# EPSRC Symposium Capstone Conference

**Mini-symposium on Model Reduction**

Organiser: Nancy Nichols

Wednesday 1 July 2009

#### Titles and Abstracts:

**Carsten Hartmann** (Frei Universitaet Berlin) *Balanced Model Reduction of Hamiltonian Systems*

We study model reduction of partially-observed linear Hamiltonian systems that are subject to an external perturbation. Large-scale systems of this kind arise in a variety of physical contexts, e.g., in molecular dynamics or structural mechanics. Common spatial decomposition methods such as the Principal Component Analysis aim at identifying a subspace of ``high-energy'' modes onto which the dynamics are projected. These modes, however, may not be relevant for the dynamics. Moreover the methods tacitly assume that all degrees of freedom can actually be observed or measured. Balanced model reduction consists in (1) transforming the system such that those degrees of freedom that are least sensitive to the perturbation also give the least output and (2) neglecting the respective unobservable/uncontrollable modes. The second step is not unique, and, from the perspective of structure-preservation, it matters how the negligible modes are eliminated; for example, projecting the dynamics onto the essential subspace destroys the underlying Hamiltonian structure. We explain how balanced model reduction can be carried out in a structure-preserving fashion, including preservation of stability and passivity.

**Nancy K. Nichols** (University of Reading) *Model Reduction by Balanced Truncation for Unstable, α-Bounded Systems* (Jt authors: C. Boess, A. Bunse-Gerstner, A.S. Lawless)

Model reduction techniques based on balanced-truncation require that the system model is stable; otherwise, the methods are able to reduce only the stable part of the system. For systems that have very large unstable parts, which often arise in the geosciences and in environmental applications, significant reductions in the model size are therefore not possible. Here we describe a new method for model reduction for unstable linear systems that are ‘α-bounded’, that is, systems for which the eigenvalues of the system matrix lie in a disc of radius α around the origin. The new method uses balanced truncation to determine a reduced order model that satisfies the first order necessary conditions for an optimal approximation in the h2,α-norm. A bound on the error between the transfer functions of the original system and the reduced order system obtained by this method is given in the h∞,α-norm. In numerical experiments, the significant unstable and stable modes of the system are captured effectively by the reduced models and the results demonstrate a superior performance using the α-bounded reduction technique in comparison to the standard balanced truncation method for systems with many unstable modes.

**Angelika Bunse-Gerstner** (Universitaet Bremen) *Interpolation-based Model Reduction for Data Assimilation *(Jt authors: C. Boess, N.K. Nichols, D. Kubalinska

Model reduction methods have already shown a potential to improve the computational efficiency of data assimilation schemes. They decrease the computational complexity considerably by generating reduced order linear dynamical systems which behave similarly to the linearisations of the model for the environmental systems used within the Gauss-Newton iteration steps. To make this approach feasible for realistic problems we need model reduction methods that can handle the very large dimensions of the original systems. Numerical methods based on rational tangential interpolation are attractive for these very large dimensions. Moreover, it can be shown that interpolation data can be chosen such that the reduced order dynamical system satisfies first order necessary conditions for an optimal approximation in the h2,α-norm. In this talk we discuss various properties as well as the computation of such reduced order models for very large systems and show experimental results. In particular it is shown how this method is related to an approximate computation of the extremely useful but computationally very expensive balanced truncation model reduction.

**Arnold Heemink** (Delft University of Technology) *Model Reduced Variational Assimilation* (Jt authors: M.U. Altaf, M.P Kaleta)

Data assimilation methods are used to combine the results of a large scale numerical model with the measurement information available in order to obtain an optimal reconstruction of the dynamic behavior of the model state. Variational data assimilation or "the adjoint method" has been used very often for data assimilation. This approach is especially attractive for model calibration problems. Using the available data, the uncertain parameters in the model are identified by minimizing a certain cost function that measures the difference between the model results and the data. In order to obtain a computational efficient procedure, the minimization is performed with a gradient-based algorithm where the gradient is determined by solving the adjoint problem.

Variational data assimilation requires the implementation of the adjoint model. Even with the use of the adjoint compilers that have become available recently this is a tremendous programming effort, that hampers new applications of the method. Therefore we propose another approach to variational data assimilation using model reduction that does not require the implementation of the adjoint of (the tangent linear approximation of) the original model. Model reduced variational data assimilation is based upon a POD (Proper Orthogonal Decomposition) approach to determine a reduced model for the tangent linear approximation of the original nonlinear forward model. Once this reduced model is available, its adjoint can be implemented very easily and the minimization process can be solved completely in reduced space with negligible computational costs. If necessary, the procedure can be repeated a few times by generating new ensembles more close to the most recent estimate of the parameters.

In the presentation we will introduce the model reduced variational data assimilation approach. The characteristics and performance of the method will be illustrated with a number of real life data assimilation applications to oil reservoir models and coastal sea models.

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

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