The alternating direction implicit (ADI) method is proposed for low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. The low-rank solution is expressed by Cholesky factor that is similar to that of Cholesky factorization for linear system of equations. The Cholesky factor is represented in a real form so that it is useful for balanced truncation of sparsely connected RLC networks. Moreover, we show how to determine the shift parameters which are required for the ADI iterations, where Krylov subspace method is used for finding the shift parameters that reduce the residual error quickly. In the illustrative examples, we confirm that the real Cholesky factor certainly provides low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. Effectiveness of the shift parameters determined by Krylov subspace method is also demonstrated.

In this paper, the performance of the induced dimension reduction (IDR) method implemented along with the method of moments (MoM) is described. The MoM is based on a combined field integral equation for solving large-scale electromagnetic scattering problems involving conducting objects. The IDR method is one of Krylov subspace methods. This method was initially developed by Peter Sonneveld in 1979; it was subsequently generalized to the IDR(s) method. The method has recently attracted considerable attention in the field of computational physics. However, the performance of the IDR(s) has hardly been studied or practiced for electromagnetic wave problems. In this study, the performance of the IDR(s) is investigated and clarified by comparing the convergence property and memory requirement of the IDR(s) with those of other representative Krylov solvers such as biconjugate gradient (BiCG) methods and generalized minimal residual algorithm (GMRES). Numerical experiments reveal that the characteristics of the IDR(s) against the parameter s strongly depend on the geometry of the problem; in a problem with a complex geometry, s should be set to an adequately small value in order to avoid the "spurious convergence" which is a problem that the IDR(s) inherently holds. As for the convergence behavior, we observe that the IDR(s) has a better convergence ability than GPBiCG and GMRES(m) in a variety of problems with different complexities. Furthermore, we also confirm the IDR(s)'s inherent advantage in terms of the memory requirements over GMRES(m).

This paper presents flexible inner-outer Krylov subspace methods, which are implemented using the fast multipole method (FMM) for solving scattering problems with mixed dielectric and conducting object. The flexible Krylov subspace methods refer to a class of methods that accept variable preconditioning. To obtain the maximum efficiency of the inner-outer methods, it is desirable to compute the inner iterations with the least possible effort. Hence, generally, inaccurate matrix-vector multiplication (MVM) is performed in the inner solver within a short computation time. This is realized by using a particular feature of the multipole techniques. The accuracy and computational cost of the FMM can be controlled by appropriately selecting the truncation number, which indicates the number of multipoles used to express far-field interactions. On the basis of the abovementioned fact, we construct a less-accurate but much cheaper version of the FMM by intentionally setting the truncation number to a sufficiently low value, and then use it for the computation of inaccurate MVM in the inner solver. However, there exists no definite rule for determining the suitable level of accuracy for the FMM within the inner solver. The main focus of this study is to clarify the relationship between the overall efficiency of the flexible inner-outer Krylov solver and the accuracy of the FMM within the inner solver. Numerical experiments reveal that there exits an optimal accuracy level for the FMM within the inner solver, and that a moderately accurate FMM operator serves as the optimal preconditioner.

The passive and sparse reduced-order modeling of a RLC network is presented, where eigenvalues and eigenvectors of the original network are used, and thus the obtained macromodel is more accurate than that provided by the Krylov subspace methods or TBR procedures for a class of circuits. Furthermore, the proposed method is applied to low pass filtering of a reduced-order model produced by these methods without breaking the passivity condition. Therefore, the proposed eigenspace method is not only a reduced-order macromodeling method, but also is embedded in other methods enhancing their performances.

Two versions of Krylov subspace order reduction techniques for VLSI interconnect reductions, including structure preserving reductions approach and adjoint networks approach, will be comparatively investigated. Also, we will propose a modified structure preserving reduction algorithm to speed up the projection construction in a linear order. The numerical experiment shows the high accuracy and low computational consumption of the modified method. In addition, it will be shown that the projection subspace generated from the structure-preserving approach and those from the adjoint networks approach are equivalent. Therefore, transfer functions of both reduced networks are identical.

The global Lanczos algorithm for solving the RLCG interconnect circuits is presented in this paper. This algorithm is an extension of the standard Lanczos algorithm for multiple-inputs multiple-outputs (MIMO) systems. A new matrix Krylov subspace will be developed first. By employing the congruence transformation with the matrix Krylov subspace, the two-side oblique projection-based method can be used to construct a reduced-order system. It will be shown that the system moments are still matched. The error of the 2q-th order system moment will be derived analytically. Furthermore, two novel model-order reduction techniques called the multiple point global Lanczos (MPGL) method and the adaptive-order global Lanczos (AOGL) method which are both based on the multiple point moment matching are proposed. The frequency responses using the multiple point moment matching method have higher coherence to the original system than those using the single point expansion method. Finally, simulation results on frequency domain will illustrate the feasibility and the efficiency of the proposed methods.

We extend the adaptive-order rational Arnoldi algorithm for multiple-inputs and multiple-outputs (MIMO) interconnect model order reductions. Instead of using the standard Arnoldi algorithm for the SISO adaptive-order reduction algorithm (AORA), we study the adaptive-order rational global Arnoldi (AORGA) algorithm for MIMO model reductions. In this new algorithm, the input matrix is treated as a vector form. A new matrix Krylov subspace, generated by the global Arnoldi algorithm, will be developed by a Frobenius-orthonormal basis. By employing congruence transformation with the matrix Krylov subspace, the one-sided projection method can be used to construct a reduced-order system. It will be shown that the system moment matching can be preserved. In addition, we also show that the transfer matrix residual error of the reduced system can be derived analytically. This error information will provide a guideline for the order selection scheme. The algorithm can also be applied to the classical multiple point MIMO Pade approximation by the rational Arnoldi algorithm for multiple expansion points. Experimental results demonstrate the feasibility and the effectiveness of the proposed method.

A new indirect approach for designing low-order linear-phase IIR filters is presented in this paper. Given an FIR filter, we utilize a new Krylov subspace projection method, called the rational Arnoldi method with adaptive orders, to synthesize an approximated IIR filter with small orders. The synthesized IIR filter can truly reflect essential dynamical features of the original FIR filter and indeed satisfies the design specifications. Also, from simulation results, it can be observed that the linear-phase property in the passband is stilled retained. This indirect approach is accomplished using the state-space realization of FIR filters, multi-point Pade approximations, the Arnoldi algorithm, and an intelligent scheme to select expansion points in the frequency domain. Such methods are quite efficient in terms of computational complexity. Fundamental developments of the proposed method will be discussed in details. Numerical results will demonstrate the accuracy and the efficiency of this two-step indirect method.

We present theoretical results on the convergence of iterative methods for the solution of linear differential-algebraic equations arising form circuit simulation. The iterative methods considered include the continuous-time and discretetime waveform relaxation methods and the Krylov subspace methods in function space. The waveform generalized minimal residual method for solving linear differential-algebraic equations in function space is developed, which is one of the waveform Krylov subspace methods. Some new criteria for convergence of these iterative methods are derived. Examples are given to verify the convergence conditions.