## Krylov Subspace Techniques for Reduced-Order Modeling of Nonlinear Dynamical Systems (2002)

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Venue: | Appl. Numer. Math |

Citations: | 59 - 4 self |

### BibTeX

@ARTICLE{Bai02krylovsubspace,

author = {Zhaojun Bai and Daniel Skoogh},

title = {Krylov Subspace Techniques for Reduced-Order Modeling of Nonlinear Dynamical Systems},

journal = {Appl. Numer. Math},

year = {2002},

volume = {43},

pages = {9--44}

}

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### Abstract

Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of large-scale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bi-linearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reduced-order bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the Volterra-Wiener representation of the bilinear system. It is shown that the two-sided Krylov subspace technique matches significant more number of multimoments than the corresponding one-side technique.

### Citations

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Asymptotic Waveform Evaluation for Timing Analysis
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Citation Context ...gular. We will assume that this is the case for all n. This formulates the framework of the asymptotic waveform evaluation (AWE) techniques as they are known in circuit simulation, first presented in =-=[70]-=- around 1990. The manuscript [20] has a complete treatment of the AWE technique and its variants. A survey of the Padé techniques for model reduction of linear systems is also presented in the earlier... |

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Citation Context ...d SyMPVL is explained in [29]. All moment-matching methods involve local approximations in nature. There is a class of global approximation methods, mostly based on the theory of balanced realization =-=[59]-=-. They sometimes are also called Gramian-based model reduction methods, or SVD-based model reduction methods [86,3]. The crux of this class of methods for applying to large-scale linear dynamical syst... |

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Citation Context ...rylov subspaces, specifically, Kn(A, r) = span{v1, v2,...,vn} and Kn(A T , l) = span{w1, w2,...,wn }. It is well-known that the Lanczos process is an elegant way to generate the desired basis vectors =-=[53]-=-. Given a matrix A, a right starting vector r and a left starting vector l, the Lanczos process generates the desired basis vectors {vi} and {wi}, known as the Lanczos vectors. Moreover, these Lanczos... |

341 |
Prima: passive reduced-order interconnect macromodeling algorithm
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Citation Context ...a rational Krylov subspace based method was proposed. For RLC, PRIMA is also widely accepted in the circuit simulation community, which combines the Arnoldi process and a direct orthogonal projection =-=[63]-=-. The connection between PRIMA and SyMPVL is explained in [29]. All moment-matching methods involve local approximations in nature. There is a class of global approximation methods, mostly based on th... |

232 |
Nonlinear Systems: Analysis, Stability and Control
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Citation Context ...proach is based on the Carleman bilinearization of a nonlinear system. The following one-sided Krylov method is similar to the method presented in [10]. By Carleman bilinearization (see, for example, =-=[12, 13]), the nonlinear syst-=-em (1.1) can be approximated by a bilinear system given in the following form � ˙�x = A�x � + N� �x u + bu, � �y = �c T (3.4) �x. The Volterra-Wiener representation of the bilinea... |

170 | The quadratic eigenvalue problem
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Citation Context ...e case that only a relatively small number of modal shapes are necessary, i.e., k ≪ N. Mathematical theory and numerical techniques for quadratic eigenvalue problems can be found in the recent survey =-=[85]-=- and references therein.s32 Z. Bai / Applied Numerical Mathematics 43 (2002) 9–44 3.2. Reduced-order modeling based on linearization The reader may notice that the modal superposition method presented... |

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Citation Context ...ularity of the Hankel matrix Mn guarantees that no breakdown occurs, see [64]. In practice, the problem is curable by a variant of the Lanczos process, for example, a look-ahead scheme is proposed in =-=[33]-=-. An implementation of the Lanczos process with a look-ahead scheme to overcome the breakdown can be found in QMRPACK [35]. ⎤sZ. Bai / Applied Numerical Mathematics 43 (2002) 9–44 17 (1) ρ1 =�r�2 (2) ... |

