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Autumn School on Future Developments in Model Order Reduction

September 21-25, 2009, Terschelling, The Netherlands

Abstracts

Thanos Antoulas (Rice University, Houston, TX, USA; Jakobs University, Bremen, Germany)

MOR: an overview of recent results in Krylov reduction methods and some new results on parametric MOR

In this talk we will first review some recent results in model reduction by means of Krylov methods. In particular, we will summarize results concerning H2-optimal and passivity preservation model reduction. In the second part we will address the issue of model reduction of systems which depend on various parameters. The main tool in our approach is the Loewner matrix pencil.

Zhaojun Bai (University of California at Davis, CA, USA)

Structure-preserving Model Order Reduction

We begin with an introductory review of Krylov subspace methods and a unified theory for structure-preserving model reduction of linear systems. We then discuss applications in interconnect circuit analysis and structure dynamics. We will discuss why structure-preservation is still not be fully acceptable and mention some open problems.

Tamara Bechtold (NXP Semiconductors, Eindhoven)

Identification of Material Properties of MEMS Devices via Parametric Model Order Reduction and Optimization

Abstract

Rudy Eid (TU Muenchen)

Interpolation-based Parametric Model Reduction using Krylov Subspaces

Abstract

Francisco Damasceno Freitas (University of Brasilia, Brazil)

Computation of Multi-bus Reduced Order Models of RLC Networks Having a Large Number of Resonant Peaks

Abstract

Martin Gutknecht (ETH Zuerich)

Block and Band Lanczos Algorithms: a Review of Options

Since the early work of Cullum and Donath (1974) and Underwood and Golub (1975, 1977), there has been, from time to time, a renewed interest in conjugate gradient methods for systems with multiple-right hand sides and in Lanzos methods with several starting vectors for solving linear systems, computing eigenvalues, or, more recently, system identification and model reduction.

While the formal definition of block versions of the Lanczos algorithm is quite easy --- at least when there are equally many right and left starting vectors (i.e., inputs and outputs) --- the true difficulties come when a dimension reduction of block Krylov space (so-called deflation) has to be treated properly, or, in the nonsymmetric case, when the block Lanczos process breaks down. Deflation for symmetric systems of equations was treated by Nikishin and Yeremin (1995) (and, to some extent, by others before). For the nonsymmetric case, the answer to both difficulties lies in an adjustment of the look-ahead Lanczos algorithm for SISO systems. The resulting nonsymmetric look-ahead (or cluster) band Lanczos algorithm has been developed in 1994--1996 in partly independent and partly cooperative efforts by Aliaga, Boley, Feldmann, Freund, Hernandez, Malhorta, and even others. The work culminates in the paper by Aliaga, Boley, Freund, and Hernandez in Math. Comp. (1999/2000, submitted 1996).

However, the detailed specification of a nonsymmetric block Lanczos algorithm (for MIMO systems) requires to choose among many options, and the solution chosen by the mentioned authors is far from optimal regarding its stability. There are trade-offs between cost, complexity, and stability. The purpose of this talk is to discuss these issues. We will in particular present the solution chosen in the dissertation of my student Damian Loher, which proved far more stable than previous ones.

Daniel Ioan (Politehnica University of Bucharest, Romania; COMSON)

Parametric Reduced Order Models for Passive Integrated Components Coupled with their EM Environment

The lecture focuses on the implementation and use of a new type of boundary conditions - Electro-Magnetic Hooks - in the Finite Integration Technique applied for the simulation of passive devices in full-wave electromagnetic field regime. This type of boundary conditions offers a new and effective possibility to carry out domain decomposition for the simulation of very large scale problems, such as complex on-chip passive devices, and to extract their parametric reduced order models.
[Extended Abstract]

Martin Kunkel (University of Hamburg)

Residual based POD Model Order Reduction of Drift Diffusion Equations in parametrized Electrical Networks

Abstract

Stefan Ludwig (Universität Hannover)

Model reduction of linear networks with embedded nonlinear time-variant elements

Abstract

Karl Meerbergen (KU Leuven, Belgium)

Krylov methods for the solution of parameterized linear systems in the simulation of structures and vibrations: theory, applications and challenges

