European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS31: Numerical methods for identification and model reduction of nonlinear systems (Part 1)
Chair: Ion Victor Gosea
25 mins
Numerical aspects of the Koopman and the dynamic mode decompositions
Zlatko Drmac, Igor Mezic, Ryan Mohr
Abstract: The Dynamic Mode Decomposition (DMD, introduced by P. Schmid) has become a tool of trade in computational data driven analysis of complex dynamical systems, e.g. fluid flows, where it can be used to decompose the flow field into component fluid structures, called DMD modes, that describe the evolution of the flow. The DMD is deeply connected with the Koopman spectral analysis of nonlinear dynamical systems, and it can be considered as a computational device in the Koopman analysis framework. Its exceptional performance motivated developments of several modifications that make the DMD an attractive method for analysis and model reduction of nonlinear systems in data driven settings. In this talk, we will present our recent results on the numerical aspects of the DMD/Koopman analysis. We show how the state of the art numerical linear algebra can be deployed to improve the numerical performances in the cases that are usually considered notoriously ill-conditioned. Further, we show how even in the data driven setting, we can work with residual bounds, which allows error estimates for the computed modes. \noindent Our presentation is based on the recent papers \cite{drmac-mezic-mohr-DFT-arxiv-2018}, \cite{DDMD-SISC-2018}, \cite{2018arXiv181112562D}.
25 mins
Model approximation and applications in fluid mechanics
Charles Poussot-Vassal
Abstract: Fluid dynamical equations are involved in multiple applications such as civil engineering, energy generation, or transportation systems. These equations generally take the form of partial differential equations that can be treated as it through dedicated software, or discretized along the spatial directions, leading to a large number of ordinary differential equations. In both cases, computing the solution of these equations can become very complicated when applied to a complex geometry (e.g. bridge, aircraft…). Depending on the application, the resulting numerical models then may become inappropriate for computer-based processing issues. This includes time simulation, performance analysis, parameter and shape optimizations, and active feedback control design. In this talk, we will illustrate how model approximation plays a pivotal role in fluid mechanical applications, in the perspective of an active control design. The talk will be illustrated using different use-cases coming from the fluid mechanics and aeroelasticity domains. We will show how model approximation may greatly enhance the simulation, analysis and the active feedback control design steps.
25 mins
Toward fitting structured nonlinear systems by means of dynamic mode decomposition
Ion Victor Gosea, Igor Pontes Duff
Abstract: With the ever-increasing availability of measured data in different scientific disciplines, the need for incorporating measurements in the identification and reduction process has steadily grown. In recent years, data-driven modeling methods have proven to be widely used in many fields such as fluid dynamics, chemical, electronic and biomedical engineering. The goal of most approaches is to use real-time analysis of measured data to construct models that can accurately identify the underlying dynamics. Methods such as Dynamic Mode Decomposition (DMD) have drawn considerable research endeavors. DMD is a data-driven method used for identifying dynamics of complex nonlinear systems. It extracts important characteristics of the underlying dynamics by means of measured time-domain data produced either by means of experiments or by numerical simulations. DMD aims at analyzing the relationship between pairs of measurements from a dynamical system. In the original methodology, the measurements are assumed to be approximately related by a linear operator. Hence, a linear discrete-time system is fitted to the given data. Often, nonlinear systems modeling physical phenomena have a particular known structure. Because of the spatial discretization, the system's dimension could be very large and hence not suitable for performing tasks such as analysis, simulation or control. Then, model reduction is needed to construct a much smaller system with the same structure and similar response characteristics as the original. We propose an identification and reduction method based on the classical DMD approach that can fit a structured nonlinear system to the measured data. We analyze multiple nonlinearities, that include for example, bilinear or quadratic. By enforcing this additional structure, more insight into extracting the nonlinear behavior of the original process is gained. Moreover, we choose to fit such terms because most systems with analytical nonlinearities (rational, trigonometrical, polynomial, etc.) can be exactly reformulated as quadratic-bilinear systems. We demonstrate the proposed algorithm for different examples, ranging from simple low-dimensional systems such as the chaotic Lorenz attractor to more complex problems in computational fluid dynamics.
25 mins
A time-domain Loewner approach for learning weakly nonlinear dynamical-system models from data
Benjamin Peherstorfer, Serkan Gugercin
Abstract: This work presents a system-theoretic approach for learning nonlinear dynamical-system models from time-domain data. Our approach builds on model reduction with Loewner, which constructs models from frequency-response measurements. We present a time-domain Loewner approach that constructs models directly from trajectories of time-domain inputs and outputs of systems of interest. We exploit the Volterra series representation of dynamical systems to infer higher-order transfer functions from time-domain data for learning linear and weakly nonlinear dynamical-system models. In contrast to traditional statistical methods that fit coefficients (weights) of linear combinations of basis (activation) functions to data, our approach learns models that represent dynamical systems. Thus, systems & control theory concepts such as stability, passivity, controllability, attractors, and eigenmodes are well defined, which is important for applications in science and engineering. Benchmark examples demonstrate that our time-domain Loewner approach learns weakly nonlinear dynamics faithfully and that the learned models generalize well beyond the training data.