## Interpolatory Projection Methods for Parameterized Model Reduction

Citations: | 3 - 2 self |

### BibTeX

@MISC{Baur_interpolatoryprojection,

author = {Ulrike Baur and Serkan Gugercin and et al.},

title = {Interpolatory Projection Methods for Parameterized Model Reduction},

year = {}

}

### OpenURL

### Abstract

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory H2 optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are optimal with respect to an H2 ⊗ L2 joint error measure. The capabilities of these approaches are illustrated by several numerical

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28 |
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Citation Context ... “good” reduced models that satisfy first-order necessary optimality conditions, in principle allowing the possibility of having a local minimizer as an outcome. Many have worked on this problem; see =-=[7, 29, 31, 36, 40, 44, 50, 51, 55]-=-. Interpolation-based H2 optimality conditions were developed first by Meier and Luenberger [40] for SISO systems. Analogous H2 optimality conditions for MIMO systems have been placed within an interp... |

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Citation Context ...tively so. Another strategy for an effective and representative choice of parameter points in higher dimensional parameter spaces (for example, say, with ν = 10) comes through the use of sparse grids =-=[9, 23, 54]-=-. This approach is based on a hierarchical basis and a sparse tensor product construction. The dimension of the sparse grid space is of order O(2 n n ν−1 ) compared to the dimension of the correspondi... |

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Citation Context ...ality conditions were developed first by Meier and Luenberger [40] for SISO systems. Analogous H2 optimality conditions for MIMO systems have been placed within an interpolation framework recently in =-=[10, 25, 46]-=-. This is summarized in the next theorem. Theorem 3.2. Suppose ˜ Hr(s) =Cr(sEr − Ar) −1Br minimizes ‖H − Hr‖H2 over all (stable) rth-order transfer functions and that the associated reduced-order penc... |

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Citation Context ...latory model reduction is an approach introduced by Skelton et al. in [13, 52, 53], which was later placed into a numerically efficient framework by Grimme [24]. Gallivan, Vandendorpe, and Van Dooren =-=[21]-=- developed a more versatile version for MIMO systems, a variant of which we describe here and then adapt to parameterized systems: Starting with a full-order system as in (3.1) and selected interpolat... |

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Citation Context ...eserving balancing-based methods such as described above. 6.2. Thermal conduction in a semiconductor chip. We consider now a model representing thermal conduction in a semiconductor chip described in =-=[35]-=-. An important requirement for a compact and efficient model of thermal conduction in this context is that it should allow flexibility in specifying boundary conditions in order to allow independent d... |

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Citation Context ...ogate in optimization methods, but this passes beyond the scope of this work. Other PMOR approaches include interpolation of the full transfer function (see [3]) and reduced basis methods (see, e.g., =-=[2, 22, 28, 32, 39]-=-). Reduced-basis methods are successful in finding an information-rich set of global ansatz functions for spatial discretization of parameterized partial differential equations (PDEs). In the setting ... |

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Citation Context ...(Vr) =rank(Wr) =r, which then determine reduced system matrices Er = W T r EVr, Ar = W T r AVr, Br = W T r B, andCr = CVr. Interpolatory model reduction is an approach introduced by Skelton et al. in =-=[13, 52, 53]-=-, which was later placed into a numerically efficient framework by Grimme [24]. Gallivan, Vandendorpe, and Van Dooren [21] developed a more versatile version for MIMO systems, a variant of which we de... |

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Citation Context ...(Vr) =rank(Wr) =r, which then determine reduced system matrices Er = W T r EVr, Ar = W T r AVr, Br = W T r B, andCr = CVr. Interpolatory model reduction is an approach introduced by Skelton et al. in =-=[13, 52, 53]-=-, which was later placed into a numerically efficient framework by Grimme [24]. Gallivan, Vandendorpe, and Van Dooren [21] developed a more versatile version for MIMO systems, a variant of which we de... |

