Showing papers on "Model order reduction published in 2011"

A Short Review on Model Order Reduction Based on Proper Generalized Decomposition

Abstract: This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition—PGD. Space and time separated representations generalize Proper Orthogonal Decompositions—POD—avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions,…) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

Certified reduced basis approximation for parametrized partial differential equations and applications

Abstract: Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems

Abstract: We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.

Model Order Reduction Techniques with Applications in Electrical Engineering

Abstract: This volume focuses on model reduction problems with particular applications in electrical engineering. Starting with a clear outline of the technique and its wide methodological background, central topics are introduced including mathematical tools, physical processes, numerical computing experience, software developments and knowledge of system theory. Several model reduction algorithms are then discussed. The aim of this work is to give the reader an overview of reduced-order model design and an operative guide. Particular attention is given to providing basic concepts for building expert systems for model reduction.

A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space

Abstract: Modern simulation scenarios require real-time or many-query responses from a simulation model. This is the driving force for increased efforts in model order reduction for high-dimensional dynamical systems or partial differential equations. This demand for fast simulation models is even more critical for parameterized problems. Several snapshot-based methods for basis construction exist for parameterized model order reduction, for example, proper orthogonal decomposition or reduced basis methods. They require the careful choice of samples for generation of the reduced model. In this article we address two types of grid-based adaptivity that can be beneficial in such basis generation procedures. First, we describe an approach for training set adaptivity. Second, we introduce an approach for multiple bases on adaptive parameter domain partitions. Due to the modularity, both methods also can easily be combined. They result in efficient reduction schemes with accelerated training times, improved approximatio...

QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems

Abstract: We present a projection-based nonlinear model order reduction method, named model order reduction via quadratic-linear systems (QLMOR). QLMOR employs two novel ideas: 1) we show that nonlinear ordinary differential equations, and more generally differential-algebraic equations (DAEs) with many commonly encountered nonlinear kernels can be rewritten equivalently in a special representation, quadratic-linear differential algebraic equations (QLDAEs), and 2) we perform a Volterra analysis to derive the Volterra kernels, and we adapt the moment-matching reduction technique of nonlinear model order reduction method (NORM) [1] to reduce these QLDAEs into QLDAEs of much smaller size. Because of the generality of the QLDAE representation, QLMOR has significantly broader applicability than Taylor-expansion-based methods [1]-[3] since there is no approximation involved in the transformation from original DAEs to QLDAEs. Because the reduced model has only quadratic nonlinearities, its computational complexity is less than that of similar prior methods. In addition, QLMOR, unlike NORM, totally avoids explicit moment calculations, hence it has improved numerical stability properties as well. We compare QLMOR against prior methods [1]-[3] on a circuit and a biochemical reaction-like system, and demonstrate that QLMOR-reduced models retain accuracy over a significantly wider range of excitation than Taylor-expansion-based methods [1]-[3]. QLMOR, therefore, demonstrates that Volterra-kernel based nonlinear MOR techniques can in fact have far broader applicability than previously suspected, possibly being competitive with trajectory-based methods (e.g., trajectory piece-wise linear reduced order modeling [4]) and nonlinear-projection based methods (e.g., maniMOR [5]).

Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

Abstract: We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, mom...

