## Krylov Projection Methods For Model Reduction (1997)

Citations: | 118 - 3 self |

### BibTeX

@TECHREPORT{Grimme97krylovprojection,

author = {Eric James Grimme},

title = {Krylov Projection Methods For Model Reduction},

institution = {},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.

### Citations

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Citation Context ...chnique is the approximate inverse approach. One constructs a P k with some sparsity pattern, so that P k (A \Gamma oe (k) E) \Gamma I is minimized with respect to some norm, e.g., the Frobenius norm =-=[96]-=-. Alternatively, and more generally, one can think of \Phi m as an operation that takes in the vectors q pm+1 , w pm+1 and outputs the vectors ~ vm , ~ z m . Hence, \Phi m can represent an iterative s... |

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Citation Context ... G T wm \Gamma ff mwm \Gamma fl mwm\Gamma1 ; end Actual implementations of the Lanczos method may encounter numerical difficulties including a loss of biorthogonality and so-called serious breakdowns =-=[66, 67, 68, 69]-=-. However, these breakdowns are less drastic and/or rarer than the breakdowns occurring in 31 explicit moment-matching. Additionally, remedies are possible [68, 69] and are discussed in Section 3.3.3.... |

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Citation Context ...aging interconnect of a circuit. This generated problem consists of fifteen identical segments connected in series (see Figure 4.3 for the structure of a segment). Using Modified Nodal Analysis (MNA) =-=[86]-=-, a set of equations of size N = 47 can be formulated to describe the interconnect. The frequency response of the interconnect between 10 8 and 10 11 Hertz is shown in Figure 4.4. The input to this sy... |

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Citation Context ... G T wm \Gamma ff mwm \Gamma fl mwm\Gamma1 ; end Actual implementations of the Lanczos method may encounter numerical difficulties including a loss of biorthogonality and so-called serious breakdowns =-=[66, 67, 68, 69]-=-. However, these breakdowns are less drastic and/or rarer than the breakdowns occurring in 31 explicit moment-matching. Additionally, remedies are possible [68, 69] and are discussed in Section 3.3.3.... |

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Citation Context ...sponse h(t) and subsequent derivatives of the impulse response at t = 0. A reduced-order model whose Markov parameterss\Gammaj equals\Gammaj for j = 1; 2; : : : ; 2M is known as a partial realization =-=[32]-=-. Because the partial realization emphasizes behavior at t = 0, such a model may be dominated by the extremely rapidly decaying dynamics of the system. Regretfully, extensions of partial realizations ... |

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Citation Context ...M ? N steps may be required to achieve the desired level of convergence in practice. Further comments on such behavior in the area of iterative solvers for linear systems of equations may be found in =-=[84, 85]-=-. The bottom line is that a loss of (bi)orthogonality tends to slow but not destroy the convergence of approximations that assume (bi)orthogonality and avoid the explicit use of (2.7). This behavior i... |

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Citation Context ...actorization of (A \Gamma oe (k) E) is computed. Elements appearing during the factorization that correspond to a certain level of fill or possess sizes that are under a certain tolerance are dropped =-=[95]-=-. A second technique is the approximate inverse approach. One constructs a P k with some sparsity pattern, so that P k (A \Gamma oe (k) E) \Gamma I is minimized with respect to some norm, e.g., the Fr... |

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Citation Context ...model development and the eventual mathematical treatment can be achieved. Models can frequently be acquired through discretizations such as the common finite difference and finite element approaches =-=[1]-=-. A range of techniques from the backward Euler method to multistep methods exists for solving the ordinary differential equations (ODE) that describe the system [2]. Stable, well-understood numerical... |

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Citation Context ...idiagonal S = W T GV and vice versa. An implementation of (2.24) with the appropriate parameter selections is the nonsymmetric Lanczos algorithm in Algorithm 2.1. The interested reader is referred to =-=[66, 67]-=- for a recent and detailed study of the nonsymmetric Lanczos method. Algorithm 2.1 Nonsymmetric Lanczos Initialize:sv 1 andsw 1 . For m = 1 to M , (S2.1.1) vm =svm =fl m where fl m = q jsw T msvm j; (... |

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Citation Context ...ical Gram-Schmidt. Computing all m terms is classical Gram-Schmidt. Computing only certain terms in the summation of (3.7) while assuming the others to be zero is known as selective orthogonalization =-=[73]-=-. More robust than classic Gram-Schmidt is two passes of Gram-Schmidt, known as reorthogonalization [74]. In this approach, one computes g m+1 via (3.7), sets ~ g m+1 = g m+1 , and computes the left-h... |

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Citation Context ...ity retention [60]. Very recently, Lanczos-based model reduction has become a popular topic in the area of high-speed circuits. Existing Lanczos algorithms were applied to the standard [47, 48], MIMO =-=[61]-=- and symmetric problems [62]. New algorithms were proposed for stability retention [63, 64]. However, through all of these application areas, the approaches remained closely tied to the classical Lanc... |

