## Model reduction based on spectral projection methods (2005)

Venue: | Dimension Reduction of Large-Scale Systems |

Citations: | 9 - 6 self |

### BibTeX

@INPROCEEDINGS{Benner05modelreduction,

author = {Peter Benner and Enrique S. Quintana-ortí},

title = {Model reduction based on spectral projection methods},

booktitle = {Dimension Reduction of Large-Scale Systems},

year = {2005},

pages = {5--45},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We discuss the efficient implementation of model reduction methods such as modal truncation, balanced truncation, and other balancing-related truncation techniques, employing the idea of spectral projection. Mostly, we will be concerned with the sign function method which serves as the major computational tool of most of the discussed algorithms for computing reduced-order models. Implementations for large-scale problems based on parallelization or formatted arithmetic will also be discussed. This chapter can also serve as a tutorial on Gramian-based model reduction using spectral projection methods. 1

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Citation Context ... or block decomposition of Z in the following way: let � � ⎡ ⎤ R11 R12 P = QRΠ, R = = ⎣ ❅ ⎦, R11 ∈ R 0 0 k×k , be a QR decomposition with column pivoting (or a rank-revealing QR decomposition (RRQR)) =-=[GV96]-=- where Π is a permutation matrix. Then the first k columns of Q form an orthonormal basis for S1 and we can transform Z to block-triangular form � ˜Z := Q T ZQ = 8 � Z11 Z12 0 Z22 , (12)swhere Λ (Z11)... |

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Citation Context ...e matrix sign function of Z is defined as sign(Z) := S � −Ik 0 0 In−k � S −1 . Note that sign(Z) is unique and independent of the order of the eigenvalues in the Jordan decomposition of Z, see, e.g., =-=[LR95]-=-. Many other definitions of the sign function can be given; see [KL95] for an overview. Some important properties of the matrix sign function are summarized in the following lemma. Lemma 3.4 Let Z ∈ R... |

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Citation Context ... Z0 ← Z, Zj+1 ← 1 2 (Zj + Z −1 j ), j = 0, 1, 2, . . .. (13) Under the given assumptions, the sequence {Zj} ∞ j=0 convergence rate and sign(Z) = lim j→∞ Zj; converges with an ultimately quadratic see =-=[Rob80]-=-. As the initial convergence may be slow, the use of acceleration techniques is recommended. There are several acceleration schemes proposed in the literature, a thorough discussion can be found in [K... |

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Citation Context ...the SR method for balanced truncation. In [LHPW87, TP87] and all textbooks treating balanced truncation, S and R are assumed to be the (square, triangular) Cholesky factors of the system Gramians. In =-=[BQQ00a]-=- it is shown that everything derived so far remains true if full-rank factors of the system Gramians are used instead of Cholesky factors. This yields a much more efficient implementation of balanced ... |

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Citation Context ...refore, counter to intuition, it should not be surprising that often, results computed by the sign function method are more accurate than those obtained by using Schur-type decompositions; see, e.g., =-=[BQ99]-=-. Example 3.6 A typical convergence history (based on �Zj −sign(Z) �F) is displayed in Figure 1, showing the fast quadratic convergence rate. Here, we computed the sign function of a dense matrix A co... |

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Citation Context ...echniques is recommended. There are several acceleration schemes proposed in the literature, a thorough discussion can be found in [KL92], and a survey and comparison of different schemes is given in =-=[BD93]-=-. For accelerating (13), in each step Zj is replaced by 1 γj Zj, where the most prominent choices for γj are briefly discussed in the sequel. 9sDeterminantal scaling [Bye87]: here, γj = | det(Zj)| 1 n... |

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Citation Context ...amians can also be transformed into diagonal matrices with the leading ˆn × ˆn submatrices equal to diag(σ1, . . .,σˆn), and . .. σn ⎥ ⎦; ˆWc ˆ Wo = diag(σ 2 1, . . .,σ 2 ˆn , 0, . . .,0); see, e.g., =-=[TP87]-=-. Using a balanced realization obtained via the transformation matrix Tb, the HSVs allow an energy interpretation of the states; see also [Van00] for a nice treatment of this subject. Specifically, th... |

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Citation Context ...ll beyond the scope of the discussed implementations of modal or balanced truncation. iss-II This is a model of the extended service module of the International Space Station, for details see Chapter =-=[CV05]-=-. (For a more complete comparison of balanced truncation based on Algorithm 4 and the SLICOT model reduction routines see [BQQ03b].) The frequency response errors for the chosen examples are shown in ... |

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Citation Context ...is choice minimizes the distance of the geometric mean of the eigenvalues of Zj from 1. Note that the determinant det (Zj) is a by-product of the computations required to implement (13). Norm scaling =-=[Hig86]-=-: here cj = � �Zj�2 �Z −1 j �2 , which has certain minimization properties in the context of computing polar decompositions. It is also beneficial regarding rounding errors as it equalizes the norms o... |

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Citation Context ...n [Var91] may provide a more accurate reduced-order model in the presence of rounding errors. It combines the SR implementation from [LHPW87, TP87] with the balancing-free model reduction approach in =-=[SC89]-=-. The BFSR algorithm only differs from the SR implementation in the procedure to obtain Tl and Tr from the SVD (31) of SRT , and in that the reduced-order model is not balanced. The main idea is that ... |

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Citation Context ... original system is highly unbalanced (and hence, the state-space transformation matrix T in (27) is ill-conditioned), the balancing-free square-root (BFSR) balanced truncation algorithm suggested in =-=[Var91]-=- may provide a more accurate reduced-order model in the presence of rounding errors. It combines the SR implementation from [LHPW87, TP87] with the balancing-free model reduction approach in [SC89]. T... |