Model reduction for circuit simulation

Abstract: Part I Invited Papers. 1 The need for novel model order reduction techniques in the electronics industry .W.H.A. Schilders. 1.1 Introduction. 1.2 Mathematical problems in the electronics industry. 1.3 Passivity and realizability. 1.4 Structure preservation. 1.5 Reduction of MIMO networks. 1.6 MOR for delay equations. 1.7 Parameterized and nonlinear MOR. 1.8 Summary: present and future needs of the electronics industry. References. 2 The SPRIM Algorithm for Structure-Preserving Order Reduction of General RCL Circuits Roland W. Freund. 2.1 Introduction. 2.2 RCL Circuit Equations. 2.3 Projection-Based Order Reduction. 2.4 The SPRIM Algorithm. 2.5 Treatment of Voltage Sources. 2.6 Numerical Examples. 2.7 Concluding Remarks. References. 3 Balancing-Related Model Reduction of Circuit Equations Using Topological Structure Tatjana Stykel. 3.1 Introduction. 3.2 Circuit equations. 3.3 Balancing-related model reduction. 3.4 Numerical methods for matrix equations. 3.5 Numerical examples. 3.6 Conclusions and open problems. References. 4 Topics in Model Order Reduction with Applications to Circuit Simulation Sanda Lefteriu and Athanasios C. Antoulas. 4.1 Introduction and Motivation. 4.2 Background. 4.3 Theoretical Aspects. 4.4 Tangential interpolation for modeling Y-parameters. 4.5 Numerical Results. 4.6 Conclusion. References. Part II Contributed Papers. 5 Forward and Reverse Modeling of Low Noise Amplifiers based on Circuit Simulations L. De Tommasi, J. Rommes, T. Beelen, M. Sevat, J. A. Croon and T. Dhaene. 5.1 Introduction. 5.2 Forward and reverse modeling: problem descriptions. 5.3 Forward Modeling. 5.3.1 Performance Figures via Surrogate Models. 5.4 Reverse Modeling with the NBI method. 5.5 Reverse modeling using transistor level simulations. 5.6 Discussion and conclusions. References. 6 Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-Hand Sides Arising in Model Reduction Peter Benner and Lihong Feng. 6.1 Introduction. 6.2 Methods Based on Recycling Krylov Subspaces. 6.3 Application to Model Order Reduction. 6.4 Simulation Results. 6.5 Conclusions. References. 7 Data-driven Parameterized Model Order Reduction Using z-domain Multivariate Orthonormal Vector Fitting Technique Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom Dhaene. 7.1 Introduction. 7.2 Background. 7.3 Parametric Macromodeling. 7.4 Choice of basis functions. 7.5 Example: Double folded stub microstrip bandstop filter. 7.6 Conclusions. References. 8 Network Reduction by Inductance Elimination M.M. Gourary, S.G.Rusakov, S.L.Ulyanov, and M.M.Zharov. 8.1 Introduction. 8.2 Elimination of RC-node by TICER. 8.3 Inductance Elimination. 8.4 Elimination of Coupled Inductances. 8.5 Eliminations under LC Couplings. 8.6 Algorithmic Aspects. 8.7 Numerical Examples. 8.8 Conclusion. References. 9 Simulation of coupled oscillators using nonlinear phase macromodels and model order reduction Davit Harutyunyan and Joost Rommes. 9.1 Introduction. 9.2 Phase noise analysis of oscillators. 9.3 Oscillator coupled to a balun. 9.4 Oscillator coupling to a transmission line. 9.5 Model order reduction. 9.6 Numerical experiments. 9.7 Conclusion. References. 10 POD Model Order Reduction of Drift-Diffusion Equations in Electrical Networks Michael Hinze, Martin Kunkel and Morten Vierling. 10.1 Introduction. 10.2 Complete coupled system. 10.3 Simulation of the full system. 10.4 Model reduction. 10.5 Numerical investigation. Appendix: Proper Orthogonal Decomposition. References. 11 Model Reduction of Periodic Descriptor Systems Using Balanced Truncation Peter Benner, Mohammad-Sahadet Hossain and Tatjana Stykel. 11.1 Introduction. 11.2 Periodic Descriptor Systems. 11.3 Periodic Gramians and Matrix Equations. 11.4 Balanced Truncation Model Reduction. 11.5 Example. 11.6 Conclusion. References. 12 On synthesis of reduced order models Roxana Ionutiu and Joost Rommes. 12.1 Introduction. 12.2 Foster synthesis of rational transfer functions. 12.3 Structure preservation and synthesis by unstamping. 12.4 Numerical examples. 12.5 Conclusions and outlook. References. 13 Model Reduction Methods for Linear Network Models of Distributed Systems with Sources Stefan Ludwig and Wolfgang Mathis. 13.1 Introduction. 13.2 Background for Model Reduction of Linear Networks. 13.3 Description of distributed sources. 13.4 Examples. 13.5 Conclusion. References. 14 Structure preserving port-Hamiltonian model reduction of electrical circuits R.V. Polyuga and A.J. van der Schaft. 14.1 Introduction. 14.2 Linear port-Hamiltonian systems. 14.3 The Kalman decomposition of port-Hamiltonian systems. 14.4 The co-energy variable representation. 14.5 Balancing for port-Hamiltonian systems. 14.6 Reduction of port-Hamiltonian systems in the general case. 14.7 Example. 14.8 Conclusions. Appendix. References. 15 Coupling of numerical and symbolic techniques for model order reduction in circuit design Oliver Schmidt Thomas Halfmann Patrick Lang. 15.1 Motivation. 15.2 Symbolic Techniques. 15.3 Hierarchical systems. 15.4 Workflow for the exploitation of the hierarchy. 15.5 Comparison to other approaches. 15.6 Summary and future work. References. 16 On Stability, Passivity and Reciprocity Preservation of ESVDMOR Peter Benner and Andre Schneider. 16.1 Introduction. 16.2 The Extended SVDMOR Approach. 16.3 Stability, Passivity, and Reciprocity. 16.4 Remarks and Outlook. References. 17 Model order reduction of nonlinear systems in circuit simulation: status and applications Michael Striebel and Joost Rommes. 17.1 Introduction. 17.2 Linear versus nonlinear model order reduction. 17.3 Some nonlinear MOR techniques. 17.4 TPWL and POD. 17.5 Numerical examples. 17.6 Discussion and outlook. References. 18 An Approach to Nonlinear Balancing and MOR Erik I. Verriest. 18.1 Static versus Dynamic Approximation. 18.2 Gramians for Linear Systems and Applications. 18.3 Metric Properties of Balanced Truncation. 18.4 Nonlinear Model Reduction. References.

