European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   MS31: Numerical methods for identification and model reduction of nonlinear systems (Part 2)
Chair: Ion Victor Gosea
13:30
25 mins
Lifting transformations and model reduction
Boris Kramer, Karen Willcox
Abstract: Lifting transformations are helpful tools to covert general nonlinear systems into structured models of polynomial forms via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model: it uses a change of variables, but introduces no approximations. We present several avenues where lifting transformations are an integral part of a model reduction strategy. We first apply proper orthogonal decomposition (POD) to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed and no hyper-reduction of the nonlinear term is needed. We also show further results on how lifting enabled balanced model reduction for a strongly nonlinear tubular reactor model.
13:55
25 mins
Interpolation-based model order reduction for parametric quadratic-bilinear systems
Xingang Cao, Joseph Maubach, Siep Weiland, Wil Schilders
Abstract: Recent years, model order reduction for weakly nonlinear systems such as bilinear and quadratic-bilinear dynamical systems has attracted lots of attention (see, e.g., [1, 4, 2, 3]). By applying Carleman linearization, many nonlinear systems can be approximated by bilinear dynamical systems. And the lifting method [1] allows one to represent many nonlinear dynamical systems by quadratic-bilinear systems. In this talk, we move one step further. Namely, we focus on parametric quadratic-bilinear systems. To reduce the state space dimension of the system, Volterra series interpolation method is considered. Then by picking suitable frequency and parameter interpolation points, Krylov subspaces can be constructed to project the original system onto a smaller dimensional space. When fixing the parameter sample points, the proposed framework allows us to interpolate the Volterra series at prescribed frequency points. When the frequency interpolation points are fixed, the reduced-order model can even match the parameter sensitivity at prescribed parameter interpolation points. While by adding more basis into the Krylov subspaces, we can even match the second-order parameter sensitivity at the sampled parameter points, which is crucial in many applications such as design optimization. To select the frequency interpolation points, the truncated H2 optimal approximation method is applied. Meanwhile, the truncated H2 norm can be considered as a cost function over the parameter domain. By minimizing this cost function, one can automatically select both the frequency and the parameter interpolation points in an iterative manner. A parametric advection-diffusion-reaction equation is tested to demonstrate the effectiveness of the proposed method.
14:20
25 mins
Driven Discovery and Control of Complex Systems: Uncovering Interpretable and Generalizable Nonlinear Models
Steven Brunton, Nathan Kutz, Jean-Christophe Loiseau, Bernd Noack
Abstract: Accurate and efficient reduced-order models are essential to understand, predict, estimate, and control complex, multiscale, and nonlinear dynamical systems. These models should ideally be generalizable, interpretable, and based on limited training data. This work develops a general framework to discover the governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity-promoting techniques and machine learning. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. This perspective, combining dynamical systems with machine learning and sparse sensing, is explored with the overarching goal of real-time closed-loop feedback control. First, we will discuss how it is possible to enforce known constraints, such as energy conserving quadratic nonlinearities in incompressible fluids, to “bake in” known physics. Next, we will demonstrate that higher-order nonlinearities can approximate the effect of truncated modes, resulting in more accurate models of lower order than Galerkin projection. Finally, we will discuss how to design sparse sensors and design models based on intrinsic measurements of the system.
14:45
25 mins
Interpolatory methods for H-infinity norm computation and H-infinity control
Nicat Aliyev, Peter Benner, Emre Mengi, Paul Schwerdtner, Matthias Voigt
Abstract: In this talk we are concerned with the computation of the H-infinity norm of large-scale dynamical systems with possibly irrational transfer functions. We discuss a subspace projection approach for solving this problem using interpolatory techniques that are well-known in model reduction. More precisely, after performing the reduction, we compute the H-infinity norm of the reduced transfer function and choose the point at which the H-infinity norm is attained as a new interpolation point to update the projection matrices. We will discuss convergence properties of this procedure and illustrate it by various examples. One focus of this talk will be on delay systems which are reduced by employing the Loewner framework, This is useful in order to get a sequence of linear reduced models whose H-infinity norm can be evaluated more efficiently. Finally, we also show how this method can be extended to the minimization of the H-infinity norm of parameter-dependent problems and how it can be applied to the design of optimal fixed-order H-infinity controllers.