Showing papers on "Model order reduction published in 2015"

Low-rank methods for high-dimensional approximation and model order reduction

Abstract: Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply to many problems in computational science which are formulated in tensor spaces, such as problems arising in stochastic calculus, uncertainty quantification or parametric analyses. Here, we present complexity reduction methods based on low-rank approximation methods. We analyze the problem of best approximation in subsets of low-rank tensors and discuss its connection with the problem of optimal model reduction in low-dimensional reduced spaces. We present different algorithms for computing approximations of a function in low-rank formats. In particular, we present constructive algorithms which are based either on a greedy construction of an approximation (with successive corrections in subsets of low-rank tensors) or on the greedy construction of tensor subspaces (for subspace-based low-rank formats). These algorithms can be applied for tensor compression, tensor completion or for the numerical solution of equations in low-rank tensor formats. A special emphasis is given to the solution of stochastic or parameter-dependent models. Different approaches are presented for the approximation of vector-valued or multivariate functions (identified with tensors), based on samples of the functions (black-box approaches) or on the models equations which are satisfied by the functions.

Reduced-Order Small-Signal Model of Microgrid Systems

Abstract: The objective of this study was to develop a reduced-order small-signal model of a microgrid system capable of operating in both the grid-tied and the islanded conditions. The nonlinear equations of the proposed system were derived in the $dq$ reference frame and then linearized around stable operating points to construct a small-signal model. The high-order state matrix was then reduced using the singular perturbation technique. The dynamic equations were divided into two groups based on the small-signal model parameters $\varepsilon$ . The slow states, which dominated the systems dynamics, were preserved, whereas the fast states were eliminated. Step responses of the model were compared to the experimental results from a hardware test to assess their accuracy and similarity to the full-order system. The proposed reduced-order model was applied to a modified IEEE-37 bus grid-tied microgrid system to evaluate systems dynamic response in grid-tied mode, islanded mode, and transition from grid-tied to islanded mode.

A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Abstract: Summary We present a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M3) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract, even from these results with real data, the numerical trends indicated by the theoretical convergence and stability analyses. Copyright © 2014 John Wiley & Sons, Ltd.

Two-Sided Projection Methods for Nonlinear Model Order Reduction

Abstract: In this paper, we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching. Using basic tensor calculus, we propose a computationally efficient way of computing reduced-order models. We further extend the idea of two-sided interpolation methods to this more general setting by employing the tensor structure of the Hessian. We investigate the use of oblique projections in order to preserve important system properties such as stability. We test one-sided and two-sided projection methods for different semi-discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition (POD).

Progressive construction of a parametric reduced-order model for PDE-constrained optimization

Abstract: Summary An adaptive approach to using reduced-order models (ROMs) as surrogates in partial differential equations (PDE)-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given ROM is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the high-dimensional model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a reduced-order basis is also presented, along with a methodology for computing sensitivities of the ROM that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared with the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%. Copyright © 2014 John Wiley & Sons, Ltd.

Data assimilation and uncertainty assessment for complex geological models using a new PCA-based parameterization

Abstract: In this paper, a recently developed parameterization procedure based on principal component analysis (PCA), which is referred to as optimization-based PCA (O-PCA), is generalized for use with a wide range of geological systems. In O-PCA, the mapping between the geological model in the full-order space and the low-dimensional subspace is framed as an optimization problem. The O-PCA optimization involves the use of regularization and bound constraints, which act to extend substantially the ability of PCA to model complex (non-Gaussian) systems. The basis matrix required by O-PCA is formed using a set of prior realizations generated by a geostatistical modeling package. We show that, by varying the form of the O-PCA regularization terms, different types of geological scenarios can be represented. Specific systems considered include binary-facies, three-facies and bimodal channelized models, and bimodal deltaic fan models. The O-PCA parameterization can be applied to generate random realizations, though our focus here is on its use for data assimilation. For this application, O-PCA is combined with the randomized maximum likelihood (RML) method to provide a subspace RML procedure that can be applied to non-Gaussian models. This approach provides multiple history-matched models, which enables an estimate of prediction uncertainty. A gradient procedure based on adjoints is used for the minimization required by the subspace RML method. The gradient of the O-PCA mapping is determined analytically or semi-analytically, depending on the form of the regularization terms. Results for two-dimensional oil-water systems, for several different geological scenarios, demonstrate that the use of O-PCA and RML enables the generation of posterior reservoir models that honor hard data, retain the large-scale connectivity features of the geological system, match historical production data, and provide an estimate of prediction uncertainty. MATLAB code for the O-PCA procedure, along with examples for three-facies and bimodal models, is included as online Supplementary Material.