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Citation Context ...tion, it is possible to implement the PVL method with an adaptive stopping criteria to determine the required number of Lanczos iterations, see [10]. Related work for error estimation can be found in =-=[48,42]-=- and recently in [62]. More efficient and accurate error estimations of the PVL approximation and its extension to the other moment-matching based Krylov techniques warrant further study. One alternat... |

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Citation Context ...tputs, respectively. In most practical cases, we can assume that m and p are much smaller than N and m � p. Linear systems arise in many applications, such as the network circuit with linear elements =-=[87]-=-, structural dynamics analysis with only lumped mass and stiffness elements [22,23], linearization of a nonlinear system around an equilibrium point [27], and a semi-discretization with respect to spa... |

125 | Microsystem Design - Senturia - 2001 |

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Citation Context ...proach is based on the Carleman bilinearization of a nonlinear system. The following one-sided Krylov method is similar to the method presented in [10]. By Carleman bilinearization (see, for example, =-=[12, 13]), the nonlinear syst-=-em (1.1) can be approximated by a bilinear system given in the following form � ˙�x = A�x � + N� �x u + bu, � �y = �c T (3.4) �x. The Volterra-Wiener representation of the bilinea... |

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Citation Context ...+ � l T r �� n+1 � �n+1 � (s − s0) 2n + O � (s − s0) 2n+1� . j=2 βj j=2 ρj Therefore, we conclude that Hn(s) is a Padé approximant of H(s). This Lanczos–Padé connection at least goes back to [40] and =-=[41]-=-. The work of [26,37] advocates the use of the Lanczos–Padé connection instead of the mathematical equivalent, but numerically unstable AWE method [70] in the circuit simulation community. The Lanczos... |

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Citation Context ...punov matrix equation of the form AX + XA T =−C, whereAis stable, and C is positive semidefinite. It is known that the solution X is positive semidefinite, namely X can be represented as X = LLT . In =-=[45]-=-, it is shown how to directly compute the Cholesky factor L of X, instead of X. Fig. 9. Accuracy: �Z(s) − Zn(s)� vs. �Z(s) − Z (c) n (s)�.sZ. Bai / Applied Numerical Mathematics 43 (2002) 9–44 29 Anot... |

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(Show Context)
Citation Context ...inconsistency by solving system gramians independently. Besides Krylov subspace based techniques, ADI based methods for solving the underlying Lyapunov equations are also presented in the recent work =-=[67,55]-=-. In contrast to the vast amount of literature on Krylov-subspace based methods for reduced-order modeling, there is little software available in the public domain. To the knowledge of the author, the... |

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- 1995
(Show Context)
Citation Context ...the earliest ones we are aware of. We have mostly discussed the treatment of single-input single-output systems. A generalization of the PVL method for multi-input multi-output systems is reviewed in =-=[31]-=-. Naturally, it is called the MPVL method since it is based on a Lanczos-type process for multiple starting vectors. Besides Lanczos process based methods, Arnoldi process based methods have also been... |

78 |
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Citation Context ...of H(s)lie in C− := {s ∈ C |ℜ(s) < 0} and • if all the poles pj of H(s)on the imaginary axis, ℜ(pj) = 0, are simple. A stable system guarantees a bounded response to a bounded input, see for example, =-=[2]-=-. If the linear dynamical system (1) describes an actual physical system, such as a functioning electronic circuit, then it will necessarily be stable. Note that for the transfer function H(s) given b... |

70 | An implementation of the QMR method based on coupled two-term recurrences
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Citation Context ...ally more accurate as illustrated in Fig. 9. Finally, we remark that, in a different context, the benefit of using a coupled, instead of a non-coupled, Lanczos process was also noted and exploited in =-=[34,44]-=-. The technique of directly computing the factorized form of a solution is also shown in other applications, such as solving a stable and non-negative definite Lyapunov matrix equation of the form AX ... |

68 |
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Citation Context ...l rights reserved. PII: S0168-9274(02)00116-2s10 Z. Bai / Applied Numerical Mathematics 43 (2002) 9–44 methods which preserve exactly a limited number of parameters of the original model. The work of =-=[25]-=- provides a survey of early work on these methods. Over the past several years, Krylov subspace based techniques have emerged as one of the most powerful tools for reduced-order modeling of large-scal... |