Linear systems with parameters arise in many applications. For exam- ple, the computation of the frequency response function (FRF) of vibrat- ing systems leads to a sequence of large linear systems with the frequency as parameter. Therefore, the computational cost of FRF's is high.
In this lecture, we review techniques based on Krylov subspaces, such as the Padé-via-Lanczos method (PVL) and the Arnoldi method. This class of methods have connections with Padé approximations, eigenvalue computations and iterative Krylov methods. The characterization based on eigenvalues helps us understand these methods and allows us to derive simple rules to understand their convergence behaviour. These methods appear to be extremely efficient when the system matrix is a linear or polynomial function of the parameter.
Special structure in the matrices may improve the computational speed. We discuss the efficient solution of systems with Rayleigh damping. We can solve large scale complex valued linear systems by a real-valued Krylov space.
We also discuss recent results on the solution of linear systems with multiple right-hand sides. The idea here is to recycle vectors from the solution with one right-hand side to solve the remaining right-hand side: we save computation time and memory.
The frequency is the most important parameter of linear systems aris- ing from vibrations. In applications, other physical quantities, such as plate thickness, or damping material properties are often additional pa- rameters. This problem is also known as parameterized modelreduction. It is a new challenging problem. We will review a few applications of this form and present numerical results.

Volker Mehrmann (TU Berlin, Germany)

Model reduction in acoustic problems

We will discuss the model reduction methods in the solution of acoustic field problems. In current industrial applications frequency response problems and second order eigenvalue problems of orders up to 10 million degrees of freedom have to be solved within an optimization loop. Based on geometric, topological or material changes the acoustic field within modern cars is then optimized on the basis of model based simulations. We will discuss the currently used methods and their properties from a numerical and computational point of view. It turns out that these methods have many deficiencies that will be analyzed described in detail.

Samuel Melchior (UC de Louvain)

A multimesh approach for H2 model reduction

Abstract

Timo Reis (Rice University, Houston, TX, USA; TU Berlin, Germany)

Passivity-preserving Balanced Truncation for Electrical Circuits

We first give an introduction to modeling of electrical circuits. Then the following problem is considered: Given a large scale electrical circuit and approximate this by a small one. The problem will be tackled as follows: First, the circuit equations will be approximated by the model reduction method of passivity-preserving model reduction for descriptor systems. Then the reduced-order model will be backinterpreted as an electrical circuit.

Rostyslav Polyuga (University of Groningen)

Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity

Model reduction of port-Hamiltonian systems by means of the Krylov methods is considered, aiming at port-Hamiltonian structure preservation. It is shown how to employ the Arnoldi method for model reduction in a particular coordinate system in order not only to preserve a specific number of the Markov parameters but also the port-Hamiltonian structure for the reduced order model. Furthermore it is shown how the Lanczos method can be applied in a structure-preserving manner to a subclass of port-Hamiltonian systems which is characterized by an algebraic condition. In fact, for the same subclass of port-Hamiltonian systems the Arnoldi method and the Lanczos method are equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function.

Michal Rewienski (Synopsys Inc, Mountain View, CA, USA)

Reduced Order Piecewise-Linear Approximation for Efficient Modeling of Nonlinear Systems: Promises, Challenges and more to explore

This lecture will review model order reduction (MOR) methodology for nonlinear systems based on trajectory piecewise-linear (TPWL) approach, including its key features, as well as more recent advances related to selection of linearization points, and stabilizing projections. Also, biggest challenges of the method, e.g. robustness, accuracy control, will be described. The presentation will include discussion of application areas where TPWL approach may be most successful, as well as more challenging high-impact areas (e.g. within transient circuit simulation) where a number of alternative efficient macromodeling strategies exist. Finally, we will take a critical look at the goals of MOR, and attempt to define a few different viewpoints on the problem of nonlinear reduced order approximation which may lead to some unexplored paths.

