Order reduction numerical methods for the algebraic Riccati equation. V. Simoncini

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1 Order reduction numerical methods for the algebraic Riccati equation V. Simoncini Dipartimento di Matematica Alma Mater Studiorum - Università di Bologna valeria.simoncini@unibo.it 1

2 The problem Find X R n n such that AX+XA XBB X+C C = 0 with A R n n, B R n p, C R s n, p,s = O(1) Rich literature on analysis, applications and numerics: Lancaster-Rodman 1995, Bini-Iannazzo-Meini 2012, Mehrmann etal We focus on the large scale case: n 1000 Different strategies (Inexact) Kleinman iteration (Newton-type method) Projection methods Invariant subspace iteration (Sparse) multilevel methods... 2

3 The problem Find X R n n such that AX+XA XBB X+C C = 0 with A R n n, B R n p, C R s n, p,s = O(1) Rich literature on analysis, applications and numerics: Lancaster-Rodman 1995, Bini-Iannazzo-Meini 2012, Mehrmann etal We focus on the large scale case: n 1000 Different strategies (Inexact) Kleinman iteration (Newton-type method) Projection methods Invariant subspace iteration (Sparse) multilevel methods... 3

4 Newton-Kleinman iteration Assume A stable. Compute sequence {X k } with X k k X (A X k BB )X k+1 +X k+1 (A BB X k )+C C +X k BB X k = 0 1: Given X 0 R n n such that X 0 = X 0, A BB X 0 is stable 2: For k = 0,1,..., until convergence 3: Set A k = A BB X k 4: Set Ck = [X k B, C ] 5: Solve A k X k+1 +X k+1 A k +C k C k = 0 Critical issues: The full matrix X k cannot be stored (sparse or low-rank approx) Need a computable stopping criterion Each iteration k requires the solution of the Lyapunov equation: (Benner, Feitzinger, Hylla, Saak, Sachs,...) 4

5 Newton-Kleinman iteration Assume A stable. Compute sequence {X k } with X k k X (A X k BB )X k+1 +X k+1 (A BB X k )+C C +X k BB X k = 0 1: Given X 0 R n n such that X 0 = X 0, A BB X 0 is stable 2: For k = 0,1,..., until convergence 3: Set A k = A BB X k 4: Set Ck = [X k B, C ] 5: Solve A k X k+1 +X k+1 A k +C k C k = 0 Critical issues: The full matrix X k cannot be stored (sparse or low-rank approx) Need a computable stopping criterion Each iteration k requires the solution of the Lyapunov equation: (Benner, Feitzinger, Hylla, Saak, Sachs,...) 5

6 Newton-Kleinman iteration Assume A stable. Compute sequence {X k } with X k k X (A X k BB )X k+1 +X k+1 (A BB X k )+C C +X k BB X k = 0 1: Given X 0 R n n such that X 0 = X 0, A BB X 0 is stable. 2: For k = 0,1,..., until convergence 3: Set A k = A BB X k 4: Set Ck = [X k B, C ] 5: Solve A k X k+1 +X k+1 A k +C k C k = 0 Critical issues: The full matrix X k cannot be stored (sparse or low-rank approx) Need a computable stopping criterion Each iteration k requires the solution of the Lyapunov equation (Benner, Feitzinger, Hylla, Saak, Sachs,...) 6

7 Galerkin projection method for the Riccati equation Given the orth basis V k for an approximation space, determine X k = V k Y k V k to the Riccati solution matrix by orthogonal projection: Galerkin condition: Residual orthogonal to approximation space V k (AX k +X k A X k BB X k +C C)V k = 0 giving the reduced Riccati equation (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 Y k is the stabilizing solution (Heyouni-Jbilou 2009) Key questions: Which approximation space? Is this meaningful from a Control Theory perspective? 7

8 Galerkin projection method for the Riccati equation Given the orth basis V k for an approximation space, determine X k = V k Y k V k to the Riccati solution matrix by orthogonal projection: Galerkin condition: Residual orthogonal to approximation space V k (AX k +X k A X k BB X k +C C)V k = 0 giving the reduced Riccati equation (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 Y k is the stabilizing solution (Heyouni-Jbilou 2009) Key questions: Which approximation space? Is this meaningful from a Control Theory perspective? 8

9 Galerkin projection method for the Riccati equation Given the orth basis V k for an approximation space, determine X k = V k Y k V k to the Riccati solution matrix by orthogonal projection: Galerkin condition: Residual orthogonal to approximation space V k (AX k +X k A X k BB X k +C C)V k = 0 giving the reduced Riccati equation (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 Y k is the stabilizing solution (Heyouni-Jbilou 2009) Key questions: Which approximation space? Is this meaningful from a Control Theory perspective? 9

10 Galerkin projection method for the Riccati equation Given the orth basis V k for an approximation space, determine X k = V k Y k V k to the Riccati solution matrix by orthogonal projection: Galerkin condition: Residual orthogonal to approximation space V k (AX k +X k A X k BB X k +C C)V k = 0 giving the reduced Riccati equation (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 Y k is the stabilizing solution (Heyouni-Jbilou 2009) Key questions: Which approximation space? Is this meaningful from a Control Theory perspective? 10

11 Dynamical systems and the Riccati equation AX+XA XBB X+C C = 0 Time-invariant linear system ẋ(t) = Ax(t)+Bu(t), x(0) = x 0 y(t) = Cx(t), u(t) : control (input) vector; y(t) : output vector x(t) : state vector; x 0 : initial state Minimization problem for a Cost functional: (simplified form) inf u J(u,x 0) J(u,x 0 ) := 0 (x(t) C Cx(t)+u(t) u(t))dt 11

12 Dynamical systems and the Riccati equation AX+XA XBB X+C C = 0 Time-invariant linear system ẋ(t) = Ax(t)+Bu(t), x(0) = x 0 y(t) = Cx(t), u(t) : control (input) vector; y(t) : output vector x(t) : state vector; x 0 : initial state Minimization problem for a Cost functional: (simplified form) inf u J(u,x 0) J(u,x 0 ) := 0 ( x(t) C Cx(t)+u(t) u(t) ) dt 12

13 Dynamical systems and the Riccati equation AX+XA XBB X+C C = 0 inf u J(u,x 0) J(u,x 0 ) := 0 ( x(t) C Cx(t)+u(t) u(t) ) dt theorem Let the pair (A,B) be stabilizable and (C,A) observable. Then there is a unique solution X 0 of the Riccati equation. Moreover, i) For each x 0 there is a unique optimal control, and it is given by u (t) = B Xexp((A BB X)t)x 0 for t 0 ii) J(u,x 0 ) = x 0 Xx 0 for all x 0 R n see, e.g., Lancaster & Rodman,

14 Order reduction of dynamical systems by projection Let V k R n d k have orthonormal columns, d k n Let T k = V k AV k, B k = V k B, C k = V k C Reduced order dynamical system: x(t) = T k x(t)+b k û(t), ŷ(t) = C k x(t) x(0) = x 0 :=V k x 0 x k (t) = V k x(t) x(t) Typical frameworks: Transfer function approximation Model reduction 14

15 The role of the projected Riccati equation Consider again the reduced Riccati equation: (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 that is T k Y +YT k YB k B k Y +C k C k = 0 ( ) Theorem. Let the pair (T k,b k ) be stabilizable and (C k,t k ) observable. Then there is a unique solution Y k 0 of (*) that for each x 0 gives the feedback optimal control û (t) = B ky k exp((t k B k B ky k )t) x 0, t 0 for the reduced system. If there exists a matrix K such that A BK is passive, then the pair (T k,b k ) is stabilizable. 15

16 The role of the projected Riccati equation Consider again the reduced Riccati equation: (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 that is T k Y +YT k YB k B k Y +C k C k = 0 ( ) Theorem. Let the pair (T k,b k ) be stabilizable and (C k,t k ) observable. Then there is a unique solution Y k 0 of (*) that for each x 0 gives the feedback optimal control û (t) = B ky k exp((t k B k B ky k )t) x 0, t 0 for the reduced system. If there exists a matrix K such that A BK is passive, then the pair (T k,b k ) is stabilizable. 16

17 The role of the projected Riccati equation Consider again the reduced Riccati equation: (V k AV k )Y +Y(V k A V k ) Y(V k BB V k )Y +(V k C )(CV k ) = 0 that is T k Y +YT k YB k B k Y +C k C k = 0 ( ) Theorem. Let the pair (T k,b k ) be stabilizable and (C k,t k ) observable. Then there is a unique solution Y k 0 of (*) that for each x 0 gives the feedback optimal control û (t) = B ky k exp((t k B k B ky k )t) x 0, t 0 for the reduced system. If there exists a matrix K such that A BK is passive, then the pair (T k,b k ) is stabilizable. 17

