E. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 (s s 1 ) m 2 (s s 1 ) m 3 (s s 1 ) H(s) m 0 m 1 (s s 1 ) m 2 (s s 1 ) m 3 (s s 1 ) such that m i = m i for i = 0; 2k 1: One shows that there always exists a kth order model that matches 2k moments of such an expansion. Di erent Interpolation Points ([9], 6] A more elaborate approach is to match the rst moments of H(s) with those of H(s) in several expansion points. For two points s 1 and s 2 this would amount to : H(s) m 0;s1 m 1;s1 (s s 1 ) m 2;s1 (s s 1 ) m 0;s2 m 1;s2 (s s 2 ) m 2;s2 (s s 2 ) H(s) m 0;s1 m 1;s1 (s s 1 ....

....s 1 ) m 2;s1 (s s 1 ) m 0;s2 m 1;s2 (s s 2 ) m 2;s2 (s s 2 ) with 2k equalities m i;s 1 = m i;s 1 for i = 0; k 1 1; m i;s 2 = m i;s 2 for i = 0; 2k 1 k 1 : This can be extended to several points fs f ; f = 1; Fg as well. Matching the Coecients ([9]) It is possible to match these coecients by projecting the system data (A; b; c) in a particular subspace of R . The projected system has lower order, but the quantities m i;s f ji = 1; I f are preserved for selected frequencies s f . The projection subspace is a union of ....

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E. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. Thesis, ECE Dept., University of Illinois at Urbana-Champaign (1997).

.... Several approaches can be taken to the model reduction problem, notably model reduction via balanced realization and Hankel norm error estimates [61] 105] and Krylov subspace projection by Pad e approximation via the Lanczos method (PVL) 46] 9] or by multipoint rational interpolation [56] [65], 128] 148] In contrast with the linear case, we are not aware of any model reduction techniques that actually build the dynamical system (2.30) 3. Spectral theory. In this section we give a general description of the spectral theory associated with QEPs, beginning with general structure and ....

E. J. Grimme, Krylov projection methods for model reduction, PhD thesis, University of Illinois at Urbana-Champaign, 1997.

....control. In fact, there is a vast literature on the general topic of dimension reduction in dynamical systems. Recently, there has been renewed interest in projection methods for model reduction. Three leading efforts in this area are Pad via Lanczos (PVL) 10] multipoint rational interpolation [14], and implicitly restarted dual Arnoldi [23] The PVL approach exploits the deep connection between the (nonsymmetric) Lanczos process and classic moment matching techniques [12] 10] for an overview see [37. The multipoint rational interpolation approach utilizes the rational Krylov method of ....

....scale problems (n 400) it is already competitive with existing dense methods. Moreover, it can provide balanced realizations where most existing methods fail. In the large scale setting, it is not clear that this approach will compete with those of (PVL) 10] or multipoint rational interpolation [14]. The error bounds and the stability of the reduced order model come at a price. The proposed implicit restarting approach involves a great deal of work associated with solving the required special Sylvester equations. However, the iterative method is based upon adjoining residual corrections to ....

E.J. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. Thesis, ECE Dept., U. of Illinois, Urbana-Champaign,(1997).

....field solvers. At the same time, effective design of complicated systems requires simple models. Hence, model reduction is now a standard procedure for obtaining simple models of complicated physical systems. Much research has been performed in the model reduction field over the past decade [1, 2, 3], intended to address three primary issues: a) Model accuracy. b) Numerically stable and computationally practical generation of models of arbitrary order. c) Generation of models that are well behaved when embedded into a simulation tool with models of other physical elements. Permission to ....

....For example, components such as interconnect do not generate energy: they are passive. Lumped RLC circuits, can be typically represented by matrices that are independent of frequency. For such lumped systems, positive realness preserving procedures such as those based on congruence transforms [2, 3] are sufficient to guarantee that the reduced models of passive full systems are passive as well. However, when accounting for high frequency effects, distributed systems represented by frequency dependent matrices are typically encountered. For example, frequency dependent matrices are ....

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Eric Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, Coordinated-Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997.

....Thus an essential feature of reduction approaches is a relatively thorough control and assessment of approximation errors that is gained by formal analysis of the reduction algorithms. The most successful algorithms for reduction of large scale linear systems have been projection based approaches[1, 2]. Algorithms such as PVL[3] Arnoldi methods[4] and PRIMA[5] obtain reduced models by projecting the linear equations describing the LTI model system into a subspace of lower dimension. The subspace chosen determines the approximation properties of the reduced model. These algorithms exploit the ....

