On the Parameter Selection Problem in the Newton-ADI Iteration for Large Scale Riccati Equations

The numerical treatment of linear-quadratic regulator problems for parabolic partial differential equations (PDEs) on infinite time horizons requires the solution of large scale algebraic Riccati equations (ARE). The Newton-ADI iteration is an efficient numerical method for this task. It includes the solution of a Lyapunov equation by the alternating directions implicit (ADI) algorithm in each iteration step. On finite time intervals the solution of a large scale differential Riccati equation is required. This can be solved by a backward differentiation formula (BDF) method, which needs to solve an ARE in each time step. Here, we study the selection of shift parameters for the ADI method. This leads to a rational min-max-problem which has been considered by many authors. Since knowledge about the complete complex spectrum is crucial for computing the optimal solution, this is infeasible for th

Quadratic alternating direction implicit iteration for the fast solution of algebraic Riccati equations

Algebraic Riccati equations (AREs) spread over many branches of signal processing and system design problems. Solution of large scale AREs, however, can be computationally prohibitive. This paper introduces a novel second order extension to the alternating direction implicit (ADI) iteration, called quadratic ADI or QADI, for the efficient solution of an ARE. QADI is simple to code and exhibits fast convergence. A Cholesky factor variant of QADI, called CFQADI, further accelerates computation by exploiting low rank matrices commonly found in physical system modeling. Application examples show remarkable efficiency and scalability of the QADI algorithms over conventional ARE solvers. © 2005 IEEE.published_or_final_versio

Fast positive-real balanced truncation via quadratic alternating direction implicit iteration

Balanced truncation (BT), as applied to date in model order reduction (MOR), is known for its superior accuracy and computable error bounds. Positive-real BT (PRBT) is a particular BT procedure that preserves passivity and stability and imposes no structural constraints on the original state space. However, PRBT requires solving two algebraic Riccati equations (AREs), whose computational complexity limits its practical use in large-scale systems. This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, which is called quadratic ADI (QADI), that efficiently solves an ARE. A Cholesky factor version of QADI, which is called CEQADI, exploits low-rank matrices and further accelerates PRBT. © 2007 IEEE.published_or_final_versio