Power distribution analysis of VLSI interconnects using model order reduction

Shin, Youngsoo; Sakurai, Takayasu

doi:10.1109/tcad.2002.1004318

Cited by 32 publications

(39 citation statements)

References 7 publications

(16 reference statements)

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“…Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].…”

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confidence: 99%

Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

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confidence: 99%

Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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“…The following representations and techniques for analyzing time-domain, dynamic performance, and frequency-domain stability of higher-order RLC-TIN systems, have been widely applied: 1) algebraic equations and differential equations, 2) signal flow graphs and transfer functions, and 3) higher-order reduction and model reduction methods [2,[4][5][6][7][8][9][10][11]. The use of higher-order reduction and model reduction approaches sometimes results in the loss of correspondence to the original physical system structure.…”

Section: Introductionmentioning

confidence: 99%

“…Several concerns regarding system modeling and simulation techniques for VLSI interconnect with an RLC higher-order tree networks have been addressed in the literature. Most analyses employ a reducedorder model or other model reduction methods to obtain simulation results for comparison with those obtained from SPICE or HSPICE tools [4][5][6][7][8][9][10][11]. It is forbidden for simulations to use SPICE-like tools to the large number of interconnect line networks.…”

Section: Introductionmentioning

confidence: 99%

“…The BG methodology, which was introduced by M.I.T. Prof. H. M. Paynter in 1959, is a graphical tool for describing and modeling subsystem interactions involving power exchange [8,9]. Applications of the BG method to model physical system structure, for a variety of practical examples, have been published in the literature [12,13,[21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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A novel structural modeling and analysis of VLSI interconnect with an RLC tree network system using a BG/SEBD approach

Chang

Kao

Chang

et al. 2011

Sci. China Inf. Sci.

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A practical and effective novel graphical structural modeling and analysis approach is presented. This approach is used for solving the state interactions modeling problem caused by the model approximations in VLSI interconnect for RLC tree network systems. The bond graph (BG) energy model with a simulink energybased block diagram (SEBD) algorithm is developed. The dynamic behaviors of an example of VLSI interconnect with a 20 th -order RLC tree are presented. Methods of sequent structuring and the delineation of dynamic interactions can be provided for users applying the BG/SEBD algorithms in the development of structural models. It is shown that the new BG/SEBD structural modeling approach requires neither mathematical higherorder reduction nor physically based model reduction. Therefore, useful information can be extracted from the BG/SEBD representation of physical RLC tree network systems. Both time-domain and frequency-domain simulation results are applied to analyze the dynamic response of energy-storing elements using BG/SEBD algorithm combined with Matlab/Simulink software. Finally, a causal analysis of the system zeros plot from the shortest causal path method is proposed, to predict the dynamic response of energy-storing elements and the system performance for design and verification purposes. The dynamic response outputs from different node points were revealed, and the obtained physical information is significant which suggests feasible directions for design (re-design) improvement toward better overall system performance.Keywords higher-order, RLC tree interconnect network (RLC-TIN), bond graph (BG), structural modeling, time-domain and frequency-domain analysis, causality analysis CitationChang R F, Kao W S, Chang C W, et al. A novel structural modeling and analysis of VLSI interconnect with an RLC tree network system using a BG/SEBD approach.

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“…Reduced-order models (ROMs) are used for efficient simulation [17,32] and control [18,28]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments, or "black box" optimization methods [31].…”

Section: Introductionmentioning

confidence: 99%

Error Estimation for Reduced Order Models of Dynamical systems

Homescu¹,

Petzold²,

Serban³

2003

View full text Add to dashboard Cite

Abstract. The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L99DOI. 10.1137/0706843921. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [2]. Reduced-order models (ROMs) are used for efficient simulation [17,32] and control [18,28]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments, or "black box" optimization methods [31].The proper orthogonal decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [23], identification of coherent structures [12,21], and control [27] and inverse problems [19]. The method has been em-

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Power distribution analysis of VLSI interconnects using model order reduction

Cited by 32 publications

References 7 publications

Error Estimation for Reduced‐Order Models of Dynamical Systems

Error Estimation for Reduced‐Order Models of Dynamical Systems

A novel structural modeling and analysis of VLSI interconnect with an RLC tree network system using a BG/SEBD approach

Error Estimation for Reduced Order Models of Dynamical systems

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