....details of a numerical implementation, but we will show how algorithms may be constructed using tools familiar from the linear model reduction problem. 2 Projection methods for LTI model reduction In this section we will review results on projection and Krylovsubspace methods for model reduction[1, 2]. As a prelude to model reduction, we define Krylov subspaces. Definition 1 (Krylov subspace) The Krylov subspace Km #A# p# generated by a matrix A and vector p, of order m, is the space spanned by the set of vectors #p#Ap#A p#####A m#1 p#. The essential elements of Krylov subspace based ....

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Eric J. Grimme, Krylov projection methods for model reduction, Ph.D. thesis, University of Illinois at UrbanaChampaign, 1997.

....0.5 and therefore Vr is in fact orthogonal to the reachability space and hence (2.23) and (2.24) can never capture the original model behavior. This example has also indirectly shown us the importance of the right hand side b and will be incorporated shortly. 2.6. 3 Arnoldi based Methods [13, 15] are examples of the several publications that have used and developed Arnoldi based model reduction methods. Here we specifically use the approach described by No Damping Case The structural dynamics can be described by the O.D.E. M i K u = bv y=cTu Y(s) cT(sZM K) lb U(s) 2.25) where M ....

E.J. Grimme, "Krylov Projection Methods for Model Reduction", PhD. Thesis, University of Illinois at Urbana-Champaign, 1994.

....the target application, interconnect synthesis, requires parameterized models valid over a wide geometric range. Second, the technique described below is a multi parameter extension of the projection subspace based moment matching methods that have proved so effective in interconnect modeling [12, 13, 10, 9, 8, 7, 11]. In the following section we present the basic background on multi parameter model order reduction for a two parameter case, and then in section three we describe the generalization to an arbitrary number of parameters. In section four, we demonstrate the effectiveness of the method on a ....

....of inputs and outputs, m, is typically much smaller than n, the number of states needed to accurately represent the electrical behavior of the interconnect. In order to generate a representation of the input output behavior given by (1) using many fewer states, one can use a projection approach [7]. In the projection approach, one first constructs an n q projection matrix V where q n, and then one generates the reduced model from the matrices of the original system using congruence transformations [10] Specifically, the reduced system is given by CV x (4) were ....

Eric Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, Coordinated-Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997.

....algorithm (PRIMA) has been developed in [5] The idea is to use an Arnoldi algorithm shown in Algorithm 1 to generate a set of orthogonal vectors, 10] These vectors are applied in a congruence transform [11] to preserve passivity. This corresponds to a reduced system with , and . In [12], such an approach is shown to match 2 moments and the main result in [5] is that this reduced order model is passive under the following conditions: 1) 2) 0 for all ; 3) 0 for all . Note that the PRIMA algorithm generates passive reduced order models which match moments at 0 for multiple ....

....Section V, moment matching about 0 described above generates poor results. For this reason we wish to consider expanding about some other point or points. Such multipoint expansions have been explored previously for explicit moment matching in [24] and for the Krylov subspace techniques in [25] [12]. Recently, a provably passive multipoint rational Arnoldi algorithm has been derived for the reduction of RLC circuits with multiple inputs and outputs [13] For expansions about points 0, the moments become Thus, to apply any multipoint scheme for the large dense systems of (31) one must be ....

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E. J. Grimme, "Krylov projection methods for model reduction," Ph.D. dissertation, Univ. Illinois, Urbana-Champaign, 1997.

....high accuracy is desired, the models generated can become excessively large and difficult to solve for a continuous range of frequencies. The need for reduced size models leads us to consider Model Order Reduction (MOR) techniques, which have been developed in the field of parameter extraction [1, 2, 3, 4, 5, 6, 7, 8, 9]. Our approach, which is based on a combination of nodal analysis formulation with a mesh analysis formulation, has significant advantages over previously reported methods, both in extraction speed and model size, making it possible to generate guaranteed passive low order models for efficient ....

....information is necessary from DC up to the highest frequency of interest in the circuit. Thus, it is essential to have models valid for a continuous range of frequencies. 3G UARANTEED PASSIVE MODEL ORDER REDUCTION AND INTERCONNECT SIMULATION Recently, Model Order Reduction (MOR) algorithms [1, 2, 5, 9, 6] have been presented to solve this problem. The basic idea of MOR techniques is to reduce the size of the system described by the circuit equations, usually written in a convenient state space form, to a much smaller one that still captures the dominant behavior of the original system. This ....

Eric Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, Coordinated-Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997.

....to, at least in the low frequency regime, a linear dynamical system with positive semi definite matrices. This positive definite result is important because it makes possible the straight forward application of the Krylov subspace based guaranteed passive model order reduction (MOR) techniques [10, 11, 12, 13, 14, 15, 16, 17]. The paper is organized as follows: in Section 2 we summarize Maxwell equations in VIE form. In Section 3, we describe the mesh analysis formulation for dielectrics. In Section 4 we show that the matrices resulting from the VIE full mesh analysis approach can be cast into a positive semidefinite ....

