Projection-based Model Order Reduction Research Articles

Stability preservation in Galerkin-type projection-based model order reduction

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.

Proper orthogonal decomposition for substructures in nonlinear finite element analysis: coupling by means of tied contact

The finite element analysis of complex structures involves a high computational effort, since the equation system to be solved includes a large number of degrees of freedom. This holds particularly in nonlinear finite element analysis. In order to reduce the numerical effort, the total system is subdivided into substructures which are analyzed separately from each other. The computing time can be further reduced, if the number of degrees of freedom of the substructures and thereby the dimension of the global equation system are decreased. In the present paper, this is achieved by projection-based model order reduction applied at the level of the substructures. The corresponding modes include internal and boundary nodes. The precomputation is carried out either directly with respect to the global system or only on one representative substructure. For the coupling, a new surface-to-surface tied contact formulation based on the penalty method is presented which couples the reduced and unreduced substructures. This is also possible for non-matching meshes. Several nonlinear numerical examples are performed which show the feasibility of the new approach.

High-Permittivity Pad Design for Dielectric Shimming in Magnetic Resonance Imaging Using Projection-Based Model Reduction and a Nonlinear Optimization Scheme.

Inhomogeneities in the transmit radio frequency magnetic field ( ) reduce the quality of magnetic resonance (MR) images. This quality can be improved by using high-permittivity pads that tailor the fields. The design of an optimal pad is application-specific and not straightforward and would therefore benefit from a systematic optimization approach. In this paper, we propose such a method to efficiently design dielectric pads. To this end, a projection-based model order reduction technique is used that significantly decreases the dimension of the design problem. Subsequently, the resulting reduced-order model is incorporated in an optimization method in which a desired field in a region of interest can be set. The method is validated by designing a pad for imaging the cerebellum at 7 T. The optimal pad that is found is used in an MR measurement to demonstrate its effectiveness in improving the image quality.

Model-order reduction of electromagnetic fields in open domains

We have developed several Krylov projection-based model-order reduction techniques to simulate electromagnetic wave propagation and diffusion in unbounded domains. Such techniques can be used to efficiently approximate transfer function field responses between a given set of sources and receivers and allow for fast and memory-efficient computation of Jacobians, thereby lowering the computational burden associated with inverse scattering problems. We found how general wavefield principles such as reciprocity, passivity, and the Schwarz reflection principle translate from the analytical to the numerical domain and developed polynomial, extended, and rational Krylov model-order reduction techniques that preserve these structures. Furthermore, we found that the symmetry of the Maxwell equations allows for projection onto polynomial and extended Krylov subspaces without saving a complete basis. In particular, short-term recurrence relations can be used to construct reduced-order models that are as memory efficient as time-stepping algorithms. In addition, we evaluated the differences between Krylov reduced-order methods for the full wave and diffusive Maxwell equations and we developed numerical examples to highlight the advantages and disadvantages of the discussed methods.

Modeling and Quantification of Model-Form Uncertainties in Eigenvalue Computations Using a Stochastic Reduced Model

A feasible, nonparametric, probabilistic approach for modeling and quantifying model-form uncertainties associated with a computational model designed for the solution of a generalized eigenvalue problem is presented. It is based on the construction of a stochastic, projection-based reduced-order model associated with a high-dimensional model using three innovative ideas: 1) the substitution of the deterministic reduced-order basis with a stochastic counterpart featuring a reduced number of hyperparameters, 2) the construction of this stochastic reduced-order basis on a subset of a compact Stiefel manifold to guarantee the linear independence of its column vectors and the satisfaction of any constraints of interest, and 3) the formulation and solution of a reduced-order inverse statistical problem to determine the hyperparameters so that the mean value and statistical fluctuations of the eigenvalues predicted using the stochastic, projection-based reduced-order model match target values obtained from available data. Consequently, the proposed approach for modeling model-form uncertainties can be interpreted as an effective approach for extracting from data fundamental information and/or knowledge that are not captured by a deterministic computational model, and incorporating them in this model. Its potential for quantifying model-form uncertainties in generalized eigencomputations is demonstrated for a natural vibration analysis of a small-scale replica of an X-56-type aircraft made of a composite material for which ground-vibration-test data are available.

Compact and Passive Time-Domain Models Including Dispersive Materials Based on Order-Reduction in the Frequency Domain

In this paper, compact time-domain (TD) models featuring materials with frequency-dependent electromagnetic (EM) properties are derived. The considered frequency-dependent material models include multiterm Debye and Lorentz models for the electric permittivity and the magnetic permeability and a multiterm Drude model for the electric conductivity. The TD models are based on finite-element systems in the frequency domain (FD). To render the model compact and computationally efficient, the dimension of the FD system is compressed with the help of projection-based model-order reduction. In contrast to older approaches, the TD transformation is performed on the FD model itself rather than on the transfer function. The result is a state-space representation, which may either be solved by custom time integrators or imported into commercial circuit simulators. The advantages of the new approach include provable passivity of the FD model, provable causality of the TD model, and the ability to reconstruct the transient EM fields.

