Numerical Initial Value Problems in Ordinary Differential Equations

Linearized reduced-order models for subsurface flow simulation

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

A model reduction technique based on the proper orthogonal decomposition is applied to a finite-element formulation of the electroquasistatic field equation. The presented formulation takes into account dielectric effects as well as nonlinear conductive effects. The implementation is based on a 2-D axial symmetric model. The applied proper orthogonal decomposition is based on the singular value decomposition. Numerical results are shown for an axial-symmetric high-voltage surge arrester consisting of constant and nonlinear conductive material.

Low-Order Electroquasistatic Field Simulations Based on Proper Orthogonal Decomposition

We present a new approach for generating fast simulation models of electromagnetic (EM) devices that contain moving components and magnetic materials with nonlinear properties. Our approach is based on generating low-dimensional simulation models that approximate the original spatially discretized models of electromagnetic field and their variations under conditions of component movement and material nonlinearity. The movement of the modeled device components is simulated by coupling the reduced-order EM field models weakly to the mechanical equations. We have successfully used our approach to generate a fast simulation model of a simple electromagnetic device with a moving component and nonlinear material properties.

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Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

This paper presents a new approach for performing fast multiobjective optimization of the design of moving nonlinear electromagnetic (EM) devices using parametric reduced order EM field models. The approach minimizes the high computational costs that arises from the frequent evaluation of large scale spatially discretized EM field models during the optimization process run. The performance of the proposed approach is demonstrated by carrying out a multiobjective design optimization of an electromagnetic device with a moving component and nonlinear material properties.

T. Fritzsche

Papers

Model-Order Reduction of Moving Nonlinear Electromagnetic Devices

Multiobjective Optimization of the Design of Nonlinear Electromagnetic Systems Using Parametric Reduced Order Models