Interpolation-based parameterized model order reduction of delayed systems

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach

Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach

Guaranteed passive parameterized model order reduction of the partial element equivalent circuit (PEEC) method

The decrease of IC feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the system under study as a function of design parameters, such as geometrical and substrate features, in addition to frequency (or time). Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters. We propose an innovative PMOR technique applicable to PEEC analysis, which combines traditional passivity-preserving model order reduction methods and positive interpolation schemes. It is able to provide parametric reduced-order models, stable, and passive by construction over a user-defined range of design parameter values. Numerical examples validate the proposed approach

Passivity-preserving parameterized model order reduction using singular values and matrix interpolation

We present a parameterized model order reduction method based on singular values and matrix interpolation. First, a fast technique using grammians is utilized to estimate the reduced order, and then common projection matrices are used to build parameterized reduced order models (ROMs). The design space is divided into cells, and a Krylov subspace is computed for each cell vertex model. The truncation of the singular values of the merged Krylov subspaces from the models located at the vertices of each cell yields a common projection matrix per design space cell. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parameterized ROM. Pertinent numerical results validate the proposed technique

Parameterized model order reduction of delayed systems using an interpolation approach with amplitude and frequency scaling coefficients

When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. We present an innovative PMOR technique for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, amplitude and frequency scaling coefficients and positive interpolation schemes. It is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach

Guaranteed passive parameterized macromodeling by using Sylvester state-space realizations

A novel state-space realization for parameterized macromodeling is proposed in this paper. A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. This technique is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parameterized macromodels. The key points of the novel state-space realizations are the choice of a proper pivot matrix and a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester realization for parameterized macromodeling

Parameterized partial element equivalent circuit method for sensitivity analysis of multiport systems

This paper presents a new technique to perform parameterized sensitivity analyses of systems that depend on multiple design parameters, such as layout and substrate features. It uses the electromagnetic (EM) method called partial element equivalent circuit to compute state space matrices at a set of design space points. These EM matrices are interpolated as functions of the design parameters. The proposed interpolation scheme allows the computation of the derivatives of the matrices, which are needed to perform the sensitivity analysis. An extensive study of the required stability and passivity properties of the system involved in the parameterized sensitivity analysis is presented. Pertinent numerical results demonstrate the robustness, accuracy, and efficiency of the proposed methodology

A Parameterization Scheme for Lossy Transmission Line Macromodels with Application to High Speed Interconnects in Mobile Devices

We introduce a novel parameterization scheme based on the generalized method of characteristics (MoC) formacromodels of transmission-line structures having a cross section depending on several free geometrical and material parameters. This situation is common in early design stages, when the physical structures still have to be finalized and optimized under signal integrity and electromagnetic compatibility constraints. The topology of the adopted line macromodels has been demonstrated to guarantee excellent accuracy and efficiency. The key factors are propagation delay extraction and rational approximations, which intrinsically lead to a SPICE-compatible macromodel stamp. We introduce a scheme that parameterizes this stamp as a function of geometrical and material parameters such as conductor-width and separation, dielectric thickness, and permettivity. The parameterization is performed via multidimensional interpolation of the residue matrices in the rational approximation of characteristic admittance and propagation operators. A significant advantage of this approach consists of the possibility of efficiently utilizing the MoC methodology in an optimization scheme and eventually helping the design of interconnects.We apply the proposed scheme to flexible printed interconnects that are typically found in portable devices having moving parts. Several validations demonstrate the effectiveness of the approac

Software Engineering and Complexity in Effective Algebraic Geometry

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434