An ℋ<sub>2</sub> ⊗ ℒ<sub>2</sub>‐Optimal Model Order Reduction Approach for Parametric Linear Time‐Invariant Systems

Hund, Manuela; Mlinarić, Petar; Saak, Jens

doi:10.1002/pamm.201800084

Cited by 12 publications

(12 citation statements)

References 2 publications

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“…The H 2 ⊗ L 2 FONC, which we first described in [15], is a direct consequence of Theorem 3.1. Hence, we restate the H 2 ⊗ L 2 FONC here as the following corollary.…”

Section: Fréchet Differentiabilitymentioning

confidence: 89%

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Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

Hund¹,

Mitchell²,

Mlinarić³

et al. 2021

Preprint

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In this paper, we generalize existing frameworks for H 2 ⊗ L 2 -optimal model order reduction to a broad class of parametric linear time-invariant systems. To this end, we derive first-order necessary optimality conditions for a class of structured reduced-order models, and then building on those, propose a stability-preserving optimization-based method for computing locally H 2 ⊗ L 2 -optimal reduced-order models. We also make a theoretical comparison to existing approaches in the literature, and in numerical experiments, show how our new method, with reasonable computational effort, produces stable optimized reduced-order models with significantly lower approximation errors.

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“…The H 2 ⊗ L 2 FONC, which we first described in [15], is a direct consequence of Theorem 3.1. Hence, we restate the H 2 ⊗ L 2 FONC here as the following corollary.…”

Section: Fréchet Differentiabilitymentioning

confidence: 89%

“…Due to the similarity of the equations in (25) with the Wilson conditions (8) for non-parametric systems, in [15] (and later in [19,Theorem 6.11]) we referred to (25) as Wilson-type optimality conditions, but for conciseness, we use FONC in this paper.…”

Section: Fréchet Differentiabilitymentioning

confidence: 99%

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

Hund¹,

Mitchell²,

Mlinarić³

et al. 2021

Preprint

Self Cite

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“…The construction of optimal reduced models requires the computation of the H 2 ⊗ L 2 (D)-norm of the considered parametric LTI system, which is given by where P(µ) solves A T P(µ) + P(µ)A + C(µ) T C(µ) = 0 [38]. Discretizing (3.15) using a suitable quadrature formula with nodes {µ i } and weights {ω i }, i = 1, .…”

Section: H 2 -Norm Computationmentioning

confidence: 99%

The Short-term Rational Lanczos Method and Applications

Palitta¹,

Pozza²,

Simoncini³

2021

Preprint

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Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In the past this procedure was abandoned because it requires twice the number of linear system solves per iteration than with the classical long-term method. We propose an implementation that allows one to obtain key rational subspace matrices without explicitly storing the whole orthonormal basis, with a moderate computational overhead associated with sparse system solves. Several applications are discussed to illustrate the advantages of the proposed procedure.

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“…Except for special cases [9,33], how one chooses optimal parameter sampling points with respect to a joint global frequency-parameter error measure has not been known until recently. In [42], Hund et al tackles this joint-optimization problem by deriving optimality conditions and then constructing model reduction bases that enforce those conditions. The most widely used approaches for global basis construction in pMOR are greedy or optimization-based sampling strategies; see [14] for a survey.…”

Section: Basic Structurementioning

confidence: 99%

Sampling-free parametric model reduction of systems with structured parameter variation

Beattie

Gugercin

Tomljanović

2018

Preprint

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We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

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An ℋ₂ ⊗ ℒ₂‐Optimal Model Order Reduction Approach for Parametric Linear Time‐Invariant Systems

Cited by 12 publications

References 2 publications

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

The Short-term Rational Lanczos Method and Applications

Sampling-free parametric model reduction of systems with structured parameter variation

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An ℋ2 ⊗ ℒ2‐Optimal Model Order Reduction Approach for Parametric Linear Time‐Invariant Systems

Cited by 12 publications

References 2 publications

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

The Short-term Rational Lanczos Method and Applications

Sampling-free parametric model reduction of systems with structured parameter variation

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An ℋ₂ ⊗ ℒ₂‐Optimal Model Order Reduction Approach for Parametric Linear Time‐Invariant Systems