Krylov-Based Model Order Reduction of Time-delay Systems

Abstract: We present a model order reduction method which allows the construction of a reduced, delay-free model of a given dimension for linear time-delay systems, whose characteristic matrix is nonlinear due to the presence of exponential functions. The method builds on the equivalent representation of the time-delay system as an infinite-dimensional linear problem. It combines ideas from a finite-dimensional approximation via a spectral discretization, on the one hand, and a Krylov-Pade model reduction approach, on the other hand. The method exhibits a good spectral approximation of the original model, in the sense that the smallest characteristic roots are well approximated and the nonconverged eigenvalues of the reduced model have a favorable location, and it preserves moments at zero and at infinity. The spectral approximation is due to an underlying Arnoldi process that relies on building an appropriate Krylov space for the linear infinite-dimensional problem. The preservation of moments is guaranteed, because the chosen finite-dimensional approximation preserves moments and, in addition, the space on which one projects is constructed in such a way that the preservation of moments carries over to the reduced model. The implementation of the method is dynamic, since the number of grid points in the spectral discretization does not need to be chosen beforehand and the accuracy of the reduced model can always be improved by doing more iterations. It relies on a reformulation of the problem involving a companion-like system matrix and a highly structured input matrix, whose structure are fully exploited.

Local/global model order reduction strategy for the simulation of quasi-brittle fracture

Abstract: This paper proposes a novel technique to reduce the computational burden associated with the simulation of localised failure. The proposed methodology affords the simulation of damage initiation and propagation whilst concentrating the computational effort where it is most needed, i.e. in the localisation zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical Proper Orthogonal Decomposition, while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly non-linear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators.

Simulation process of flexible multibody systems with non-modal model order reduction techniques

Abstract: One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In this work, nonmodal model reduction techniques for flexible multibody systems within the floating frame of reference framework are considered. While traditionally in the multibody community modal techniques in many different forms are used, here other methods from system dynamics and mathematics are in the focus. For the reduction process, finite element data and user inputs are necessary. Prior to the reduction process, the user first needs to choose boundary conditions fitting the chosen reference frame before defining the appropriate in- and outputs. In this work, four different possibilities of modeling appropriate interface points to reduce the number of inputs and outputs are presented.

