In this paper, a parallel high-order envelope-following (EF) method is presented. The proposed method exploits the high-order and A-stable Obreshkov formula (ObF) to provide superior accuracy and speedup for the EF technique. Utilizing ObF provides accurate and faster analysis while keeping the same accuracy as the conventional low-order integration methods. In addition, a parallel method that is based on multiple-shooting (MS) techniques is used. Using MS allows partitioning and solving the sensitivity equations in parallel. It is also shown that with proper partitioning scheme, high CPU scalability can be achieved.

The high-order A- and L-stable Obreshkov-based method was recently proposed for simulating the transient response of general nonlinear circuits in the time-domain. This method has consistently resulted in more than one order-of-magnitude speedup compared with the traditional methods used in the SPICE engine. Nonetheless, the formulation of this Obreshkov-based approach assumed that the memory elements in the circuits are characterized by a linear constituent equation. The goal of this paper is to generalize this approach to handle memory elements that are described by nonlinear constituent equations.

This paper describes a new approach to accelerate the simulation of the steady-state response of nonlinear circuits using the harmonic-balance (HB) technique. The approach presented in this paper focuses on the direct factorization of the Jacobian matrix, of the HB nonlinear equations, using a graphical processing unit (GPU) platform. The computational core of the proposed approach is based on developing a block-wise version of the KLU factorization algorithm, where scalar arithmetic operations are replaced by block-aware matrix operations. For a large number of harmonics, or excitation tones, or both, the Block-KLU (BKLU) approach effectively raises the ratio of floating-point operations to other operations and, therefore, becomes an ideal vehicle for implementation on a GPU-based platform. Motivated by this fact, we develop a GPU-based framework to implement the BKLU. The proposed approach yields speedup by up to 89 times over conventional direct factorization on CPU.

This paper presents structural characterization and performance enhancement strategies for the recently proposedA-stable and L-stable high-order integration methods based on the Obreshkov formula. It is demonstrated that although the Jacobian matrix in these methods has a bigger size than the Jacobian matrix in classical low-order methods, it enjoys a special structure that can be used to develop efficient factorization techniques. In addition, the paper proposes a method to reduce the number of Newton-Raphson iterations needed to converge in the large Jacobian domain.

This paper describes a new$A$- and$L$-stable integration method for simulating the time-domain transient response of nonlinear circuits. The proposed method, which is based on the Obreshkov formula, can be made of arbitrary high order while maintaining the$A$-stability property. The new method allows for the adoption of higher order integration methods for the transient analysis of electronic circuits while enabling them to take larger step sizes without violating the stability, leading to faster simulations. The method can be run in an$L$-stable mode to handle circuits with extremely stiff equations. Necessary theoretical foundations, implementation details, error-control mechanisms, and computational results are presented.

Computation of passive and compact macromodels of distributed interconnects has gained considerable importance during the recent years. Method of characteristics (MoC) is widely used for macromodeling of transmission lines, however, it may not be guaranteed passive. This paper presents a new algorithm for passivity enforcement of MoC-based macromodels of multiconductor transmission lines. The algorithm is based on the first-order perturbation of the related delay differential equations and can handle single as well as coupled interconnects. Necessary theoretical foundations and validating numerical results are presented.

This paper presents a robust method for guaranteed passive macromodeling of high-speed interconnects characterized by multiconductor transmission lines with frequency-dependent per-unit-length (FD-PUL) parameters, using integrated-congruence-transformation-based model-order-reduction techniques. For this purpose, a new formulation is developed to directly incorporate state-space equations corresponding to the positive-real (PR) rational functions representing the frequency dependence of PUL parameters, in Telegrapher's equations. The proposed formulation overcomes the need for explicit lumped-circuit realization with all positive elements for PR rational functions of FD-PUL parameters. Proof of passivity and numerical examples are presented to validate the theory.