65 | Balancing for nonlinear systems
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Citation Context ...r orthogonal decomposition (POD) methods. Methods of balanced truncation extend the success of balanced truncation of linear systems to nonlinear systems. The interested reader is referred to [6] and =-=[14]-=-. The latest work includes [7] and [11]. Means of applying Krylov subspace techniques for adaptively extracting accurate reduced-order models of large-scale nonlinear dynamical systems is a relatively... |

63 | A subspace approach to balanced truncation for model reduction of nonlinear control systems
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Citation Context ...) methods. Methods of balanced truncation extend the success of balanced truncation of linear systems to nonlinear systems. The interested reader is referred to [6] and [14]. The latest work includes =-=[7]-=- and [11]. Means of applying Krylov subspace techniques for adaptively extracting accurate reduced-order models of large-scale nonlinear dynamical systems is a relatively open problem. There has been ... |

61 | Dooren, Asymptotic waveform evaluation via a restarted Lanczos method
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Citation Context ... �n+1 � (s − s0) 2n + O � (s − s0) 2n+1� . j=2 βj j=2 ρj Therefore, we conclude that Hn(s) is a Padé approximant of H(s). This Lanczos–Padé connection at least goes back to [40] and [41]. The work of =-=[26,37]-=- advocates the use of the Lanczos–Padé connection instead of the mathematical equivalent, but numerically unstable AWE method [70] in the circuit simulation community. The Lanczos-based Padé approxima... |

60 | Model reduction of state space systems via an implicitly restarted Lanczos method
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(Show Context)
Citation Context ... the MPVL method since it is based on a Lanczos-type process for multiple starting vectors. Besides Lanczos process based methods, Arnoldi process based methods have also been studied extensively. In =-=[43]-=-, an implicitly restarted Lanczos method was developed. In [42], a rational Krylov subspace based method was proposed. For RLC, PRIMA is also widely accepted in the circuit simulation community, which... |

58 | Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation
- Freund
- 1999
(Show Context)
Citation Context ...subspace based techniques have emerged as one of the most powerful tools for reduced-order modeling of large-scale systems. We would like to call the reader’s attention to recent surveys on the topic =-=[27,29,3]-=-, which are complimentary to this work. In order to introduce first-time readers to this topic, we will begin with a tutorial of Krylov subspace techniques for reduced-order modeling of linear dynamic... |

56 |
Reduction to tridiagonal form and minimal realizations
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Citation Context ...k ≈ 0 considering the finite precision arithmetic) at step 7 in Fig. 1. This is called breakdown. Our assumption of the nonsingularity of the Hankel matrix Mn guarantees that no breakdown occurs, see =-=[64]-=-. In practice, the problem is curable by a variant of the Lanczos process, for example, a look-ahead scheme is proposed in [33]. An implementation of the Lanczos process with a look-ahead scheme to ov... |

48 | Approximation of large-scale dynamical systems: An overview
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(Show Context)
Citation Context ...subspace based techniques have emerged as one of the most powerful tools for reduced-order modeling of large-scale systems. We would like to call the reader’s attention to recent surveys on the topic =-=[27,29,3]-=-, which are complimentary to this work. In order to introduce first-time readers to this topic, we will begin with a tutorial of Krylov subspace techniques for reduced-order modeling of linear dynamic... |

48 | Krylov-subspace methods for reduced-order modeling in circuit simulation
- Freund
(Show Context)
Citation Context ...subspace based techniques have emerged as one of the most powerful tools for reduced-order modeling of large-scale systems. We would like to call the reader’s attention to recent surveys on the topic =-=[27,29,3]-=-, which are complimentary to this work. In order to introduce first-time readers to this topic, we will begin with a tutorial of Krylov subspace techniques for reduced-order modeling of linear dynamic... |

