while including model error, is dramatically reduced from … Read more

Preconditioning and Globalizing Conjugate Gradients in Dual Space for Quadratically Penalized Nonlinear-Least Squares Problems

When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each … Read more

Covariance regularization in inverse space

This paper proposes to apply Gaussian graphical models to estimate the large-scale normal distribution in the context of data assimilation from a relatively small number of data from the satellite. Data assimilation is a field which fits simulation models to observation data developed mainly in meteorology and oceanography. The optimization problem tends to be huge … Read more