Reduced-Order Modeling (ROM) for Simulation and Optimization

We discuss an interpolatory model reduction framework for quadratic-bilinear (QB) descriptor systems, arising especially from the semi-discretization of the Navier–Stokes equations. Several recent results indicate that directly applying interpolatory model reduction frameworks, developed for systems of ordinary differential equations, to descriptor systems, may lead to an unbounded error between the original and reduced-order systems, e.g., in the \(\mathscr {H}_2\)-norm, due to an inappropriate treatment of the polynomial part of the original system. Thus, the main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QB ordinary differential equations (QBODEs) to aforementioned descriptor systems while ensuring bounded error. For this, we first aim at transforming the descriptor system into an equivalent ODE system by means of projectors for which standard model reduction techniques can be applied. Subsequently, we discuss how to construct optimal reduced systems corresponding to an equivalent ODE, without requiring explicit computation of the expensive projection used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier–Stokes equations.

Modern simulation scenarios frequently require multi-query or real-time responses of simulation models for statistical analysis, optimization, or process control. However, the underlying simulation models may be very time-consuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modeling of functions from vectorial input to vectorial output spaces. These appear, for instance, in simulation of coupled models or in the case of approximating general input–output maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular, we recall the vectorial kernel orthogonal greedy algorithm (VKOGA) for approximating vector-valued functions. We collect some recent convergence statements that provide sound foundation for these algorithms, in particular quasi-optimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with non-symmetric greedy kernel approximation schemes. The results indicate better stability and overall more accurate models in situations where the input data locations are not equally distributed.

In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.

This paper introduces an efficient sequential application of reduced order models (ROMs) to solve linear quadratic optimal control problems with initial value controls. The numerical solution of such a problem requires Hessian-times-vector multiplications, each of which requires solving a linearized state equation with initial value given by the vector and solving a second-order adjoint equation. Projection-based ROMs are applied to these differential equations to generate a Hessian approximation. However, in general, no fixed ROM well-approximates the application of the Hessian to all possible vectors of initial data. To improve a basic ROM, Heinkenschloss and Jando: Reduced-Order Modeling for Time-Dependent Optimization Problems with Initial Value Controls (SIAM Journal on Scientific Computing, 40(1), A22–A51, 2018, https://​doi.​org/​10.​1137/​16M1109084) introduce an augmentation of the basic ROM by the right-hand side of the optimality system. This augmented ROM substantially improves the accuracy of the computed control, but this accuracy may still not be enough. The proposed sequential application of the augmented ROM can compute an approximate control with the same accuracy as the one obtained using only the expensive full-order model, but at a fraction of the cost.

Using isothermal Euler equations and a network graph to model gas flow in a pipeline network is a classical description, and we prove that any direct space discretization results in a system of index 2 nonlinear differential algebraic equations (DAE). Those are hard to simulate, and model order reduction techniques are not very developed for this system class. However, we can show that a simple approximation results in an index 1 system of nonlinear differential algebraic equations, which is easier to simulate and we can show that a structured projection leads to a reduced system that also typically has index 1. We validate the use of this model and its advantage for fast simulation, including model order reduction, in some numerical examples.

The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that contain non-symmetric components. These are usually introduced during the mass production process and are modeled by small perturbations in the geometry (e.g., eccentricity) or the material parameters. While model order reduction for symmetric machines is clear and does not need special treatment, the non-symmetric setting adds additional challenges. An adaptive strategy based on proper orthogonal decomposition is developed to overcome these difficulties. Equipped with an a posteriori error estimator, the obtained solution is certified. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Many new promising model order reduction (MOR) methods and algorithms were developed during the last decade. Industry and academic research institutions intend to test, validate, compare, and use these new promising MOR techniques with their own models. Therefore, an MOR toolbox bridging the gap between theoretical, algorithmic, and numerical developments to an end-user-oriented program, usable by non-experts, was developed called ‘Model Order Reduction of Elastic Multibody Systems’ (Morembs). A C++ implementation as well as a Matlab implementation including an intuitive graphical user interface is available. Import from various FE programs is possible, and the reduced elastic bodies can be exported to a variety of programs to simulate the compact models. In the course of the various projects, many improvements on the algorithmic side were added. As we learned over the years, there is not one ‘optimal’ MOR method. ‘Optimal’ MOR depends on circumstances, like boundary conditions, excitation spectra, further model usage. The toolbox is now used, e.g., in solid mechanics, biomechanics, vehicle dynamics, control of flexible structures, or crash simulations. In all these use cases, the toolbox allows the user to facilitate their well-known modeling and simulation environment. Only the critical MOR process during preprocessing is performed with Morembs, which helps to compare the various MOR techniques to find the most suited one.

An increasing number of disruptive innovations with high economic and social impact shape our digitalizing world. Speed and extending scope of these developments are limited by available tools and paradigms to master exploding complexities. Simulation technologies are key enablers of digitalization. They enable digital twins mirroring products and systems into the digital world. Digital twins require a paradigm shift. Instead of expert centric tools, engineering and operation require autonomous assist systems continuously interacting with its physical and digital environment through background simulations. Model order reduction (MOR) is a key technology to transfer highly detailed and complex simulation models to other domains and life cycle phases. Reducing the degree of freedom, i.e., increasing the speed of model execution while maintaining required accuracies and predictability, opens up new applications. Within this contribution, we address the advantages of model order reduction for model-based system engineering and real-time thermal control of electric motors.