66 |
The Davidson method
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Citation Context ...mentary to that in the first and third rows. This difference follows from the fact that both the first row and the 179 Table 9.1: Classifying Existing PIES i m = 1 i m = oe m i m = oe m + ffi m T = I =-=[98, 19, 112, 113, 105]-=- [109, 114, 110] Tm = (A \Gamma oe m I) \Gamma1 [103] Tm = (A \Gamma oe m I) [107] [110] second column of Table 9.1 tend to require exact ES preconditioners. As long as exact ES preconditioners are be... |

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Citation Context ...rnoldi, is a two-sided technique that constructs orthogonal bases for unions of Krylov subspaces. The utilized Krylov subspaces of this method are adaptations of those seen in an eigenvalue technique =-=[15]-=-. Due to its emphasis on orthogonalization, this method is extremely robust and is an important contribution to Krylov-based model reduction. However, this robustness is not cheap. A second algorithm,... |

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Citation Context ...opic in the area of high-speed circuits. Existing Lanczos algorithms were applied to the standard [47, 48], MIMO [61] and symmetric problems [62]. New algorithms were proposed for stability retention =-=[63, 64]-=-. However, through all of these application areas, the approaches remained closely tied to the classical Lanczos algorithm. These approaches did not emphasize or exploit the fundamental structure in p... |

65 |
Computing interior eigenvalues of large matrices
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Citation Context ...rs, Pm = (A \Gamma oe m I) \Gamma1 , it is not appropriate in general. 9.1.2.3 Strategy 3, Tm = (A \Gamma oe m I) This choice was proposed as a more efficient avenue for treating interior eigenvalues =-=[107]-=-. To evaluate this transformation, again rewrite it in terms of a second, equivalent 175 eigenvalue problem. Define the quantities, S T S = Y T m (A \Gamma oe m I)Ym ; ~ Y = (A \Gamma oe m I)Ym S; ~ x... |

64 |
Padé-Type Approximation and General Orthogonal Polynomials
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Citation Context ...ations present in the literature. 2.4.1 History The methods forming the foundation for this work are relatively old. The history of Pad'e approximation, for example, spans more than one hundred years =-=[38]-=-. The algorithm of Lanczos, an important Krylov-based iteration, is nearing its fiftieth anniversary [39]. Yet, as evident by this dissertation and its many recent references, the understanding and ap... |

61 |
Generalizations of Davidson’s method for computing eigenvalues of sparse symmetric matrices
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Citation Context ...t noteworthy is the stability problem. A variety of approaches already exists for computing the nontrivial solutions, eigenvaluessn and eigenvectors x n , to (1.1) over some s region (see for example =-=[15, 18, 19]-=-). Variations on these approaches appear in some of the model-reduction techniques proposed in this dissertation. A survey of pertinent eigenvalue techniques and their connections with model-reduction... |

61 |
Model reduction using a projection formulation
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Citation Context ...hat partial realizations could be generated through the Lanczos algorithm [32]. Adaptations of Krylov subspaces were proposed in 1987 to generate Pad'e approximations and shifted Pad'e approximations =-=[49]-=-. Beyond the mathematical connections, the Lanczos method was utilized for model reduction in many application areas. The first of these areas chronologically was apparently structural dynamics. Even ... |

60 |
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Citation Context ...e large and sparse; they typically contain lumped parameters of a largescale system. The scalar s 2C denotes complex frequency. Such a matrix pencil arises in several problems in linear system theory =-=[9, 10]-=-. For example, finding the poles of a dynamic system entails computing the generalized eigenvalues of (A; E), i.e., finding the valuessn 2C and vectors x n 2C N \Theta1 such that (A \Gammasn E)x n = 0... |

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Citation Context ...olation points oe k . The impact of various DS preconditioners (various choices for the oe (k) ) is considered in Chapter 6. The use of multiple varying preconditioners has appeared in the literature =-=[70, 71]-=- for solving fixed systems of linear equations. Similarities can be seen between these algorithms and the model-reduction algorithms developed in Chapter 4. The modelreduction problem is frequency dep... |

55 |
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Citation Context ...ence and finite element approaches [1]. A range of techniques from the backward Euler method to multistep methods exists for solving the ordinary differential equations (ODE) that describe the system =-=[2]-=-. Stable, well-understood numerical linear algebra algorithms, e.g., a reduction to Schur form by orthogonal transformations, dominate the low-level mathematical operations [3]. When combined in vario... |

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Citation Context ...Hankel singular values not retained in the reduced-order model [29]. Unfortunately, implementing balanced truncation involves the solution of Lyapunov equations and thus, a cost of O(N 3 ) operations =-=[30]-=-. The invariant properties of importance in this work are the coefficients of some power series expansion of h(s). The solution techniques proposed determine a reduced-order model that accurately matc... |