Daniel Rixen (TU Delft)

Modal Truncation Augmentation and Moment Matching: two faces of the same coin

Modal Truncation augmentation and generalized mode acceleration corrections are techniques proposed in the field of reduction for structural models. In fact those concepts uses the idea of "Load Dependent Vectors" also called Wilson-Ritz vectors proposed more than 25 years ago. In summary the modal truncation augmentation uses the Krylov vectors generated starting from static solutions in order to enrich standard eigenmode basis. Hence it is clear that the ingredients of those methods are very similar to the reduction basis used in moment matching approaches. This will be outlined in the first part of the presentation. In the second part of the presentation we will show how the modal truncation augmentation can be efficiently used in substructuring techniques. We will show that, when combined with reduction of the interface dofs of the substructures, the addition of truncation correction modes can efficiently improve the accuracy of the reduced modeled for the higher frequency range.

Joost Rommes (NXP Semiconductors, Eindhoven, The Netherlands; O-MOORE-NICE!)

Eigenvalue problems and model order reduction in electronics industry

Physical structures and processes are modeled by dynamical systems in a wide range of application areas. During the design of very large-scale integration (VLSI) chips, for instance, dynamical systems are used to describe the low-level circuit behavior. Since these dynamical systems can become very large for modern chips, the essential simulation before production may consume hours or days of computing time. Hence there is need for efficient mathematical approaches that limit the computing time while preserving the accuracy. Model order reduction (MOR) is often seen as a natural way to deal with this, but a closer look at practical examples learns that in some cases MOR is not applicable (due to required output) or not needed (due to existence of other techniques). In this lecture we will consider several examples from electronics industry, describe what is needed for fast and accurate simulation, and explain various techniques, including linear solvers, eigensolvers, concepts from graph theory and combinatorics, and MOR.

Janne Roos (Helsinki University of Technology, Finland)

PRIOR: Passive, Reciprocal, and Infinity-Observing Reduction

In this talk, we introduce a new Krylov-subspace MOR method, PRIOR, which combines many features of the MOR methods PRIMA, SPRIM, ENOR, and SAPOR with some fresh new ideas. PRIOR preserves passivity and reciprocity of the original RLC-circuit block, and matches, just like SPRIM, twice as many moments as PRIMA. In addition to the implicit matching of the z-parameter moments of Z(s)=M0+M1*s+M2*s^2+... at s=0 (M0 giving the exact DC solution corresponding to t=oo), PRIOR can simultaneously match the moments of Z(1/s)=N0+N1*(1/s)+N2*(1/s)^2+... at s=oo (N0 giving the exact time response at t=0). That is, PRIOR has asymptotically correct frequency-domain and time-domain behavior. Moreover, since PRIOR forms the equations in the second-order form, the resulting reduced-order model can be synthesized as an RLC netlist using the RLCSYN method. Simulation examples are shown to demonstrate the desired effect of simultaneous moment matching at s=0 and s=oo.

Patricio Rosen (TU Eindhoven)

Factorization of indefinite systems associated with RLC circuits

In this presentation we deal with large systems coming from linear RLC circuits. One of the characteristics of these systems is that in the general case they are very large, indefinite and non symmetric, making their solution no-trivial. Based on the Schilders factorization we construct factorizations of the circuit equations, for RL and RLC circuits. Furthermore we study the application of these factorizations for implementing direct solvers. The time complexity of such algorithms is analyzed and compared to the time complexity of standard direct solvers. This is done analytically and trough some study cases.

Jacquelien Scherpen (University of Groningen, The Netherlands)

Dissipativity preserving balanced truncation and port-Hamiltonian structure preserving exact order reduction

In the first part of the lecture a general framework of dissipativity preserving balanced truncation methods will be treated that includes positive- and bounded real balanced truncation methods. Other types of balanced truncation, such as LQG balancing, and their fit into the framework will be discussed as well. The methods are treated for a class of continuous time nonlinear input affine systems, and the linear case will be treated as a special case. In the second part of the lecture additional structure is added, i.e., port-Hamiltonian systems are considered. For these type of systems a way to reduce a non-minimal port-Hamiltonian system to a minimial port-Hamiltonian system is treated. The potential application and extension of these methods to approximate reduction methods while preserving the port-Hamiltonian structure will be discussed.

Oliver Schmidt (Fraunhofer Institut, Kaiserslautern)

Numerical and symbolic MOR techniques using hierarchical circuit structure

Besides purely numerical model order reduction methods there also exist symbolic ones, which are indeed costly to compute, but particularly for nonlinear DAEs they additionally allow a deeper insight into the functional relations between the circuit?s components. Some symbolic methods for model order reduction will be presented.