48 |
Matrix interpretations and applications of the continued fraction algorithm
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(Show Context)
Citation Context ...)= Hn(s) + � l T r �� n+1 � �n+1 � (s − s0) 2n + O � (s − s0) 2n+1� . j=2 βj j=2 ρj Therefore, we conclude that Hn(s) is a Padé approximant of H(s). This Lanczos–Padé connection at least goes back to =-=[40]-=- and [41]. The work of [26,37] advocates the use of the Lanczos–Padé connection instead of the mathematical equivalent, but numerically unstable AWE method [70] in the circuit simulation community. Th... |

46 | A review on the inverse of symmetric tridiagonal and block tridiagonal matrices
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Citation Context ...τn1(σ ) and τ1n(σ ) are the (1,n) and (n, 1) elements of the inverse of the tridiagonal matrix I − σ T n. This is in agreement with the rapid decay phenomenon observed in the inverse of a band matrix =-=[58]-=-. Fig. 4 shows typical convergence behavior of the factor |σ 2τn1(σ )τ1n(σ )| for a fixed σ . The direct computation of the second factor γn+1(σ ) would cost just as much as computing the original tra... |

45 |
Efficient linear circuit analysis by Pade approximation via the Lanczos process
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(Show Context)
Citation Context ... �n+1 � (s − s0) 2n + O � (s − s0) 2n+1� . j=2 βj j=2 ρj Therefore, we conclude that Hn(s) is a Padé approximant of H(s). This Lanczos–Padé connection at least goes back to [40] and [41]. The work of =-=[26,37]-=- advocates the use of the Lanczos–Padé connection instead of the mathematical equivalent, but numerically unstable AWE method [70] in the circuit simulation community. The Lanczos-based Padé approxima... |

38 |
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Citation Context ...s) = B T (G + sC) −1 B = �B T� I + (s − s0)A �−1�B, where A = M−1CM−T and �B = M−1B. To exploit the symmetry of the transfer function Z(s), we can use a symmetric band Lanczos process, as proposed in =-=[75,28]-=-. Given A T = A and m starting vectors �B = �Q 1 =[˜q 1 ˜q 2 ... ˜q m ],ann-step symmetric band Lanczos process generates a sequence of linearly independent Lanczos vectors Qn =[q1 q2 ... qn ], such t... |

37 | QMRPACK: a package of QMR algorithms
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(Show Context)
Citation Context ...riant of the Lanczos process, for example, a look-ahead scheme is proposed in [33]. An implementation of the Lanczos process with a look-ahead scheme to overcome the breakdown can be found in QMRPACK =-=[35]-=-. ⎤sZ. Bai / Applied Numerical Mathematics 43 (2002) 9–44 17 (1) ρ1 =�r�2 (2) η1 =�l�2 (3) v 1 = r/ρ 1 (4) w1 = l/η1 (5) for k = 1, 2,...,ndo (6) δk = w T k vk (7) αk = w T k Avk/δk (8) βk = (δk/δ k−1... |

32 | Projection frameworks for model reduction of weakly nonlinear systems
- Phillips
- 2000
(Show Context)
Citation Context ... construction of a Krylov projection subspace. The approach is based on the Carleman bilinearization of a nonlinear system. The following one-sided Krylov method is similar to the method presented in =-=[10]. By Carleman bilin-=-earization (see, for example, [12, 13]), the nonlinear system (1.1) can be approximated by a bilinear system given in the following form � ˙�x = A�x � + N� �x u + bu, � �y = �c T (3... |

32 | Relatively robust representations of symmetric tridiagonals
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(Show Context)
Citation Context ... �Z(s) − Z (c) n (s)�.sZ. Bai / Applied Numerical Mathematics 43 (2002) 9–44 29 Another example is to compute eigenvalues of a tridiagonal matrix T in high relative precision. As shown in recent work =-=[65]-=-, it is better to work with the LDL T form of T , instead of T directly. Those efforts highlight ingenuity in the work of numerical analysis and scientific computing. 2.11. Other reduced-order modelin... |