Soft Computing Approach for Model Order Reduction of Linear Time Invariant Systems

Abstract: Most of the physical systems can be represented by mathematical models. The mathematical procedure of system modeling often leads to a comprehensive description of a process in the form of higher-order differential equations which are difficult to use either for analysis or for controller synthesis. It is, therefore, useful and sometimes necessary to find the possibility of some equations of the same type but of lower order that may be considered to adequately reflect almost all essential characteristics of the system under consideration. This paper proposes a new method for order reduction of higher-order linear time invariant systems based on stability equation method and particle swarm optimization algorithm. Reduced-order model will definitely be stable if the original model is stable. The superiority of the proposed method is illustrated by numerical examples of single-input, single-output systems and multiple-input and multiple-output systems. The results are compared with well-known methods available in the literature.

Low-rank tensor methods for model order reduction

Abstract: Parameter-dependent models arise in many contexts such as uncertainty quantification, sensitivity analysis, inverse problems or optimization. Parametric or uncertainty analyses usually require the evaluation of an output of a model for many instances of the input parameters, which may be intractable for complex numerical models. A possible remedy consists in replacing the model by an approximate model with reduced complexity (a so called reduced order model) allowing a fast evaluation of output variables of interest. This chapter provides an overview of low-rank methods for the approximation of functions that are identified either with order-two tensors (for vector-valued functions) or higher-order tensors (for multivariate functions). Different approaches are presented for the computation of low-rank approximations, either based on samples of the function or on the equations that are satisfied by the function, the latter approaches including projection-based model order reduction methods. For multivariate functions, different notions of ranks and the corresponding low-rank approximation formats are introduced.

Model reduction for stochastic systems

Abstract: To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. If one uses the well-known Galerkin scheme, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations we consider balanced truncation as a model order reduction technique. This approach is well-known from deterministic control theory and successfully employed in practice for decades. So, we generalize balanced truncation for controlled linear systems with Levy noise, discuss properties of the reduced order model, provide an error bound, and give some examples.

pyMOR - Generic Algorithms and Interfaces for Model Order Reduction

Abstract: 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.

A Fully Adaptive Scheme for Model Order Reduction Based on Moment Matching

Abstract: A fully adaptive model order reduction scheme based on moment matching is proposed to derive the reduced-order models of linear time-invariant (LTI) systems. According to the given error tolerance, the order of the reduced-order model as well as the expansion points for the transfer function is automatically determined on the fly during the process of model reduction. In this sense, the reduced-order model is automatically obtained without assigning the number of moments and expansion points in a priori , which is a prerequisite for the standard implementation of model reduction based on moment matching. The proposed adaptive scheme is found to be efficient when it is tested on various LTI systems.

Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system

Abstract: The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced. At the same time the induced error can be bounded by fast evaluable a-posteriori error bounds. In the offline-phase the proposed algorithms make use of the decomposed problem structure. Rigorous a-posteriori error bounds are developed, indicating the accuracy of certain lifting operators used in the offline-phase as well as the accuracy of the reduced coupled system. Also, a strategy separately bounding pressure and velocity errors is extended. Numerical experiments dealing with groundwater flow scenarios demonstrate the efficiency of the approach as well as the limitations regarding a-posteriori error estimation.