Model order reduction and low-dimensional representations for random linear dynamical systems

We consider linear dynamical systems of ordinary differential equations or differential algebraic equations. Physical parameters are substituted by random variables for an uncertainty quantification. We expand the state variables as well as a quantity of interest into an orthogonal system of basis functions, which depend on the random variables. For example, polynomial chaos expansions are applicable. The stochastic Galerkin method yields a larger linear dynamical system, whose solution approximates the unknown coefficients in the expansions. The Hardy norms of the transfer function provide information about the input–output behaviour of the Galerkin system. We investigate two approaches to construct a low-dimensional representation of the quantity of interest, which can also be interpreted as a sparse representation. Firstly, a standard basis is reduced by the omission of basis functions, whose accompanying Hardy norms are relatively small. Secondly, a projection-based model order reduction is applied to the Galerkin system and allows for the definition of new basis functions within a low-dimensional representation. In both cases, we prove error bounds on the low-dimensional approximation with respect to Hardy norms. Numerical experiments are demonstrated for two test examples.

Projection-Based Model Order Reduction Methods for the Estimation of Vector-Valued Variables of Interest

We propose and compare goal-oriented projection based model order reduction methods for the estimation of vector-valued functionals of the solution of parameter-dependent equations. The first projection method is a generalization of the classical primal-dual method to the case of vector-valued variables of interest. We highlight the role played by three reduced spaces: the approximation space and the test space associated to the primal variable, and the approximation space associated to the dual variable. Then we propose a Petrov-Galerkin projection method based on a saddle point problem involving an approximation space for the primal variable and an approximation space for an auxiliary variable. A goal-oriented choice of the latter space, defined as the sum of two spaces, allows us to improve the approximation of the variable of interest compared to a primal-dual method using the same reduced spaces. Then, for both approaches, we derive computable error estimates for the approximations of the variable of interest and we propose greedy algorithms for the goal-oriented construction of reduced spaces. The performance of the algorithms are illustrated on numerical examples and compared to standard (non goal-oriented) algorithms.

Dynamical Model Reduction Method for Solving Parameter-Dependent Dynamical Systems

We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution of the full-order model at some selected parameters values. The approximation obtained by Galerkin projection is the solution of a reduced dynamical system with a modified flux which takes into account the time dependency of the reduced spaces. An a posteriori error estimate is derived and a greedy algorithm using this error estimate is proposed for the adaptive selection of parameters values. The resulting method can be interpreted as a dynamical low-rank approximation method with a subspace point of view and a uniform control of the error over the parameter set.

The Minimum Norm Least-Squares Solution in Reduction by Krylov Subspace Methods

In recent years, model order reduction (MOR) of interconnect system has become an important technique to reduce the computation complexity and improve the verification efficiency in the nanometer VLSI design. The Krylov subspaces techniques in existing MOR methods are efficient, and have become the methods of choice for generating small-scale macro-models of the large-scale multi-port RCL networks that arise in VLSI interconnect analysis. Although the Krylov subspace projection-based MOR methods have been widely studied over the past decade in the electrical computer-aided design community, all of them do not provide a best optimal solution in a given order. In this paper, a minimum norm least-squares solution for MOR by Krylov subspace methods is proposed. The method is based on generalized inverse (or pseudo-inverse) theory. This enables a new criterion for MOR-based Krylov subspace projection methods. Two numerical examples are used to test the PRIMA method based on the method proposed in this paper as a standard model.

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

PEBL-ROM: Projection-error based local reduced-order models

Projection-based model order reduction (MOR) using local subspaces is becoming an increasingly important topic in the context of the fast simulation of complex nonlinear models. Most approaches rely on multiple local spaces constructed using parameter, time or state-space partitioning. State-space partitioning is usually based on Euclidean distances. This work highlights the fact that the Euclidean distance is suboptimal and that local MOR procedures can be improved by the use of a metric directly related to the projections underlying the reduction. More specifically, scale-invariances of the underlying model can be captured by the use of a true projection error as a dissimilarity criterion instead of the Euclidean distance. The capability of the proposed approach to construct local and compact reduced subspaces is illustrated by approximation experiments of several data sets and by the model reduction of two nonlinear systems.

PyMOR -- Generic Algorithms and Interfaces for Model Order Reduction

Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach.