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Citation Context ...lowable parameter ranges within the parameter space. There are methods to address this difficulty. One possible approach is the so-called greedy selection algorithm of Bui-Thanh, Willcox, and Ghattas =-=[8]-=-. Even though the final reduced-order model of [8] proves to be a high quality approximation, the optimization algorithm that needs to be solved at each step could be computationally expensive, possib... |

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Citation Context ...ality conditions were developed first by Meier and Luenberger [40] for SISO systems. Analogous H2 optimality conditions for MIMO systems have been placed within an interpolation framework recently in =-=[10, 25, 46]-=-. This is summarized in the next theorem. Theorem 3.2. Suppose ˜ Hr(s) =Cr(sEr − Ar) −1Br minimizes ‖H − Hr‖H2 over all (stable) rth-order transfer functions and that the associated reduced-order penc... |

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Citation Context ...ingular for each i =1,...,r, then the reduced system Hr(s) = Cr(sEr − Ar) −1Br defined by Ar = WT r AVr, Er = WT r EVr, Br = WT r B, and Cr = CVr solves the tangential interpolation problem (3.3). In =-=[4]-=-, Beattie and Gugercin showed how to solve the tangential interpolation problem posed in (3.3) for a substantially larger class of transfer functions—those having a coprime factorization of the form H... |

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Citation Context ...ogate in optimization methods, but this passes beyond the scope of this work. Other PMOR approaches include interpolation of the full transfer function (see [3]) and reduced basis methods (see, e.g., =-=[2, 22, 28, 32, 39]-=-). Reduced-basis methods are successful in finding an information-rich set of global ansatz functions for spatial discretization of parameterized partial differential equations (PDEs). In the setting ... |

9 |
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Citation Context ...ear dependencies are removed only up to thresholds associated with machine precision. The construction of truncation matrices is similar to the trajectory piecewise approximation methods suggested in =-=[43, 47]-=-. Effectiveness of this algorithm is illustrated with several numerical examples in section 6. Algorithm 5.1. Piecewise H2 Optimal Interpolatory PMOR. 1. Select L parameter vectors {p (1) , p (2) ,...... |

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Citation Context ... basis and a sparse tensor product construction. The dimension of the sparse grid space is of order O(2 n n ν−1 ) compared to the dimension of the corresponding full grid space given by O(2 νn ). See =-=[3, 42]-=- for other approaches to parameterized model reduction using sparse grids. Heuristics such as these can provide effective choices for interpolation points. However, in the absence of compelling heuris... |

5 | H2optimal model reduction for large scale discrete dynamical MIMO systems - Bunse-Gerstner, Kubalinska, et al. |

5 |
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Citation Context ...efining matrices in [49, 12, 27]), or they are formulated in terms of series expansions and term-by-term matching of moments. Explicit moment matching is conceptually simple (if painful), and indeed, =-=[6]-=- considers a framework of parametric dependence that is quite general; this approach could be extended also to the situations we consider. However, such approaches have led to strategies that are then... |

5 | Efficient block-based parameterized timing analysis covering all potentially critical paths
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Citation Context ... model may be used to compute parameter sensitivities more cheaply than the original model and will exactly match the original model sensitivities at every parameter interpolation point, ˆp. See also =-=[30, 48]-=- for recent methods that use sensitivity data and PMOR type methods. There are also interesting consequences for optimization with respect to p of objective functions depending on H(s, p) (or on the o... |

4 |
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(Show Context)
Citation Context ...ear dependencies are removed only up to thresholds associated with machine precision. The construction of truncation matrices is similar to the trajectory piecewise approximation methods suggested in =-=[43, 47]-=-. Effectiveness of this algorithm is illustrated with several numerical examples in section 6. Algorithm 5.1. Piecewise H2 Optimal Interpolatory PMOR. 1. Select L parameter vectors {p (1) , p (2) ,...... |

3 |
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