Second order adjoints for solving PDE-constrained optimization problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods. In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems

Low rank updates in preconditioning the saddle point systems arising from data assimilation problems

The numerical solution of saddle point systems has received a lot of attention over the past few years in a wide variety of applications such as constrained optimization, computational fluid dynamics and optimal control, to name a few. In this paper, we focus on the saddle point formulation of a large-scale variational data assimilation problem, where the computations involving the constraint blocks are supposed to be much more expensive than those related to the (1, 1) block of the saddle point matrix. New low-rank limited memory preconditioners exploiting the particular structure of the problem are proposed and analysed theoretically. Numerical experiments performed within the Object-Oriented Prediction System are presented to highlight the relevance of the proposed preconditioners

Discrete Second Order Adjoints in Atmospheric Chemical Transport Modeling

Atmospheric chemical transport models (CTMs) are essential tools for the study of air pollution, for environmental policy decisions, for the interpretation of observational data, and for producing air quality forecasts. Many air quality studies require sensitivity analyses, i.e., the computation of derivatives of the model output with respect to model parameters. The derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through adjoint sensitivity analysis. While the traditional (first order) adjoint models give the gradient of the cost functional with respect to parameters, second order adjoint models give second derivative information in the form of products between the Hessian of the cost functional and a user defined vector. In this paper we discuss the mathematical foundations of the discrete second order adjoint sensitivity method and present a complete set of computational tools for performing second order sensitivity studies in three-dimensional atmospheric CTMs. The tools include discrete second order adjoints of Runge Kutta and of Rosenbrock time stepping methods for stiff equations together with efficient implementation strategies. Numerical examples illustrate the use of these computational tools in important applications like sensitivity analysis, optimization, uncertainty quantification, and the calculation of directions of maximal error growth in three-dimensional atmospheric CTMs

Variational inverse methods for transport problems

Variational inverse data assimilation schemes are developed for three types of parameter identification problems in transport models: (1) the tracer inverse for the Lagrangian mean transport velocity in a long-term advection-diffusion transport model; (2) determination of inflow salinity open boundary condition in an intra-tidal salinity transport model; and (3) determination of settling velocity and resuspension rate for a cohesive sediment transport model. The gradient of the cost function with respect to the control variables is obtained by the adjoint model. A series of twin experiments are conducted to test the inverse models for the three types of problems. Results show that variational data assimilation can successfully retrieve poorly known parameters in transport models. The first problem is associated with the long-term advective transport, represented by the Lagrangian mean transport velocity which can be decomposed into two parts: the Eulerian transport velocity and the curl of a 3-D vector potential A. The optimal long-term advective transport field is obtained through adjusting the vector potential using a variational data assimilation method. Experiments are performed in an idealized estuary. Results show that the variational data assimilation method can successfully retrieve the effective Lagrangian mean transport velocity in a long-term transport model. Results also show that the smooth best fit model state can still be retrieved using a penalty method when observations are too sparse or contain noisy signals. A variational inverse model for optimally determining open boundary condition is developed and tested in a 3-D intra-tidal salinity transport model. The maximum inflow salinity open boundary value and its recovery time from outflow condition are treated as control variables. Effects of scaling, preconditioning, and penalty are investigated. It is shown that proper scaling and preconditioning can greatly speed up the convergence rate of the minimization process. The spatial oscillations in the recovery time of the inflow boundary condition can be effectively eliminated by an penalty technique. A variational inverse model is developed to estimate the settling velocity and resuspension constant. The settling velocity &w\sb{lcub}s{rcub}& and resuspension constant &M\sb{lcub}o{rcub}& are assumed to be constant in the whole model domain. The inverse model is tested in an idealized 3-D estuary and the James River, a tributary of the Chesapeake Bay. Experimental results demonstrate that the variational inverse model can be used to identify the poorly known parameters in cohesive sediment transport modeling

A Feasibility Study of Dynamical Assimilation of Tide Gauge Data in the Chesapeake Bay

The feasibility of dynamical assimilation of surface elevation from tide gauges is investigated to estimate the bottom drag coefficient and surface stress as a first step in improving modeled tidal and wind-driven circulation in the Chesapeake Bay. A two-dimensional shallow water model and an adjoint variational method with a limited memory quasi-Newton optimization algorithm are used to achieve this goal. Assimilation of tide gauge observations from ten permanent stations in the Bay and use of a two-dimensional model adequately estimate the bottom drag coefficient, wind stress and surface elevation at the Bay mouth. Subsequent use of these estimates in the circulation model considerably improves the modeled surface elevation in the entire Bay. Assimilation of predicted tidal elevations yields a drag coefficient, defined in the hydraulic way, varying between 2.5 x 10-4 and 3.1 x 10-3. The bottom drag coefficient displays a periodicity corresponding to the spring-neap tide cycle. From assimilation of actual tide gauge observations, it is found that the fortnightly modulation is altered during frontal passage. Furthermore, the response of the sea surface to the wind forcing is found to be more important in the lower Bay than in the upper Bay, where the barometric pressure effect could be more important. In addition, identical twin experiments with model generated data show that a penalty term has to be added to the simple cost function defined as the distance between modeled and observed surface elevation in order to assure smoothness of the surface elevation field at the Bay mouth. Classical scaling of the parameters to bring them to the same order of magnitude was not effective in accelerating the convergence during the assimilation procedure and yielded larger errors in the estimated parameters