A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

In this paper we study the approximation of a distributed optimal control problem for linear para\-bolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches

Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach

Modeling of waste water treatment plant via system ID & model reduction technique

This paper investigates the application of Model Order Reduction (MOR) technique to Waste Water Treatment Plant (WWTP) system. The mathematical model of WWTP is obtained by using System Identification. In this paper, Prediction Error Estimate of Linear or Nonlinear Model (PEM) is proposed as the System Identification method which is used to find the parameter of linear or nonlinear system in state-space model from an experimental input­ output data WWTP. The result shows that the estimated model of WWTP is a high order system with good best fit with 91.56% and80.19% compared to the original experimental model. To simplify the obtained model,the MOR technique is proposed to reduce the high order system to lower order system while still retaining the characteristics of the original system. In this paper, the balanced truncation and Frequency Weighted Model Reduction (FWMR) are proposed to obtain a lower order WWTP model. The result shows that by MOR techniques, the higher WWTP system can be simplified to lower order system with a low error of the reduced system. The result of reduced model will be represented in sigma graph and numerical value

Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction

We present a novel technique to perform the model-order reduction (MOR) of multiport systems under the effect of statistical variability of geometrical or electrical parameters. The proposed approach combines a deterministic MOR phase with the use of the Polynomial Chaos (PC) expansion to perform the variability analysis of the system under study very efficiently. The combination of MOR and PC techniques generates a final reduced-order model able to accurately perform stochastic computations and variability analysis. The novel proposed method guarantees a high-degree of flexibility, since different MOR schemes can be used and different types of modern electrical systems (e. g., filters and connectors) can be modeled. The accuracy and efficiency of the proposed approach is verified by means of two numerical examples and compared with other existing variability analysis techniques

Towards Time-Limited H2\mathcal H_2-Optimal Model Order Reduction

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are $\mathcal H_2$-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited $\mathcal H_2$-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited $\mathcal H_2$-norm. Based on these optimality conditions, we propose an iterative scheme, which, upon convergence, aims at satisfying these conditions approximately. Then, we analyze how far away the obtained ROM due to the proposed algorithm is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to the unrestricted IRKA in the finite time interval of interest