Proper orthogonal decomposition closure models for fluid flows: Burgers equation

This paper puts forth several closure models for the proper orthogonal decomposition (POD) reduced order modeling of fluid flows. These new closure models, together with other standard closure models, are investigated in the numerical simulation of the Burgers equation. This simplified setting represents just the first step in the investigation of the new closure models. It allows a thorough assessment of the performance of the new models, including a parameter sensitivity study. Two challenging test problems displaying moving shock waves are chosen in the numerical investigation. The closure models and a standard Galerkin POD reduced order model are benchmarked against the fine resolution numerical simulation. Both numerical accuracy and computational efficiency are used to assess the performance of the models

emgr - The Empirical Gramian Framework

System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework - emgr - implements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction

Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation

We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.Comment: 19 pages, 5 figures, 2 table

Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction