Jing-Rebecca Li

Low Rank Solution of Lyapunov Equations

This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX+XAT=-BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems

We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newtons method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.

An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect

In this paper we present a new algorithm for computing reduced-order models of interconnect which utilizes the dominant controllable subspace of the system. The dominant controllable modes are computed via a new iterative Lyapunov equation solver, Vector ADI. This new algorithm is as inexpensive as Krylov subspace-based moment matching methods, and often produces a better approximation over a wide frequency range. A spiral inductor and a transmission line example show this new method can be much more accurate than moment matching via Arnoldi.

Low-Rank Solution of Lyapunov Equations

This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX+XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

Efficient model reduction of interconnect via approximate system gramians

Krylov-subspace based methods for generating low-order models of complicated interconnect are extremely effective, but there is no optimality theory for the resulting models. Alternatively, methods based on truncating a balanced realization (TBR), in which the observability and controllability gramians have been diagonalized, do have an optimality property but are too computationally expensive to use on complicated problems. In this paper we present a method for computing reduced-order models of interconnect by projection via the orthogonalized union of the approximate dominant eigenspaces of the systems controllability and observability gramians. The approximate dominant eigenspaces are obtained efficiently using an iterative Lyapunov equation solver, Vector ADI, which requires only linear matrix-vector solves. A spiral inductor and a transmission line example are used to demonstrate that the new method accurately approximates the TBR results and gives much more accurate wideband models than Krylov subspace-based moment matching methods.

A Fast Time Stepping Method for Evaluating Fractional Integrals

We evaluate the fractional integral

On the numerical solution of the heat equation I

I^\alpha[f](t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-\tau)^{\alpha-1}\,f(\tau)\,d\tau

High Order Accurate Methods for the Evaluation of Layer Heat Potentials

,

Strongly consistent marching schemes for the wave equation

0<\alpha<1

PEEC Model of a Spiral Inductor Generated by Fasthenry

, at time steps