However, the development during the last years shows that model descriptions based on DAEs for certain applications cannot model the occuring physical effects accurately enough. Thus, supplementary model descriptions based on PDEs are necessary. For discretized PDEs, dedicated numerical MOR techniques, e.g. reduction via the Arnoldi algorithm, already exist.

Further, by standard graph theoretical methods like, e.g., modified nodal analysis for transforming the circuit netlist into an equation system, the structure information of the in general hierarchical system is lost. Therefore a modelling approach is aspired that transmits this information into the set of equations. Thus, subsystems of the circuit can be reduced separately either with numerical or symbolic methods, according to their complexity level, and finally be combined to a reduced overall system. This approach intuitively facilitates the overall circuit?s analysis.

An appropriate workflow that permits the exploitation of the circuit?s hierarchical structure will be presented. Starting with a hierarchical netlist of interacting subsystems, their sensitivities within the connecting structure will be investigated. For this and since the already existing symbolic methods implemented in Analog Insydes cannot be applied directly, a technique to set up the subsystems for the use of these methods has been developed and will be presented. According to the sensitivity analysis appropriate error bounds for the separate reduction of the subsystems will be chosen to finally reduce the entire circuit. For the subsystem reduction adequate numerical or symbolic reduction methods can be chosen according to the subsystems? complexity levels.

Finally, all the presented methods will be applied to some examples and compared to the direct approach, i.e. the reduction of the entire circuit examples not exploiting the hierarchical structure.

Luis Miguel Silveira (INESC ID / Cadence Research Labs / IST - TU Lisbon, Lisbon, Portugal)

Algorithms for Structure Preserving, Parameterized Model Order Reduction

In this talk we review algorithms for model order reduction of parameterized systems. We discuss several different approaches in terms of their flexibility and efficiency. Issues such as structure preservation, parameter dependence and model evaluation are discussed and analyzed. Special attention is devoted to sampling based techniques for model generation. We show how these may allow better accuracy control, enable independent adaptive order determination with respect to each parameter. In this context, the issue of efficient sample selection is also discussed and several techniques will be presented and compared.

Alexander Steenhoek (TU Delft)

Model Order Reduction for Coupled Problems

The demand for model order reduction is generally driven by the complexity and the large scale of models. In several fields of engineering, for example in the dynamic design of MicroSystems, this complexity is induced by the multiphysical background of models. For the separate physical fields solution techniques and reduction bases are well-known, but this does not necessarily give a good basis for the coupled problem as well. This was especially observed for the cross-coupled transient response or transfer functions, whereas this cross-coupling of physics is often the underlying working principle of MicroSystem designs. We can present an investigation to reduction bases for the coupled problem that start from bases for the uncoupled phsyical fields; a correction on the uncoupled bases is obtained from a first order perturbation and is proposed to improve on the representation of the coupling effect. Under certain conditions for the frequency spectra of the uncoupled physical fields we can give a physical interpretation of the obtained correction.

Fatih Yetkin (Istanbul Technical University)

Parallel Model Order Reduction via Matlab Distributed Toolbox

Abstract

Yao Yue (KU Leuven)

Optimization of large-scale systems with model reduction

Numerical parameter studies of acoustic problems arising from applications such as airplane engines and insulation panels along motor ways or in houses are usually extremely expensive,since for each parameter value, an entire frequency response function needs to be computed. The computation of a single frequency response function for fixed parameter values is by itself already quite expensive. Parameter studies are often carried out in order to choose the "optimal" values of the parameters. The computational cost for the frequency response function has been dramatically reduced by a factor of ten or more by using model order reduction methods. However, little work has been done to introduce MOR in optimization. Our research group has recently started research on exploiting the properties of model reduction methods in optimization problems arising from acoustics. This talk presents our first results. We have tried to use two MOR methods in optimization: SOAR and PIMTAP. SOAR does MOR on one parameter, while PIMTAP does MOR on multiple parameters. We first compare SOAR and PIMTAP in accuracy and performance. Then, we analyze the feasibility of derivative computation via reduced model and how to combine MOR with optimization algorithms. Numerical results show that using MOR in optimization could drastically reduce the optimization time.