Reduced-order modeling for data assimilation

One of the bottlenecks of current procedures for the generation and distribution of green (wind or solar) energy is the accurate and timely simulation of processes in the ocean and atmosphere that can be used in short term planning and real time control of energy systems. A particular difficulty is the real time construction of physically plausible model initializations and 'controls/inputs' to bring simulations into coherence with available observations when observation locations and observations are coming in at variable times and locations.

The currently best approach for fixed observation times and locations are variational data assimilation techniques. These methods use a four dimensional model that is adapted to the incoming observations using a combination of different filtering techniques and numerical integration of the dynamical system. In order to make these methods efficient in real time data assimilation they have to be combined with appropriate model order reduction methods. A major difficulty in these techniques is the combination of approximate transfer functions and approximate initial and boundary conditions as well as the construction of guaranteed error estimates and the capturing of essential features of the original model.

The so-called representer approach formulates the data assimilation problem as the numerical solution of a large-scale nonlinear optimal control problem and incorporates the assimilation of the model to the observations, via an extended ensemble Kalman filter, and the adaptation of the initial data in one approach. Adding further assumptions and linearization this optimization problem usually reduces to a linear quadratic optimal control problem which is solved via the solution of a boundary value problem with Hamiltonian structure.

Currently the solution of the optimal control/data assimilation problem is computationally extremely demanding. Since the solution depends on the measuring times and measured values, which are chosen consecutively, the computations have to be carried out in real time between two measurements. This prohibits any attempts to optimize the measuring times, which would require the solution of the full optimal control problem inside an optimization loop.

Also in the representer approach the solution of the arising BVP is extremely costly. Thus this approach is only practical if it can be combined with appropriate model order reduction techniques. It is, however, a major open problem to achieve structure preserving reduced order models with guaranteed error bounds for nonlinear Hamiltonian boundary value problems (BVP).

The classical model order reduction approaches cannot be used since the optimality system is not stable. Moreover, the usual model order reduction for the forward problem suffers from the fact that the reduced order adjoint equation may be a very bad approximation of the true adjoint equation and that this approach may lead to very large errors, since the boundary conditions are not captured accurately.

In order to deal with these current difficulties we develop new model reduction techniques for the representer approach.

The successful completion of this project will a) enable the pre-computation of a reduced order model, which allows real time computations, b) makes the optimization of the measuring times and positions possible. Moreover, c) the computation of error estimates will make the approach feasible for adaptive on-line computation.

Model order reduction techniques for Hamiltonian initial value problems are also useful in other applications and have recently been studied in many contexts.