Reduction Method for Finite Element Nonlinear Dynamic Analysis of Shells

Abstract: DOI: 10.2514/1.J051003Theprojectionoftheequationsofmotiononasuitablemodalbasiscandrasticallyreducethenumberofdegreesoffreedom of a nonlinear dynamic !nite element analysis. A reduction method based on enriching a set of vibrationmodeswithsecond-ordermodescalculatedviaaperturbationtechniqueispresented.Thesesecond-ordermodesarereadily calculated via the solution of corresponding linear problems. The results indicate that the second-ordermodes are displacement !elds that are essential for an appropriate representation of the nonlinear behavior of thestructure. Problems exhibiting strong geometrical nonlinearities under relatively high dynamic loads aresuccessfully handled by forming the reduction basis with vibration modes and their corresponding second-ordermodes, calculated at two different static equilibrium con!gurations.

Parameter Preserving Model Order Reduction for MEMS Applications

Abstract: Model order reduction techniques are known to work reliably for finite element-type simulations of micro-electro-mechanical systems devices. These techniques can tremendously shorten computational times for transient and harmonic analyses. However, standard model reduction techniques cannot be applied if the equation system incorporates time-varying matrices or parameters that are to be preserved for the reduced model. However, design cycles often involve parameter modification, which should remain possible also in the reduced model. In this article we demonstrate a novel parameterization method to numerically construct highly accurate parametric ordinary differential equation systems based on a small number of systems with different parameter settings. This method is demonstrated to parameterize the geometry of a model of a micro-gyroscope, where the relative error introduced by the parameterization lies in the region of . We also present recent developments on semi-automatic order reduction methods that...

Efficient Balancing-Based MOR for Large-Scale Second-Order Systems

Abstract: Large-scale structure dynamics models arise in all areas where vibrational analysis is performed, ranging from control of machine tools to microsystems simulation To reduce computational and resource demands and be able to compute solutions and controls in acceptable, that is, applicable, time frames, model order reduction (MOR) is applied Classically modal truncation is used for this task The reduced-order models (ROMs) generated are often relatively large and often need manual modification by the addition of certain technically motivated modes That means they are at least partially heuristic and cannot be generated fully automatic Here, we will consider the application of fully automatic balancing-based MOR techniques Our main focus will be on presenting a way to efficiently compute the ROM exploiting the sparsity and second-order structure of the finite element method (FEM) semi-discretization, following a reduction technique originally presented in [V Chahlaoui, KA Gallivan, A Vandendorpe, a

Multipoint Full-Wave Model Order Reduction for Delayed PEEC Models With Large Delays

Abstract: The increase of operating frequencies requires 3-D electromagnetic (EM) 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 EM methods and model order reduction (MOR) techniques are used to reduce such a high complexity. When signal waveform rise times decrease and the corresponding frequency content increases, or the geometric dimensions become electrically large, time delays must be included in the modeling. A PEEC formulation, which include delay elements called τ PEEC method, becomes necessary and leads to systems of neutral delayed differential equations (NDDE). The reduction of large NDDE is still a very challenging research topic, especially for electrically large structures, where delays among coupled elements cannot be neglected or easily approximated by rational basis functions. We propose a novel model order technique for τ PEEC models that is able to accurately reduce electrically large systems with large delays. It is based on an adaptive multipoint expansion and MOR of equivalent first-order systems. The neutral delayed differential formulation is preserved in the reduced model. Pertinent numerical examples based on τ PEEC models validate the proposed MOR approach.

Adaptive Reduced Basis Methods for Nonlinear Convection–Diffusion Equations

Abstract: Many applications from science and engineering are based on parametrized evolution equations and depend on time-consuming parameter studies or need to ensure critical constraints on the simulation time. For both settings, model order reduction by the reduced basis methods is a suitable means to reduce computational time. In this proceedings, we show the applicability of the reduced basis framework to a finite volume scheme of a parametrized and highly nonlinear convection-diffusion problem with discontinuous solutions. The complexity of the problem setting requires the use of several new techniques like parametrized empirical operator interpolation, efficient a posteriori error estimation and adaptive generation of reduced data. The latter is usually realized by an adaptive search for base functions in the parameter space. Common methods and effects are shortly revised in this presentation and supplemented by the analysis of a new strategy to adaptively search in the time domain for empirical interpolation data.

