Nonlinear multigrid for the solution of large‐scale Riccati equations in low‐rank and ℋ︁‐matrix format

The algebraic matrix Riccati equation AX+XAT−XFX+C=0, where matrices A, B, C, F ∈ ℝn × n are given and a solution X ∈ ℝn × n is sought, plays a fundamental role in optimal control problems. Large-scale systems typically appear if the constraint is described by a partial differential equation (PDE). We provide a nonlinear multigrid algorithm that computes the solution X in a data-sparse, low-rank format and has a complexity of (n), subject to the condition that F and C are of low rank and A is the finite element or finite difference discretization of an elliptic PDE. We indicate how to generalize the method to ℋ-matrices C, F and X that are only blockwise of low rank and thus allow a broader applicability with a complexity of (nlog(n)p), p being a small constant. The method can also be applied to unstructured and dense matrices C and X in order to solve the Riccati equation in (n2). Copyright © 2008 John Wiley & Sons, Ltd.

[1]  Kirsten Morris,et al.  SOLUTION OF ALGEBRAIC RICCATI EQUATIONS ARISING IN CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[2]  A. Laub.  Invariant Subspace Methods for the Numerical Solution of Riccati Equations , 1991 .

[3]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[4]  Tongxing Lu,et al.  Solution of the matrix equation AX−XB=C , 2005, Computing.

[5]  J. D. Roberts,et al.  Linear model reduction and solution of the algebraic Riccati equation by use of the sign function , 1980 .

[6]  R. Curtain.  Infinite-Dimensional Linear Systems Theory , 1978 .

[7]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[8]  P. Dooren.  A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[9]  V. Mehrmann,et al.  Defect correction method for the solution of algebraic Riccati equations , 1988 .

[10]  A. A. Samarskii,et al.  Numerical Methods for Grid Equations , 2018 .

[11]  P. Benner,et al.  Solving large-scale control problems , 2004, IEEE Control Systems.

[12]  Thilo Penzl,et al.  A Multi-Grid Method for Generalized Lyapunov Equations , 2005 .

[13]  Fredi Tröltzsch,et al.  Theoretical and Numerical Aspects of Controlled Cooling of Steel Profiles , 1996 .

[14]  Stephen G. Nash,et al.  Numerical Methods for Grid Equations: Vol. II: Iterative Methods , 1990 .

[15]  Chun-Hua Guo,et al.  Analysis and modificaton of Newton's method for algebraic Riccati equations , 1998, Math. Comput..

[16]  On maximum norm convergence of multigrid methods for two-point boundary value problems , 1992 .

[17]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[18]  T. Stykel.  Generalized Alternating Direction Implicit Method for Projected Generalized Lyapunov Equations , 2004 .

[19]  Thilo Penzl.  Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case , 2000 .

[20]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[21]  J. S. Gibson,et al.  The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces , 1979 .

[22]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[23]  Enrique S. Quintana-Ortí,et al.  Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search , 2000, Parallel Comput..

[24]  Y. Zhou,et al.  On the decay rate of Hankel singular values and related issues , 2002, Syst. Control. Lett..

[25]  Jacob K. White,et al.  Low-Rank Solution of Lyapunov Equations , 2004, SIAM Rev..

[26]  Karl Kunisch,et al.  The linear regulator problem for parabolic systems , 1984 .

[27]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[28]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[29]  Wen-Chyuan Yueh.  EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES , 2005 .

[30]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[31]  P. Dorato,et al.  Dynamical systems and their applications: Linear theory , 1979, Proceedings of the IEEE.

[32]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[33]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[34]  C. Stéphanos.  Sur une extension du calcul des substitutions linéaires , 1900 .

[35]  Robert E. Mahony,et al.  Cubically Convergent Iterations for Invariant Subspace Computation , 2004, SIAM J. Matrix Anal. Appl..

[36]  Rafael Mayo,et al.  Parallel Solution of Large-Scale and Sparse Generalized Algebraic Riccati Equations , 2006, Euro-Par.

[37]  Ronald Kriemann,et al.  Parallel Black Box Domain Decomposition Based H-LU Preconditioning , 2005 .

[38]  Eugene L. Wachspress,et al.  Solution of lyapunov equations by alternating direction implicit iteration , 1991 .

[39]  Boris N. Khoromskij,et al.  Solution of Large Scale Algebraic Matrix Riccati Equations by Use of Hierarchical Matrices , 2003, Computing.

[40]  Lars Grasedyck,et al.  Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation , 2004, Numer. Linear Algebra Appl..

[41]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem , 1991 .

[42]  I. Rosen,et al.  A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations , 1995 .

[43]  Wolfgang Hackbusch,et al.  A Multigrid Method to Solve Large Scale Sylvester Equations , 2007, SIAM J. Matrix Anal. Appl..

[44]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[45]  Thilo Penzl,et al.  A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations , 1998, SIAM J. Sci. Comput..