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Citation Context ... [50, 51, 52]. Later work in the field utilized the Lanczos method for Pad'e approximation [53] including MIMO systems [54, 55]. The next wave of application work took place in the control literature =-=[56, 57, 58]-=-. A large amount of existing work was repeated, although new results did appear in the areas of error analysis [59] and stability retention [60]. Very recently, Lanczos-based model reduction has becom... |

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Citation Context ...ity to efficiently solve shifted systems of equations is desirable in certain ODE solvers. A few techniques do exist for iteratively solving shifted systems of equations over a single Krylov subspace =-=[20, 21]-=-. However, these methods are restricted to the case E = I and are limited in their choices of preconditioner. Suitable preconditioners for various regions of the shifted problem are considered in [22]... |

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Citation Context ...ently, Lanczos-based model reduction has become a popular topic in the area of high-speed circuits. Existing Lanczos algorithms were applied to the standard [47, 48], MIMO [61] and symmetric problems =-=[62]-=-. New algorithms were proposed for stability retention [63, 64]. However, through all of these application areas, the approaches remained closely tied to the classical Lanczos algorithm. These approac... |

40 |
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Citation Context ...nced truncation. Balanced truncation possesses the desirable feature that the H1 norm of the modeling error is bounded by the sum of the Hankel singular values not retained in the reduced-order model =-=[29]-=-. Unfortunately, implementing balanced truncation involves the solution of Lyapunov equations and thus, a cost of O(N 3 ) operations [30]. The invariant properties of importance in this work are the c... |

39 |
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Citation Context ... the MIMO problem is that the same number of moments need not be matched for every subsystem. The standard Krylov-based approaches to the MIMO problem assume K = 1 and fix J b l = J c l = J for all l =-=[54, 55, 61, 79]-=-. Such an approach is known as a block method, because the individual vectors in B or C are treated identically during the construction of the projection matrices. Although a block approach is perhaps... |

35 |
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Citation Context ...ity to efficiently solve shifted systems of equations is desirable in certain ODE solvers. A few techniques do exist for iteratively solving shifted systems of equations over a single Krylov subspace =-=[20, 21]-=-. However, these methods are restricted to the case E = I and are limited in their choices of preconditioner. Suitable preconditioners for various regions of the shifted problem are considered in [22]... |

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Citation Context ...opic in the area of high-speed circuits. Existing Lanczos algorithms were applied to the standard [47, 48], MIMO [61] and symmetric problems [62]. New algorithms were proposed for stability retention =-=[63, 64]-=-. However, through all of these application areas, the approaches remained closely tied to the classical Lanczos algorithm. These approaches did not emphasize or exploit the fundamental structure in p... |

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Citation Context ...1 . In this case, ~ ym+1 in (S9.2.3) is approximatelysx dm and no new direction is obtained. To avoid this difficulty, one can perturb i m slightly away from oe m . A popular choice in the literature =-=[109]-=- is to select i m+1 = oe m+1 + ffi m+1 , where 178 Algorithm 9.2 Davidson's Method Initialize: an orthogonal vector y 1 = Y 1 ; For m = 1 to M , (S9.2.1) Compute (sdm ;sx dm ) from Y T m AYm ; (S9.2.2... |

34 |
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(Show Context)
Citation Context ...ortunately, and perhaps surprisingly, a poorly conditioned (A \Gamma oeE) does not lead to catastrophic results when attempting to match information at s = oe. This conditioning issue was examined in =-=[72]-=- with respect to related concerns in the method of inverse iteration for eigenvalue problems. Quoting [72], The period when inverse iteration was first considered was notable for exaggerated fears con... |

33 |
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(Show Context)
Citation Context ...M ? N steps may be required to achieve the desired level of convergence in practice. Further comments on such behavior in the area of iterative solvers for linear systems of equations may be found in =-=[84, 85]-=-. The bottom line is that a loss of (bi)orthogonality tends to slow but not destroy the convergence of approximations that assume (bi)orthogonality and avoid the explicit use of (2.7). This behavior i... |

28 | Projection and deflation methods for partial pole assignment in linear state feedback - Saad - 1988 |

28 | Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems
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(Show Context)
Citation Context ...he column space of Y becomes a Krylov subspace, colsp fY g = K(P (A \Gamma iI); y 1 ); and Ps(A \Gamma oeI) \Gamma1 need only be computed once. Symmetric Lanczos type methods can be used to compute Y =-=[103, 104, 105]-=-. Fortuitously, some form of the reduced-order pencil (Y T TAY;Y T TY ) is implicitly generated by an a Lanczos-type algorithm. In practice, the Lanczos method is restarted multiple times. The Lanczos... |