Physics-Based Passivity-Preserving Parameterized Model Order Reduction for PEEC Circuit Analysis

Abstract: 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.

SparseRC: Sparsity Preserving Model Reduction for RC Circuits With Many Terminals

Abstract: A novel model order reduction (MOR) method, SparseRC, for multiterminal RC circuits is proposed. Specifically tailored to systems with many terminals, SparseRC employs graph-partitioning and fill-in reducing orderings to improve sparsity during model reduction, while maintaining accuracy via moment matching. The reduced models are easily converted to their circuit representation. These contain much fewer nodes and circuit elements than otherwise obtained with conventional MOR techniques, allowing faster simulations at little accuracy loss.

Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos

Abstract: Structure preserving model reduction of single-input single-output port-Hamiltonian systems is considered by employing the rational Krylov methods. The rational Arnoldi method is shown to preserve (for the reduced order model) not only a specific number of the moments at an arbitrary point in the complex plane but also the port-Hamiltonian structure. Furthermore, it is shown how the rational Lanczos method applied to a subclass of port-Hamiltonian systems, characterized by an algebraic condition, preserves the port-Hamiltonian structure. In fact, for the same subclass of port-Hamiltonian systems the rational Arnoldi method and the rational Lanczos method turn out to be equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function.

Optimal second order reduction basis selection for nonlinear transient analysis

Abstract: Effective Model Order Reduction (MOR) for geometrically nonlinear structural dynamics problems can be achieved by projecting the Finite Element (FE) equations on a basis constituted by a set of vibration modes and associated second order modal derivatives. However, the number of modal derivatives gener- ated by such approach is quadratic with respect to the number of chosen vibration modes, thus quickly making the dimension of the reduction basis large. We show that the selection of the most important second order modes can be based on the convergence of the underlying linear modal truncation approximation. Given a cer- tain time dependency of the load, this method allows to select the most significant modal derivatives set before computing it.

Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems

Abstract: In this article, we discuss a model order reduction method for multiple-input and multiple-output discrete-time bilinear control systems. Similar to the continuous-time case, we will show that a system can be characterised by a series of generalised transfer functions. This will be achieved by a multivariate Z-transform of kernels corresponding to an explicit solution formula for discrete-time systems. We will further address the problem of generalised tangential interpolation which naturally comes along with this approach. We will introduce a reasonable generalisation of the linear ℋ2-norm. Based on this concept, we discuss the choice of interpolation points. Furthermore, an efficient discretisation of continuous-time systems is provided. The performance of the proposed method is evaluated in some numerical examples and compared with the method of balanced truncation for bilinear systems.

SparseRC : sparsity preserving model reduction for RC circuits with many terminals

Abstract: A novel model order reduction (MOR) method for multi-terminal RC circuits is proposed: SparseRC. Specifically tailored to systems with many terminals, SparseRC employs graph-partitioning and fill-in reducing orderings to improve sparsity during model reduction, while maintaining accuracy via moment matching. The reduced models are easily converted to their circuit representation. These contain much fewer nodes and circuit elements than otherwise obtained with conventional MOR techniques, allowing faster simulations at little accuracy loss.

Fast Frequency Sweep of FEM Models via the Balanced Truncation Proper Orthogonal Decomposition

Abstract: A fast frequency sweep method for wideband antennas and infinite arrays based on a singular value decomposition (SVD)-Krylov model reduction method for frequency-domain tangential vector finite elements (TVFEMs) is presented. Reduced models are constructed using balanced congruence transformations constructed from the dominant invariant subspace of the system's Hankel matrix. Traditionally, forming such matrix requires the intensive computation of Gramians; the proposed method only forms their low-rank Cholesky factors via a novel adaptive proper orthogonal decomposition (POD) sampling strategy, leading to significant savings. Unlike some other model reduction methods, balanced truncation POD (BT-POD) is directly applicable to lossy and dispersive electromagnetic models. Numerical studies on large-scale wideband antennas and infinite arrays show that the method is stable, error controllable and, without memory overheads capable of up to two orders-of-magnitude speed-ups.