This paper describes a new algorithm for the analysis of high-speed interconnects characterized by frequency-dependent per-unit-length parameter matrices. The proposed technique is based on using the idea of integrated congruence transform to construct a reduced-order system for the interconnect. The reduced-order system offers simple time-domain ordinary differential equations that can be embedded into conventional circuit simulators. The proposed algorithm provides a new approach to perform implicit moment matching on a system described by a mixed set of linear differential algebraic equations. This feature offers a high accuracy in the reduced system over a wide frequency range. The passivity of the reduced-order system is guaranteed. Numerical results demonstrate the validity of the proposed technique.

This paper presents a new algorithm for computing the sensitivity of steady-state responses of nonlinear circuits with respect to an arbitrary network parameter. The proposed algorithm is based on circuit-reduction techniques obtained through nonlinear-projection approaches. The main idea in the proposed sensitivity-computation algorithm is based on projecting the adjoint system of equations onto a subspace of smaller dimension. Continuation methods are then applied on the reduced system to trace the solution trajectory of the adjoint system in the reduced space. We show that once the steady-state solution has been obtained using nonlinear order reduction, the computational cost required to compute sensitivity is only a limited number of forward/backward substitutions. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed algorithm.

This paper presents a new algorithm based on integrated congruence transform for efficient simulation of nonuniform transmission lines. The proposed algorithm introduces the concept of model-order reduction (MOR) via implicit usage of the Hilbert-space moments in distributed networks. The key idea in the proposed algorithm is the development of an orthogonalization procedure that does not require the explicit computation of the Hilbert-space moments in order to find their spanning orthogonal basis. The proposed orthogonalization procedure can thus be used to compute an orthogonal basis for any set of elements that are related through a differential operator in a generalized Hilbert space, without the need to have these elements in an explicit form. The proposed algorithm also addresses the problem of MOR of nonuniform transmission lines, through defining a weighted inner product and norm mappings over the Hilbert space of the moments. Numerical examples demonstrate more accurate numerical approximation capabilities over using the moments explicitly.

This paper presents a new algorithm based on Integrated Congruence transform for the analysis of both uniform and nonuniform transmission lines. The key advantage of the proposed algorithm is that constructing a spanning orthonormal basis for the space-dependent moments is done without computing these moments explicitly. The proposed algorithm thus carries the numerical efficiency of Krylov-based projection techniques of lumped RLC networks to the domain of the distributed transmission line networks. The proposed algorithm can be used to construct an orthogonal basis for any set of moments related through a differential operator.

This paper presents a new approach for model-order reduction of linear time varying system based on expanding the time-varying system in the right half plane of the s-domain. The proposed algorithm is developed through introducing Krylov subspace-based reduction to time-varying transfer functions. The proposed algorithm does not require solution of large system of equations to construct a basis for the time-varying moments. Instead, it computes such a basis through time-domain integration of the corresponding linear time-varying differential algebraic equations. Numerical experiments show that expanding in the right-half plane compresses the transient phase of the response of these equations by several orders of magnitude.

Signal integrity analysis has become imperative for high-speed designs. In this paper, we present a new technique to advance Krylov-space based passive model-reduction algorithms to include lossy coupled transmission lines described by Telegrapher's equations. In the proposed scheme, transmission line subnetworks are treated with closed-form stamps obtained using matrix-exponential Padé, where the coefficients describing the model are computed a priori and analytically. In addition, a technique is given to ensure that the contribution of these stamps to the modified nodal analysis (MNA) formulation leads to guaranteed passive macromodel.

A new algorithm based on model reduction using the Krylov subspace technique is proposed to compute the DC solution of large nonlinear circuits. The proposed method combines continuation methods with model reduction techniques. Thus it enables the application of the continuation methods to an equivalent reduced-order set of nonlinear equations instead of the original system. This results in a significant reduction in the computational expense as the size of the reduced equations is much less than that of the original system. The reduced order system is obtained by projecting the set of nonlinear equations, whose solution represents the DC operating point, into a subspace of a much lower dimension. It is also shown that both the reduced-order system and the original system share the first q derivatives w.r.t. the circuit variable used to parameterize the family of the solution trajectories generated by the continuation method.