32 |
Model Reduction and Control of Flexible Structures Using Krylov Vectors
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(Show Context)
Citation Context ...reduced-order model is no longer in a second-order form. For engineering design and control of dynamical systems, it is highly desirable to have a reduced-order model preserving the second-order form =-=[83]-=-. 3.3. Reduced-order modeling based on second-order systems In this section, we discuss a Krylov subspace method which produces a reduced-order model of the second-order form. This is based on the wor... |

29 |
der Vorst (Eds.), Templates for the solution of algebraic eigenvalue problems: A practical guide, Society for Industrial and Applied Mathematics
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Citation Context ...odal shapes is necessary, i.e., k ≪ N. The problem of finding a few modal shapes Sk within a certain frequency range is one of the well-known algebraic eigenvalue problems in numerical linear algebra =-=[4]-=-. 2.2. Reduced-order modeling The desired attributes of reduced-order modeling of the linear dynamical system (1) include replacing the full-order system by a system of the same type but with a much s... |

27 |
Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency
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Citation Context ...that all zeros of H(s) must also in C−. Passivity (and positive realness) is a very important concept in system and control theory. Since the introduction of the concept of positive realness by Brune =-=[16]-=- in 1931, there is a large volume of work concerned with characterizing and testing the positive realness. A history and summary of these works can be found in [2,14] and references therein. It is des... |

26 | Error estimation of the padé approximation of transfer functions via the lanczos process. Electronic transactions on Numerical Analysis
- Bai, Ye
- 1998
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Citation Context ...arious Krylov methods and their applications in model reduction for state-space control models in control system theory is presented in [13]. The presentation style here partially follows the work of =-=[11]-=-. In the following, we present two examples, one from circuit simulation and one from structural dynamics, as empirical validation of the efficiency of the PVL method. We note that in both cases, we o... |

24 | Emerging simulation approaches for micromachined devices,” IEEE transactions on computer-aided design of integrated circuits and systems
- Mukherjee, Fedder, et al.
- 2000
(Show Context)
Citation Context ...onlinear dynamical systems of the form (1.1) include the simulation of time-varying nonlinear circuit elements by independent excitation source [4, 3], and MEMS, such as 1 (1.1)smicro-pressure sensor =-=[8]-=-. The modeling of the dynamical behavior of a voltage-controlled parallelplate electrostatic actuator also derives a set of state equations of the form (1.1) [15, p.138]. Such an electrostatic actuato... |

24 |
Essentials of Padé Approximants
- Jr
- 1975
(Show Context)
Citation Context ...s0 if it matches with the moments of H(s)as far as possible. Precisely, it is required that H(s)= Hn(s) + O � (s − s0) 2n� . (10) For a thorough treatment of Padé approximants, we refer the reader to =-=[12]-=-. Note that equation (10) presents 2n conditions on the 2n degrees of freedom that describe any function Hn(s) ∈ Rn−1,n. Specifically, let Hn(s) = Pn−1(s) Qn(s) = an−1s n−1 +···+a1s + a0 bnsn + bn−1s ... |

24 |
Oblique projection methods for large scale model reduction
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- 1995
(Show Context)
Citation Context ...tion, it is possible to implement the PVL method with an adaptive stopping criteria to determine the required number of Lanczos iterations, see [10]. Related work for error estimation can be found in =-=[48,42]-=- and recently in [62]. More efficient and accurate error estimations of the PVL approximation and its extension to the other moment-matching based Krylov techniques warrant further study. One alternat... |

21 |
Padé Approximation and Its
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- 1981
(Show Context)
Citation Context ...ϕn−k(s) are free. These parameters are chosen such that the first 2n − m moments of H(s) and �Hn(s) are matched. A rational function of this form is called a partial Padé approximation as proposed in =-=[15]-=-. Finally, this partial Padé approximation of the transfer function H(s) can be interpreted as a partial inverse eigenvalue problem, namely, find a vector z such that partial eigenvalues of �T n = T n... |

