Interpolation Based Parametric Model Order Reduction

In this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time

Symplectic Model Reduction of Hamiltonian Systems

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure

Passivity-preserving parameterized model order reduction using singular values and matrix interpolation

We present a parameterized model order reduction method based on singular values and matrix interpolation. First, a fast technique using grammians is utilized to estimate the reduced order, and then common projection matrices are used to build parameterized reduced order models (ROMs). The design space is divided into cells, and a Krylov subspace is computed for each cell vertex model. The truncation of the singular values of the merged Krylov subspaces from the models located at the vertices of each cell yields a common projection matrix per design space cell. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parameterized ROM. Pertinent numerical results validate the proposed technique