Projection-based model reduction for contact problems

Abstract: To be feasible for computationally intensive applications such as parametric studies, optimization and control design, large-scale finite element analysis requires model order reduction. This is particularly true in nonlinear settings that tend to dramatically increase computational complexity. Although significant progress has been achieved in the development of computational approaches for the reduction of nonlinear computational mechanics models, addressing the issue of contact remains a major hurdle. To this effect, this paper introduces a projection-based model reduction approach for both static and dynamic contact problems. It features the application of a non-negative matrix factorization scheme to the construction of a positive reduced-order basis for the contact forces, and a greedy sampling algorithm coupled with an error indicator for achieving robustness with respect to model parameter variations. The proposed approach is successfully demonstrated for the reduction of several two-dimensional, simple, but representative contact and self contact computational models.

3-D Distributed Memory Polynomial Behavioral Model for Concurrent Dual-Band Envelope Tracking Power Amplifier Linearization

Abstract: This paper presents a new 3-D behavioral model to compensate for the nonlinear distortion arising in concurrent dual-band (DB) Envelope Tracking (ET) Power Amplifiers (PAs). The advantage of the proposed 3-D distributed memory polynomial (3D-DMP) behavioral model, in comparison to the already published behavioral models used for concurrent dual-band envelope tracking PA linearization, is that it requires a smaller number of coefficients to achieve the same linearity performance, which reduces the overall identification and adaptation computational complexity. The proposed 3D-DMP digital predistorter (DPD) is tested under different ET supply modulation techniques. Moreover, further model order reduction of the 3D-DMP DPD is achieved by applying the principal component analysis (PCA) technique. Experimental results are shown considering a concurrent DB transmission of a WCDMA signal at 1.75 GHz and a 10-MHz bandwidth LTE signal at 2.1 GHz. The performance of the proposed 3D-DMP DPD is evaluated in terms of linearity, drain power efficiency, and computational complexity.

Constraint reduction procedures for reduced‐order subsurface flow models based on POD–TPWL

Abstract: Summary The properties and numerical performance of reduced-order models based on trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) are assessed. The target application is subsurface flow modeling, although our findings should be applicable to a range of problems. The errors arising at each step in the POD–TPWL procedure are described. The impact of constraint reduction on accuracy and stability is considered in detail. Constraint reduction entails projection of the overdetermined system into a low-dimensional subspace, in which the system is solvable. Optimality conditions for constraint reduction, in terms of error minimization, are derived. Galerkin and Petrov–Galerkin projections are shown to correspond to optimality in norms that involve weighting with the Jacobian matrix. Two new treatments, inverse projection and weighted inverse projection, are suggested. These methods minimize error in appropriate norms, although they require substantial preprocessing computations. Numerical results are presented for oil reservoir simulation problems. Galerkin projection provides reasonable accuracy for simpler oil–water systems, although it becomes unstable in more challenging cases. Petrov–Galerkin projection is observed to behave stably in all cases considered. Weighted inverse projection also behaves stably, and it provides the highest accuracy. Runtime speedups of 150–400 are achieved using these POD–TPWL models. Copyright © 2015 John Wiley & Sons, Ltd.

SVD-based improvements for component mode synthesis in elastic multibody systems

Abstract: The method of elastic multibody systems (EMBS) is used for dynamic simulations, where a multitude of deformable bodies interact. An important step to achieve models yielding reasonable computational effort is the model order reduction for the elastic parts. Advanced schemes, such as Craig-Bampton, Krylov-subspace methods and SVD-based methods, do not only focus on homogeneous solutions but also take the loading situation into account. Thus, it is possible to generate models of low dimension but good approximation quality. The modular approach of EMBS causes that exact boundary conditions on the elastic parts may be unknown during the reduction step, thus making it difficult to find suitable load cases. Input–output based methods, especially balanced truncation, can fail to deliver a good approximation in this context. In this contribution, it is shown how these loading conditions can be obtained in a general manner in the EMBS-framework. Using the presented novel methodology, input–output based reduction schemes, such as frequency weighted balanced truncation, can be correctly applied and lead to a greatly improved approximation quality in frequency and time domain compared to existing methods. The benefits of the presented method are demonstrated on a large, industrially relevant model.

