This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin.
Project SE1:
Reduced order modeling for data assimilation
Duration: 
June 2014  May 2017 
Project leaders: 
V. Mehrmann, C. Schröder 
Department of Mathematics Technical University of Berlin Strasse des 17. Juni 136 10623 Berlin, Germany 

Tel: +49 (0)30  314 25 736 (office) /  314 21 264  
email: mehrmann(at)math.tuberlin.de  
Tel: +49 (0)30  314 23 439 (office) /  314 25 104  
email: schroed(at)math.tuberlin.de  
Responsible: 
M. Voigt 
Department of Mathematics, Technical University of Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany 

Tel: +49 (0)30  314 24 767  
email: mvoigt(at)math.tuberlin.de  
External cooperations: 
AiF project IGF16799N: Rechnergestützte Verfahren zur Entwicklung geräuscharmer Bremsen (associated project) C. Beattie (Virginia Tech) S. Gugercin (Virginia Tech) T. Reis (U Hamburg) 
Support: 
Einstein Center for Mathematics (ECMath) 
ECMath project website: 
Project SE1 
Background
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 socalled representer approach formulates the data assimilation problem as the numerical solution of a largescale 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.
back to topThe Problems
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.
back to topThe Goal
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 precomputation 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 online computation.
Model order reduction techniques for Hamiltonian initial value problems are also useful in other applications and have recently been studied in many contexts.
back to topHow we do it
We follow a recent idea which derives the optimality condition for the full data assimilation problem as an extremely large nonlinear (locally selfadjoint) BVP. If one can efficiently employ structure preserving model reduction techniques for this BVP that also address the proper incorporation of the boundary values, then one could automatically preserve the essential features of the original data assimilation problem.
We exploit our expertise in the the solution theory of optimal control problems (which has recently been extended to fully general continuous and discretetime descriptor systems), methods for the solution of large scale even eigenvalue problems, and sparse representation techniques for subspaces associated with selfadjoint eigenvalue problems. We collaborate with researchers from FU Berlin, U Bremen, Virginia Tech, U Bath as well as from the UK Met Office and CERFACS Toulouse.
back to topHighlights
In the beginning of the project the focus has been set on the analysis of linearquadratic optimal control problems for differentialalgebraic equations, which is one of the core concepts of the project. In particular, we were able to derive feasibility conditions and the optimal solutions for a large class of such problems. The conclusions drawn from these results are very useful for better understanding the structure of linearquadratic optimal control problems and for developing numerical algorithms. Moreover, a new approach for numerically computing reachable sets of descriptor control systems has been developed. This approach has been extended to second order systems via appropriate projections. The structure of the linearquadratic optimal control problems appears in the data assimilation problem in the form of Hamiltonian boundary value problems. We have developed a balanced truncation like scheme for the reduction of such BVPs and a method for the determination of appropriate boundary values.
back to topPublications
Refereed Articles in Journals

P. Benner, P. Losse, V. Mehrmann, and M. Voigt
Numerical linear algebra methods for linear differentialalgebraic equations
In A. Ilchmann and T. Reis, editors, Surveys in DifferentialAlgebraic Equations III, Differ.Algebr. Equ. Forum, chapter 3, pages 117175, SpringerVerlag, Cham, Switzerland. 
V. Mehrmann and H. Xu
Structure preserving deflation of infinite eigenvalues in structured pencils
Electron. Trans. Numer. Anal., 44:124, 2015. 
V. Mehrmann and C. Schröder
Eigenvalue analysis and model reduction in the treatment of disc brake squeal
SIAM News, Volume 49, Number 1, January/February 2016. 
T. Reis, O. Rendel, and M. Voigt
The KalmanYakubovichPopov inequality for differentialalgebraic systems
Linear Algebra Appl., 485:153193, 2015. 
T. Reis and M. Voigt
The KalmanYakubovichPopov inequality for differentialalgebraic systems: existence of nonpositive solutions
Systems Control Lett., 86(1):18, 2015.
Proceedings

S. Datta and V. Mehrmann
Computation of state reachable points of descriptor systems
In Proc. 53rd IEEE Conference on Decision and Control, pages 63896394, Los Angeles, CA, USA, 2014. 
N. Gräbner, S. Quraishi, C. Schröder, V. Mehrmann, and U. von Wagner
New numerical methods for the complex eigenvalue analysis of disk break squeal
In Proc. EuroBrake 2014, Lille, France, 2014.
Submitted articles

C. Beattie, V. Mehrmann, and H. Xu
PortHamiltonian realizations of linear time invariant systems
Preprint 23/2015, Institut für Mathematik, TU Berlin, Germany, 2015. Available from http://www3.math.tuberlin.de/preprints/files/BeaMX15_ppt.pdf. 
S. Datta and V. Mehrmann
Computation of state reachable points of linear time invariant descriptor systems
Preprint 17/2014, Institut für Mathematik, TU Berlin, Germany, 2014. Available from http://www3.math.tuberlin.de/preprints/files/DatM14_ppt.pdf. 
N. Gräbner, N. Hoffman, V. Mehrmann, S. Quraishi, C. Schröder, M. Tiedemann, and U. von Wagner
Rechnergestützte Verfahren zur Entwicklung geräuscharmer Bremsen : Abschlussbericht zu dem IGFVorhaben
TIBReport 846109786, LeibnizInformationszentrum Technik und Naturwissenschaften, 2015, Available from https://www.tib.eu/de/suchen/id/TIBKAT%3A846109786/Rechnergest%C3%BCtzteVerfahrenzurEntwicklungger%C3%A4uscharmer/. 
N. Gräbner, V. Mehrmann, S. Quraishi, C. Schröder, and U. von Wagner
Numerical methods for parametric model reduction in the simulation of disk brake squeal
Preprint 16/2015, Institut für Mathematik, TU Berlin, Germany, 2014. 
T. Reis and M. Voigt
Innerouter factorization for differentialalgebraic systems
Hamburger Beiträge zur angewandten Mathematik 201531, Fachbereich Mathematik, Universität Hamburg, 2015. Available from http://preprint.math.unihamburg.de/public/papers/hbam/hbam201531.pdf
Theses

D. Bankmann
On LinearQuadratic Control Theory of Implicit Difference Equations
Masterarbeit, Institut für Mathematik, TU Berlin, Germany, November 2015.