E. J. Grimme. Krylov projection methods for model reduction. PhD Thesis, Univ. of Illinois at Urbana-Champaign, 1997.

....bases for the subspaces S k (10) and Z k (13) similar to the two sided Arnoldi algorithm [26] This is more expensive than the biorthogonalisation process. 1. 7 Related Work A comprehensive treatment of Krylov subspace methods for model order reduction is made by Grimme in his PhD thesis [20]. An overview of Krylov subspace methods for model order reduction in circuit theory is given by Freund [11] The rst proof about the connection between the two sided Lanczos and Pad# for the single input single output was given by Gragg [19] Feldmann and Freund introduced the Lanczos process in ....

....by Pad# via Lanczos [2] The rational Lanczos algorithm discussed by Gallivan, Grimme and Van Dooren in their later work [18] is a multipoint Pad# approximation; the method builds up a biorthogonal pair of bases. Multipoint Pad# and Pad# type methods are also discussed by Grimme in his PhD thesis [20]. Nguyen and Li discuss a block rational Lanczos algorithm [22] Silveria, Kamon, Elfadel and White create a passive reduced order model through an L orthogonal Arnoldi algorithm [33] the algorithm is further discussed by Elfadel and Ling [7] Odabasioglu, Celik and Pileggi[23] use the Arnoldi ....

E. J. Grimme. Krylov Projection Methods For Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, 1997.

.... approximation, which is a rational approximation [9] Feldmann and Freund introduced the Lanczos process in circuit simulation [5, 6] and, at about the same, so did Gallivan, Grimme and Van Dooren [8] A comprehensive treatment of Krylov subspace methods for model order reduction is made by Grimme [10]. An overview of Krylov subspace methods for model order reduction in circuit theory is given by Freund [7] 2 Subspace Methods 2.1 Introduction In this section we will show how to solve the eigenproblem Au = u; 6) the linear system of equations Ax = b (7) and the model order reduction ....

....vg: 25) Some of the Krylov subspace methods are: 1. The Arnoldi method 2. The Hermitian Lanczos method In the following we state some properties of the Krylov subspaces. A more complete discussion can be found in [19] for the eigenvalue problem, in [20] for linear systems of equations, and in [10] for model order reduction. Proposition 3. A vector x 2 K k can be written as x = p(A)v, where p is a polynomial of degree not exceeding k 1. Proposition 4. Let be the lowest degree of a polynomial p such that p(A)v = 0. Then K is invariant under A and K k = K for all k . This means that K ....

E. J. Grimme. Krylov Projection Methods For Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, 1997.

....stable systems, all of these three approaches are guaranteed to preserve stability and provide bounds on the approximation error. Recently much research has been done to establish a connection between Krylov subspace projection methods used in numerical linear algebra and model reduction [2] 3] [4], 5] 7] 12] The implicitly restarting algorithm [11] has been applied to obtain stable reduced models [6] The approximate balancing method introduced in [2] iteratively computes a k th order approximately balanced system without computing the full order balanced model. In this note, we ....

E.J. Grimme,Krylov Projection Methods for Model Reduction, Ph.D. Thesis, ECE Dept., U. of Illinois, Urbana-Champaign,(1997).

.... Several approaches can be taken to the model reduction problem, notably model reduction via balanced realization and Hankel norm error estimates [61] 105] and Krylov subspace projection by Pad e approximation via the Lanczos method (PVL) 46] 9] or by multipoint rational interpolation [56] [65], 128] 148] In contrast with the linear case, we are not aware of any model reduction techniques that actually build the dynamical system (2.30) 3. Spectral theory. In this section we give a general description of the spectral theory associated with QEPs, beginning with general structure and ....

E. J. Grimme, Krylov projection methods for model reduction, PhD thesis, University of Illinois at Urbana-Champaign, 1997.

....high accuracy is desired, the models generated can become excessively large and difficult to solve for a continuous range of frequencies. The need for reduced size models leads us to consider Model Order Reduction (MOR) techniques, which have been developed in the field of parameter extraction [1, 2, 3, 4, 5, 6, 7, 8, 9]. Our approach, which is based on a combination of nodal analysis formulation with a mesh analysis formulation, has significant advantages over previously reported methods, both in extraction speed and model size, making it possible to generate guaranteed passive low order models for efficient ....

....information is necessary from DC up to the highest frequency of interest in the circuit. Thus, it is essential to have models valid for a continuous range of frequencies. 3G UARANTEED PASSIVE MODEL ORDER REDUCTION AND INTERCONNECT SIMULATION Recently, Model Order Reduction (MOR) algorithms [1, 2, 5, 9, 6] have been presented to solve this problem. The basic idea of MOR techniques is to reduce the size of the system described by the circuit equations, usually written in a convenient state space form, to a much smaller one that still captures the dominant behavior of the original system. This ....