Interpolation of Inverse Operators for Preconditioning Parameter-Dependent Equations

We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. Approximations of the Frobenius norm using random matrices are introduced in order to handle large matrices. The resulting statistical estimators of the Frobenius norm yield quasi-optimal projections that are controlled with high probability. Strategies for the adaptive selection of interpolation points are then proposed for different objectives in the context of projection-based model order reduction methods: the improvement of residual-based error estimators, the improvement of the projection on a given reduced approximation space, or the recycling of computations for sampling based model order reduction methods.

Model order reduction in porous media flow simulation using quadratic bilinear formulation

In this work, the saturation equation in the two phase flow model is transformed into quadratic bilinear form by introducing auxiliary variables. This new formulation is the first step for applying training-free reduced order modeling. There is no approximation involved in this transformation and it only increases the size of the problem linearly, before applying any model order reduction technique. Although this might seem counterintuitive to increase the size of the system, it can improve the basis selection and consequently the reduced system. Also, this formulation allows certain properties of the original model to be preserved in the reduced-order model, such as stability and passivity. A projection-based model order reduction on this form of system will yield a reduced-order model without the need to use anymore approximation such as discrete empirical interpolation method for evaluating nonlinear terms. Although in this work only the proper orthogonal decomposition is applied, this formulation is the first step for training free model order reduction after addressing the existing challenges. One of the main challenges is the dependency of the saturation equation on flux which is changing at different time steps, and thus makes the saturation equation time varying. The numerical results of applying the proposed formulation with POD model order reduction on a two-phase immiscible flow show a substantial reduction in the computational complexity.

Extended Hamiltonian Hessenberg Matrices arise in Projection based Model Order Reduction

AbstractWe show that extended Hamiltonian Hessenberg matrices arise naturally in projection‐based model order reduction. Therefore we reduce a large dynamical system by projecting it on an extended Krylov subspace. The eigenvalues of the reduced order model can then be computed directly by applying the extended Hamiltonian QR algorithm. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A New Method for Accurate and Efficient Residual Computation in Adaptive Model-Order Reduction

Projection-based model-order reduction is a powerful methodology for solving parameter-dependent linear systems of equations. The efficient computation of the residual norm is of paramount importance in adaptive model reduction schemes because it is heavily used in error indicators and a posteriori error bounds. These guide the adaptive selection of expansion points in multi-point methods and serve as stopping criteria for subspace enrichment. This paper demonstrates that the standard algorithm for fast residual norm computation leads to premature stagnation, and it presents a new approach of improved accuracy.

주파수응답에 대한 투영기반 모델차수축소법의 비교

본 논문에서는 대표적 투영기반 모델차수축소법인 크리로프 부공간 모델차수축소법(KSM)과 모드중첩법(MTM)을 고려하여 주파수응답해석에 대한 수치적 정확도와 효율성을 비교하였다. 두 모델차수축소법의 수치 정확도 비교를 위하여 주파수응답해석 결과, 축소차수 및 관심주파수에 따른 상대오차를 고려하였으며 이후에 오차수렴지표를 통한 자동적인 축소차수의 결정이 가능 여부를 확인하였다. 효율성 비교를 위해서는 각 축소모델의 주파수응답 해석시간 및 축소차수에 따른 변환행렬 생성시간을 비교하였다. 자동차 현가장치에 대한 유한요소모델을 적용예제로 선정하여 수치 비교를 수행하였다.

Control of Parameter-dependent High-order Systems using Parametric Model Reduction

Abstract In this paper controllers and observers are designed for high-dimensional parameter-dependent LTI systems. The application of parametric model order reduction by matrix interpolation is proposed in order to use common methods of control. In the offline phase, the parameter space is sampled and a set of locally reduced systems is obtained using projection-based model order reduction, which results in a database of system matrices. In the online phase, a reduced system can be calculated for a desired parameter vector by interpolating the system matrices from the database. Controllers and observers can be obtained for the interpolated system using common methods of control design. The approach is demonstrated through a practical example on a test rig for a gantry crane operating with different loads.

Accurate Capacitance Extraction in Equivalent Capacitor Substitution Method (ECSM) for Monopole Antenna Calibration

Free-space methods for EMC antenna calibrations at Open Area Test Site (OATS) are not suitable for monopole antennas because of corresponding large wavelengths. The Equivalent Capacitor Substitution Method (ECSM) is adopted in ANSI and CISPR standards. One of major uncertainty factors of ECSM is the accuracy of capacitance. A novel capacitance extraction method based on full-wave impedance modelling is proposed in this paper, which is more accurate than conventional analytical formulae and experienced values, especially for practical monopole antennas, with a telescopic rod, or with a finite ground plane. A projection based model order reduction method is then utilized to achieve a compact macro-model. The capacitance value can be determined from the input impedance which is given by the macro-model output. The scheme and computational methodology of a developed calculation tool are introduced, together with some practical calibration data which indicates the effectiveness of the proposed methods.