Dynamical systems and their numerical simulation are very important for investigating physical and technical problems. The more accuracy is desired, the more equations are needed to reach the desired level of accuracy. This leads to large-scale dynamical systems. The problem is that computations become infeasible due to the limitations on time and/or memory in large-scale settings. Another important issue is numerical ill-conditioning. These are the main reasons for the need of model reduction, i.e. replacing the original system by a reduced system of much smaller dimension. Then one uses the reduced models in order to simulate or control processes. The main goal of this thesis is to investigate an interpolation-based approach to the weighted-H2 model reduction problem. Nonetheless, first we will discuss the regular (unweighted) H2 model reduction problem. We will re-visit the interpolation conditions for H2-optimality, also known as Meier-Luenberger conditions, and discuss how to obtain an optimal reduced order system via projection. After having introduced the H2-norm and the unweighted-H2 model reduction problem, we will introduce the weighted-H2 model reduction problem. We will first derive a new error expression for the weighted-H2 model reduction problem. This error expression illustrates the significance of interpolation at the mirror images of the reduced system poles and the original system poles, as in the unweighted case. However, in the weighted case this expression yields that interpolation at the mirror images of the poles of the weighting system is also significant. Finally, based on the new weighted-H2 error expression, we will propose an iteratively corrected interpolation-based algorithm for the weighted-H2 model reduction problem. Moreover we will present new optimality conditions for the weighted-H2 approximation. These conditions occur as structured orthogonality conditions similar to those for the unweighted case which were derived by Antoulas, Beattie and Gugercin. We present several numerical examples to illustrate the effectiveness of the proposed approach and compare it with the frequency-weighted balanced truncation method. We observe that, for virtually all of our numerical examples, the proposed method outperforms the frequency-weighted balanced truncation method. / Master of Science

Dynamical systems are useful tools for modeling many complex physical phenomena. In many situations, we do not have access to the governing equations to create these models. Instead, we have access to data in the form of input-output measurements. Data-driven approaches use these measurements to construct reduced order models (ROMs), a small scale model that well approximates the true system, directly from input/output data. Frequency domain data-driven methods, which require access to values (and in some cases to derivatives) of the transfer function, have been very successful in constructing high-fidelity ROMs from data. However, at times this frequency domain data can be difficult to obtain or one might have only access to time-domain data. Recently, Burohman et al. [2020] introduced a framework to approximate transfer function values using only time-domain data. We first discuss improvements to this method to allow a more efficient and more robust numerical implementation. Then, we develop an algorithm that performs optimal-H2 approximation using purely time-domain data; thus significantly extending the applicability of H2-optimal approximation without a need for frequency domain sampling. We also investigate how well other established frequency-based ROM techniques (such as the Loewner Framework, Adaptive Anderson-Antoulas Algorithm, and Vector Fitting) perform on this identified data, and compare them to the optimal-H2 model. / Master of Science / Dynamical systems are useful tools for modeling many phenomena found in physics, chemistry, biology, and other fields of science. A dynamical system is a system of ordinary differential equations (ODEs), together with a state to output mapping. These typically result from a spatial discretization of a partial differential equation (PDE). For every dynamical system, there is a corresponding transfer function in the frequency domain that directly links an input to the system with its corresponding output. For some phenomena where the underlying system does not have a known governing PDE, we are forced to use observations of system input-output behavior to construct models of the system. Such models are called data-driven models. If in addition, we seek a model that can well approximate the true system while keeping the number of degrees of freedom low (e.g., for fast simulation of the system or lightweight memory requirements), we refer to the resulting model as a reduced order model (ROM). There are well established ROM methods that assume access to transfer function input-output data, but such data may be costly or impossible to obtain. This thesis expands upon a method introduced by Burohman et al. [2020] to infer values and derivatives of the transfer function using time domain input-output data. The first contribution of this thesis is to provide a robust and efficient implementation for the data informativity framework. We then provide an algorithm for constructing a ROM that is optimal in a frequency domain sense from time domain data. Finally, we investigate how other established frequency domain ROM techniques perform on the learned frequency domain data.