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Citation Context ... t ≥ 0, of order r, r ≪ n, and associated TFM ˆ G(s) = Ĉ(sI − Â)−1 ˆ B + ˆ D which approximates G(s). Model reduction of discrete-time LTI systems can be formulated in an analogous manner; see, e.g., =-=[OA01]-=-. Most of the methods and approaches discussed here carry over to the discretetime setting as well. Here, we will focus our attention on the continuous-time setting, the discrete-time case being discu... |

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Citation Context ...� S −1 . Note that sign(Z) is unique and independent of the order of the eigenvalues in the Jordan decomposition of Z, see, e.g., [LR95]. Many other definitions of the sign function can be given; see =-=[KL95]-=- for an overview. Some important properties of the matrix sign function are summarized in the following lemma. Lemma 3.4 Let Z ∈ R n×n with Λ (Z) ∩ jR = ∅. Then: a) (sign(Z)) 2 = In, i.e., sign(Z) is ... |

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Citation Context ...0]. As the initial convergence may be slow, the use of acceleration techniques is recommended. There are several acceleration schemes proposed in the literature, a thorough discussion can be found in =-=[KL92]-=-, and a survey and comparison of different schemes is given in [BD93]. For accelerating (13), in each step Zj is replaced by 1 γj Zj, where the most prominent choices for γj are briefly discussed in t... |

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Citation Context ... partition I × I by an H-tree TI×I, where a problem dependent admissibility condition is used to decide whether a block t × s ⊂ I × I allows for a low rank approximation of this block. Definition 5.1 =-=[GH03]-=- The set of hierarchical matrices is defined by H(TI×I, k) := {M ∈ R I×I | rank(M|t×s) ≤ k for all admissible leaves t × s of TI×I}. Submatrices of M ∈ H(TI×I, k) corresponding to inadmissible leaves ... |

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Citation Context ...al dynamics. In particular, the model reduction method in [CB68] and its relatives, called nowadays substructuring methods, which combine the modal analysis with a static compensation following Guyan =-=[Guy68]-=-, are frequently used. We will not elaborate on these type of methods, but 17 � .swill only focus on the basic principles of modal truncation and how it can be implemented using spectral projection id... |

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Citation Context ...ods and approaches discussed here carry over to the discretetime setting as well. Here, we will focus our attention on the continuous-time setting, the discrete-time case being discussed in detail in =-=[BQQ03a]-=-. Balancing-related model reduction methods are based on finding an appropriate coordinate system for the state-space in which the chosen Gramian matrices of the system are ∗ Fakultät für Mathematik, ... |

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Citation Context ...on ˜ XW is obtained by solving (38) using Newton’s method with exact line search as described in [Ben97] with the sign function method used for solving the Lyapunov equations in each Newton step; see =-=[BQQ01]-=- for details. The Lyapunov equation for R is solved using the sign function iteration from subsection 3.3. 4.3.4 Further Riccati-Based Truncation Methods There is a variety of other balanced truncatio... |

27 |
Balanced truncation model reduction for large-scale systems in descriptor form
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Citation Context ...lly only differ in the way the factors of the Gramians are computed. Approximation methods suitable for sparse systems based mainly on Smith- and ADI-type methods are discussed in Chapters [GL05] and =-=[MS05]-=-. These allow the computation of the factors at a computational cost and a memory requirement proportional to the number of nonzeros in A. Thus, implementations of balanced truncation based on these i... |

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Citation Context ...vice module of the International Space Station, for details see Chapter [CV05]. (For a more complete comparison of balanced truncation based on Algorithm 4 and the SLICOT model reduction routines see =-=[BQQ03b]-=-.) The frequency response errors for the chosen examples are shown in Figure 4. For the implementations of balanced truncation, we only plotted the error curve for btsr as the graphs produced by the o... |

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Citation Context ...ing the Bartels-Stewart or Hammarling’s method for computing the system Gramians: – the SLICOT [BMS + 99] implementation of balanced truncation, called via a mex-function from the Matlab function bta =-=[Var01]-=-, – the Matlab Control Toolbox (Version 6.1 (R14SP1)) function balreal followed by modred, – the Matlab Robust Control Toolbox (Version 3.0 (R14SP1)) function balmr. The examples that we chose to comp... |

22 |
Balanced stochastic realization
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Citation Context ...cial matrix D = [ǫIp 0] [Glo86]. Balanced stochastic truncation (BST) is a model reduction method based on truncating a balanced stochastic realization. Such a realization is obtained as follows; see =-=[Gre88]-=- for details. Define the power spectrum Φ(s) = G(s)G T (−s), and let W be a square minimum phase right spectral factor of Φ, satisfying Φ(s) = W T (−s)W(s). As D has full row rank, E := DD T is positi... |

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Citation Context ... suggested in [Hig86, KL92] to approximate this norm by the Frobenius norm or to use the bound (see, e.g., [GV96]) �Zj�2 ≤ � �Zj�1�Zj�∞. (14) Numerical experiments and partial analytic considerations =-=[BQQ04d]-=- suggest that norm scaling is to be preferred in the situations most frequently encountered in the sign function-based calculations discussed in the following; see also Example 3.6 below. Moreover, th... |

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Balanced realization and model reduction for unstable systems
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Citation Context ...described in more detail in [BCQQ04] where also some numerical examples are given. An extension of this approach using balancing for appropriately defined Gramians of unstable systems is discussed in =-=[ZSW99]-=-. This approach can also be implemented using sign function-based spectral projection techniques similar to the ones used so far. Alternative model reduction techniques for unstable systems based on c... |