Cable Model Order Reduction for HVDC Systems Interoperability Analysis

Abstract: In order to predict the stability of cable-based offshore Voltage Source Converter High Voltage Direct Current (VSC HVDC) systems, an accurate cable model is needed to represent the characteristics of the transmission system. This paper presents a model consisting of cascaded pi-sections with multiple longitudinal parallel branches, allowing for a detailed representation in the frequency domain. Furthermore, a cable model order reduction for system interoperability studies is proposed based on the identification of interacting modes. Simulation results demonstrate how a classical cascaded pi-model is unable to accurately represent the interacting modes and show how the state-space representation of the proposed cable model can be reduced whilst preserving an accurate representation of the converter interactions.

Methodology for Obtaining Linear State Space Building Energy Simulation Models

Abstract: Optimal climate control for building systems is facilitated by linear, low-order models of the building structure and of its Heating, Ventilation and Air Conditioning (HVAC) systems. However, obtaining these models in a practical form is often difficult, which greatly hampers the commercial implementation of model predictive controllers. This work describes a methodology for obtaining a linear State Space Model (SSM) of Building Energy Simulation (BES) models, consisting of walls, windows, floors and the zone air. The methodology uses the Modelica library IDEAS to develop a BES model, including its non-linearities, and automates its linearisation. The Dymola function linearize2 is used to generate the state space formulation, facilitating further mathematical manipulations, or simulation in different environments. Optionally this model can then be reduced for control purposes using model order reduction (MOR) techniques. The methodology is illustrated for the zone air temperature in an office building. For this case, the absolute error between the non-linear BES and its SSM remains under 1 K and its yearly average is 0.21 K. The original 50 states SSM could furthermore be reduced to 16 states without significant loss of accuracy.

A comparison of model order reduction techniques for electrochemical characterization of Lithium-ion batteries

Abstract: Model order reduction has become a rather common approach to approximate complex first-principles electrochemical models (described by systems of linear or nonlinear partial differential equations) into low-order dynamic system models, for control or estimation design.

Computational Methods for Model Reduction of Large-Scale Sparse Structured Descriptor Systems

Abstract: Currently descriptor systems, i.e., the systems whose dynamics obey differentialalgebraic equations (DAEs), play important roles in various disciplines of science and technology. In general, such systems are generated by finite element or finite difference methods. If the grid resolution becomes very fine, because many details must be resolved, the systems become very large. Moreover they are sparse, i.e., most of the elements in the matrices of the system are zero, which are not stored. A high dimensional system will always be complex, requiring a great deal of memory, thereby hindering computational performance significantly in simulation. Sometimes the systems are too large to store due to memory restrictions. Therefore, we seek to reduce the complexity of the model by applying model order reduction (MOR), i.e., we seek an approximation of the original model that well-approximates the behavior of the original model, yet is much faster to evaluate. We investigate efficient model reduction of sparse large-scale descriptor systems. We focus on the balancing based method balanced truncation (BT). A balanced truncation based method for such systems is introduced by Stykel (see, e.g., her PhD thesis, published in 2002). The author discusses a general framework of the BT method for a descriptor system. In general, the method is based on explicit computation of the spectral projectors onto the left and right deflating subspaces of the matrix pencil corresponding to the finite and infinite eigenvalues. Although these projectors are available for particular systems, computation is expensive. In this thesis, we focus on how to avoid computing such kind of projectors explicitly. Besides balanced truncation, the idea of avoidance of the projectors is extended to interpolation of transfer function, via iterative rational Krylov algorithms (IRKA) and projection onto dominant eigenspace, of the Gramian (PDEG) based model reduction methods. First, we discuss the model reduction problem for index 2 first order unstable descriptor systems arising from spatially discretized linearized Navier-Stokes equations. We apply our algorithms to the linearization of the von Karman vortex shedding at a moderate Reynolds number. We demonstrate that the resulting reduced model can be used to accurately simulate the unstable linearized model and to design a stabilizing controller. Future work will include the realization of the resulting control law for the full nonlinear model. Second, we investigate model reduction of a finite element model of a spindle head configuration in a machine tool. The special feature of this spindle head is that it is partially driven by a set of piezo actuators. Due