21 |
Padé techniques for model reduction in linear system theory: a survey
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- 1986
(Show Context)
Citation Context ...d 1990. The manuscript [20] has a complete treatment of the AWE technique and its variants. A survey of the Padé techniques for model reduction of linear systems is also presented in the earlier work =-=[17]-=-. It is well-known that in practice, the Hankel matrix Mn is generally extremely ill-conditioned. Therefore, the computation of Padé approximants using explicit moments is inherently numerically unsta... |

20 | A quadratic method for nonlinear model order reduction
- Chen, White
- 2000
(Show Context)
Citation Context ...brium at 0, i.e., f(0) = 0. Examples of the origins of nonlinear dynamical systems of the form (1.1) include the simulation of time-varying nonlinear circuit elements by independent excitation source =-=[4, 3]-=-, and MEMS, such as 1 (1.1)smicro-pressure sensor [8]. The modeling of the dynamical behavior of a voltage-controlled parallelplate electrostatic actuator also derives a set of state equations of the ... |

20 |
Asymptotic Waveform Evaluation
- Chiprout, Nakhla
- 1994
(Show Context)
Citation Context ...s the case for all n. This formulates the framework of the asymptotic waveform evaluation (AWE) techniques as they are known in circuit simulation, first presented in [70] around 1990. The manuscript =-=[20]-=- has a complete treatment of the AWE technique and its variants. A survey of the Padé techniques for model reduction of linear systems is also presented in the earlier work [17]. It is well-known that... |

20 | Model Reduction of Large Linear Systems via Low Rank System Gramians
- Li
- 2000
(Show Context)
Citation Context ...inconsistency by solving system gramians independently. Besides Krylov subspace based techniques, ADI based methods for solving the underlying Lyapunov equations are also presented in the recent work =-=[67,55]-=-. In contrast to the vast amount of literature on Krylov-subspace based methods for reduced-order modeling, there is little software available in the public domain. To the knowledge of the author, the... |

19 | The SyMPVL algorithm and its applications to interconnect simulation
- Freund, Feldmann
- 1997
(Show Context)
Citation Context ...sion n is defined as follows: Zn(s) = R T� �−1Rn, n I n + (s − s0)T n where Rn = Q T n �B. This is referred to as the SyMPVL method to denote a symmetric matrix Padé approximation via Lanczos process =-=[32]-=-. Since A is positive semidefinite, T n is also positive semidefinite. As a result, in exact arithmetic, the reduced-order model Zn(s) is stable and passive. It inherits the essential properties of th... |

19 |
Calculation of Gauss quadratures with multiple free and fixed knots
- Golub, Kautsky
- 1983
(Show Context)
Citation Context ... problem, namely, find a vector z such that partial eigenvalues of �T n = T n + ze T n and � T ′ n = T ′ n + z′ e T n−1 are prescribed. This rank-one updating strategy generalizes methods proposed in =-=[39]-=- on computation of certain Gaussian-type quadratures. 2.10. Reduced-order modeling in finite precision In this section, we discuss the robustness issue of the reduced-order modeling techniques in the ... |

19 |
Implicitly restarted Krylov subspace methods for stable partial realizations
- Jaimoukha, Kasenally
- 1997
(Show Context)
Citation Context ...g two large-scale Lyapunov matrix equations for the system gramians. Low rank approximations to the system gramians have been proposed by using Lanczos and Arnoldi-based Krylov subspace techniques in =-=[48, 47]-=-. In the latest work [82], a low rank approximation to a cross gramian is proposed which overcomes the possible inconsistency by solving system gramians independently. Besides Krylov subspace based te... |

18 | Gramian based model reduction of large-scale dynamical systems. Numerical analysis
- Dooren
- 1999
(Show Context)
Citation Context ...ss of global approximation methods, mostly based on the theory of balanced realization [59]. They sometimes are also called Gramian-based model reduction methods, or SVD-based model reduction methods =-=[86,3]-=-. The crux of this class of methods for applying to large-scale linear dynamical systems lies in solving two large-scale Lyapunov matrix equations for the system gramians. Low rank approximations to t... |