Eric Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, Coordinated-Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997.

....methods have also been proposed. By considering multiple expansion points s 1 ; s 2 ; with their corresponding system matrices, A 1 ; A 2 ; one generates a subspace spanned by union of block Krylov basis vectors. This family of methods called rational Krylov schemes is described in [5] and, while being relatively expensive to apply, has the potential of producing the most compact models. As an example we show the reduction of a package model. The package is originally modeled by a linear circuit consisting of more than 4000 RLC elements. The simulation of the integrated ....

E. J. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, University of Illinois at UrbanaChampaign, Urbana, Illinois, 1997.

....to show that our method generates arbitrarily accurate and guaranteed stable reduced order models for general RLC circuits. It should be noted that the method presented is one of a general class of preconditioned rational Krylov methods, other variants of which can be used to guarantee passivity [10, 11]. In the next section we briefly describe background on RLC circuit formulation, model order reduction, Pad e approximation, and Arnoldi methods. Then in Section 3, we present a guaranteed stability theory comprising two steps: a coordinate transformation requiring the computation of a matrix ....

Eric Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, CoordinatedScience Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997.

....be matched. If we were to consider a Taylor series expansion of the transfer function about some non zero value of s, a model could be obtained which would give a better approximation of the system dynamics for higher frequencies. These multiple frequency point Arnoldi methods are described in [13]. In order to calculate the basis, we consider input vectors which correspond to a particular blade having a unit displacement or velocity and all other blades fixed. Although vectors must be constructed for each of the N blades being perturbed in turn, the calculation need only be performed once, ....

E. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, CoordinatedScience Laboratory, University of Illinois at Urbana-Champaign, 1997.

....are required. We developed methods incorporating statistical considerations to improve robustness to noise. Likewise, in the future, we expect to develop methods for model order reduction, to both improve algorithm stability, and to offer a tradeoff between performance and computation (see [17, 18, 21]) We have also explored further the incorporation of nonlinear (and sparsity based) regularization techniques [30, 31, 32, 33] These steps provide yet additional ways to further enhance our methods for obtaining high resolution aero optics images. D ....

E. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, Directed by K. Gallivan, 1997.

....is important to note that the system matrices only enter the modeling problem through its moments explicitly. Unfortunately, these explicit moment matching methods exhibit numerical instabilities, which was first pointed out in [8] and later on in [7] The reader is referred to those papers and to [14] for a detailed discussion. The numerical difficulties come from the construction of the Hankel and Loewner matrices involved and the ill conditioning of the associated linear systems. Both [8] and [7] point out that moment matching via Krylov projection methods is a preferred numerical ....

....to produce a reduced system that matched multiple moments at multiple frequency values. We now give a basic theorem describing the relationships between Krylov subspaces, the iterative algorithms for constructing these subspaces, and model reduction via rational interpolation. It was proven in [14] and [15] and extends [26] to multipoint approximations. Theorem 1 If K [ k=1 K J b k i (oe (k) E Gamma A) Gamma1 E; oe (k) E Gamma A) Gamma1 B j V (9) and K [ k=1 K J c k i (oe (k) E Gamma A) GammaT E T ; oe (k) E Gamma A) GammaT C T j Z (10) ....

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E. Grimme (1997), Krylov Projection Methods for Model Reduction, PhD thesis, University of Illinois at Urbana-Champaign, IL.

....all of these explicit moment matching methods are known to exhibit numerical instabilities, particularly as the dimension of the reducedorder model M grows. The source of these difficulties was pointed out in [17] and in the independent work of [18] The reader is referred to those papers and to [19] for a detailed discussion. The difficulties center around numerical problems when constructing the Hankel matrices involved and, even when accurate matrices are available, the ill conditioning of the associated linear systems. Both [17] and [18] point out that moment matching via the Lanczos ....

....procedure and modifying it to produce a reduced system that matched multiple moments at multiple frequency values. 3 General Projection Formulation A large amount of the credit for connecting Krylov projection with Pad e approximation belongs to Villemagne and Skelton. 37] However, recently in [19] and [38] the relationships between Krylov subspaces, the iterative algorithms for constructing these subspaces, and model reduction via rational interpolation has been explored in great detail. The results clearly demonstrate that the Lanczos type methods are certainly not the only choice for ....

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E. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, 1997.

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E. Grimme. Krylov Projection Methods for Model Reduction. PhD thesis, University of Illinois at Urbana-Champaign, 1997.

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E. Grimme, Krylov projection methods for model reduction, Ph.D. thesis, University of Illinois, 1997.

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E. J. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1997.

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E. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. Thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois, 1994.

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