Model Reduction and Simulation of Nonlinear Circuits via Tensor Decomposition

Abstract: Model order reduction of nonlinear circuits (especially highly nonlinear circuits) has always been a theoretically and numerically challenging task. In this paper, we utilize tensors (namely, a higher order generalization of matrices) to develop a tensor-based nonlinear model order reduction algorithm we named TNMOR for the efficient simulation of nonlinear circuits. Unlike existing nonlinear model order reduction methods, in TNMOR high-order nonlinearities are captured using tensors, followed by decomposition and reduction to a compact tensor-based reduced-order model. Therefore, TNMOR completely avoids the dense reduced-order system matrices, which in turn allows faster simulation and a smaller memory requirement if relatively low-rank approximations of these tensors exist. Numerical experiments on transient and periodic steady-state analyses confirm the superior accuracy and efficiency of TNMOR, particularly in highly nonlinear scenarios.

Modal Matching for LPV Model Reduction of Aeroservoelastic Vehicles

Abstract: A model order reduction method is proposed for models of aeroservoelastic vehicles in the linear parameter-varying (LPV) systems framework, based on state space interpolation of modal forms. The dynamic order of such models is usually too large for control synthesis and implementation since they combine rigid body dynamics, structural dynamics and unsteady aerodynamics. Thus, model order reduction is necessary. For linear timeinvariant (LTI) models, order reduction is often based on balanced realizations. For LPV models, this requires the solution of a large set of linear matrix inequalities (LMIs), leading to numerical issues and high computational cost. The proposed approach is to use well developed and numerically stable LTI techniques for reducing the LPV model locally and then to transform the resulting collection of systems into a consistent modal representation suitable for interpolation. The method is demonstrated on an LPV model of the body freedom flutter vehicle, reducing the number of states from 148 to 15. The accuracy of the reduced order model (ROM) is confirmed by evaluating the ν-gap metric with respect to the full order model and by comparison to another ROM obtained by state-of-the-art LPV balanced truncation techniques.

Uncertainty Quantification for Maxwell's Equations Using Stochastic Collocation and Model Order Reduction

Abstract: Modeling and simulation are important for the design process of new semiconductor structures. Difficulties proceed from shrinking structures, increasing working frequencies, and uncertainties of materials and geometries. Therefore, we consider the time-harmonic Maxwell’s equations for the simulation of a coplanar waveguide with uncertain material parameters. To analyze the uncertainty of the system, we use stochastic collocation with Stroud and sparse grid points. The results are compared to a Monte Carlo simulation. Both methods rely on repetitive runs of a deterministic solver. Hence, we compute a reduced model by means of proper orthogonal decomposition to reduce the computational cost. The Monte Carlo simulation and the stochastic collocation are both applied to the full and the reduced model. All results are compared concerning accuracy and computation time.

Model-Order Reduction of Multiple-Input Non-Linear Systems Based on POD and DEI Methods

Abstract: The proper orthogonal decomposition combined with the discrete empirical interpolation method is investigated in order to reduce a finite-element model of a multiple-input non-linear device. The non-linear reduced problem is solved using the Newton–Raphson method. The transient state of a three-phase transformer with a variable load is studied with the proposed reduction method for different supply voltage conditions.