Model Order Reduction of Nonlinear Dynamical Systems

Abstract: Author(s): Gu, Chenjie | Advisor(s): Roychowdhury, Jaijeet | Abstract: Higher-level representations (macromodels, reduced-order models) abstract away unnecessary implementation details and model only important system properties such as functionality. This methodology -- well-developed for linear systems and digital (Boolean) circuits -- is not mature for general nonlinear systems (such as analog/mixed-signal circuits). Questions arise regarding abstracting/macromodeling nonlinear dynamical systems: What are ``important'' system properties to preserve in the macromodel? What is the appropriate representation of the macromodel? What is the general algorithmic framework to develop a macromodel? How to automatically derive a macromodel from a white-box/black-box model? This dissertation presents techniques for solving the problem of macromodeling nonlinear dynamical systems by trying to answer these questions. We formulate the nonlinear model order reduction problem as an optimization problem and present a general nonlinear projection framework that encompasses previous linear projection-based techniques as well as the techniques developed in this dissertation. We illustrate that nonlinear projection is natural and appropriate for reducing nonlinear systems, and can achieve more compact and accurate reduced models than linear projection.The first method, ManiMOR, is a direct implementation of the nonlinear projection framework. It generates a nonlinear reduced model by projection on a general-purpose nonlinear manifold. The proposed manifold can be proven to capture important system dynamics such as DC and AC responses. We develop numerical methods that alleviates the computational cost of the reduced model which is otherwise too expensive to make the reduced order model of any value compared to the full model.The second method, QLMOR, transforms the full model to a canonical QLDAE representation and performs Volterra analysis to derive a reduced model. We develop an algorithm that can mechanically transform a set of nonlinear differential equations to another set of equivalent nonlinear differential equations that involve only quadratic terms of state variables, and therefore it avoids any problem brought by previous Taylor-expansion-based methods. With the QLDAE representation, we develop the corresponding model order reduction algorithm that extends and generalizes previously-developed Volterra-based technique.The third method, NTIM, derives a macromodel that specifically captures timing/phase responses of a nonlinear system. We rigorously define the phase response for a non-autonomous system,and derive the dynamics of the phase response. The macromodel emerges as a scalar, nonlinear time-varying differential equation that can be computed by performing Floquet analysis of the full model. With the theory developed, we also present efficient numerical methods to compute the macromodel.The fourth method, DAE2FSM, considers a slightly different problem -- finite state machine abstraction of continuous dynamical systems. We present an algorithm that learns a Mealy machine from a set of differential equations from its input-output trajectories. The algorithm explores the state space in a smart way so that it can identify the underlying finite state machine using very few information about input-output trajectories.

Reduced-Order Transfer Matrices From RLC Network Descriptor Models of Electric Power Grids

Abstract: This paper compares the computational performances of four model order reduction methods applied to large-scale electric power RLC networks transfer functions with many resonant peaks. Two of these methods require the state-space or descriptor model of the system, while the third requires only its frequency response data. The fourth method is proposed in this paper, being a combination of two of the previous methods. The methods were assessed for their ability to reduce eight test systems, either of the single-input single-output (SISO) or multiple-input multiple-output (MIMO) type. The results indicate that the reduced models obtained, of much smaller dimension, reproduce the dynamic behaviors of the original test systems over an ample range of frequencies with high accuracy.

PartMOR: Partitioning-Based Realizable Model-Order Reduction Method for RLC Circuits

Abstract: This paper presents a robust partitioning-based model-order reduction (MOR) method, PartMOR, suitable for reduction of very large RLC circuits or RLC-circuit parts of a non-RLC circuit. The MOR is carried out on a partitioned circuit, which enables the use of low-order moments and macromodels of few elements, while still preserving good accuracy for the reduction. As the method produces a positive-valued, passive, and stable reduced-order RLC circuit (netlist-in-netlist-out), it can be used in conjunction with any standard analysis tool or circuit simulator without modification. It is shown that PartMOR achieves excellent reduction results in terms of accuracy and reduced CPU time for RLC, RC, and RL circuits.