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 + XA T = −BB T. The coefficient matrix A is assumed to be large, and the rank of the righthand side −BB T 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.
International audienceWe 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 Newton's 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
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 + XA T = −BB T . The coefficient matrix A is assumed to be large, and the rank of the right-hand side −BB T 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.
We evaluate the fractional integralat time steps t = Δt, 2Δt, . . . , NΔt by making use of the integral representation of the convolutionconstruct an efficient Q-point quadrature of this integral representation and use it as a part of a fast time stepping method. The new method has algorithmic complexity O(NQ) and storage requirement O(Q). The number of quadrature nodes Q is independent of N and grows like O − log − log Δt 2 , where is the quadrature error tolerance and Δt is the size of the time step. The (possible) singularity of f near τ = 0 is taken into account. This new method is particularly well-suited for long time simulations.
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