18 Projected optimal control vs approximate control Our projected optimal control function: û (t) = B k Y k exp((t k B k B k Y k )t) x 0, t 0 with X k = V k Y k V k Commonly used approximate control function: If X is some approximation to X, then ũ(t) := B X x(t) where x(t) := exp((a BB X)t)x0 û ũ They induce different actions on the functional J, even for X = X k 18

19 X k = V k Y k V k Projected optimal control vs approximate control Residual matrix: R k := AX k +X k A X k BB X k +C C Projected optimal control function: û (t) = B k Y kexp((t k B k B k Y k)t) theorem. Assume that A BB X k is stable and that ũ(t) := B X k x(t) approx control. Then J(ũ,x 0 ) Ĵk(û, x 0 ) = E k, with E k R k 2α x 0 x 0, where α > 0 is such that e (A BB X k )t e αt for all t 0. Note: J(ũ,x 0 ) Ĵk(û, x 0 ) is nonzero for R k 0 19

20 On the choice of approximation space Approximate solution X k = V k Y k V k, with (V k AV k )Y k +Y k (V k A V k ) Y k (V k BB V k )Y k +(V k C )(CV k ) = 0 Krylov-type subspaces: (extensively used in the linear case) K k (A,C ) := Range([C,AC,...,A k 1 C ]) (Polynomial) EK k (A,C ) := K k (A,C )+K k (A 1,A 1 C ) (EKS, Rational) RK k (A,C,s) := k 1 Range([C,(A s 2 I) 1 C,..., (A s j+1 I) 1 C ]) j=1 (RKS, Rational) Matrix BB not involved Parameters s j (adaptively) chosen in field of values of A 20

21 On the choice of approximation space Approximate solution X k = V k Y k V k, with (V k AV k )Y k +Y k (V k A V k ) Y k (V k BB V k )Y k +(V k C )(CV k ) = 0 Krylov-type subspaces: (extensively used in the linear case) K k (A,C ) := Range([C,AC,...,A k 1 C ]) (Polynomial) EK k (A,C ) := K k (A,C )+K k (A 1,A 1 C ) (EKS, Rational) RK k (A,C,s) := k 1 Range([C,(A s 2 I) 1 C,..., (A s j+1 I) 1 C ]) j=1 (RKS, Rational) Matrix BB not involved (nonlinear term!) Parameters s j (adaptively) chosen in field of values of A 21

22 On the choice of approximation space Approximate solution X k = V k Y k V k, with (V k AV k )Y k +Y k (V k A V k ) Y k (V k BB V k )Y k +(V k C )(CV k ) = 0 Krylov-type subspaces: (extensively used in the linear case) K k (A,C ) := Range([C,AC,...,A k 1 C ]) (Polynomial) EK k (A,C ) := K k (A,C )+K k (A 1,A 1 C ) (EKS, Rational) RK k (A,C,s) := k 1 Range([C,(A s 2 I) 1 C,..., (A s j+1 I) 1 C ]) j=1 (RKS, Rational) Matrix BB not involved (nonlinear term!) Parameters s j (adaptively) chosen in field of values of A 22

23 Performance of solvers Problem: A: 3D Laplace operator, B, C randn matrices, tol=10 8 (n,p,s) = (125000,5,5) its inner its time space dim rank(x f ) Newton X 0 = ,..., GP-EKS GP-RKS (n,p,s) = (125000,20,20) its inner its time space dim rank(x f ) Newton X 0 = ,..., GP-EKS GP-RKS GP=Galerkin projection (V.Simoncini & D.Szyld & M.Monsalve, 2014) 23

24 A numerical example on the role of BB Consider the Toeplitz matrix A = toeplitz( 1,2.5,1,1,1), C = [1, 2,1, 2,...],B = RKSM 10 0 absolute residual norm space dimension Parameter computation: Left: adaptive RKS on A Right: adaptive RKS on A BB X k (Lin & Simoncini 2015) 24

25 A numerical example on the role of BB Consider the Toeplitz matrix A = toeplitz( 1,2.5,1,1,1), C = [1, 2,1, 2,...],B = RKSM 10 4 RKSM absolute residual norm absolute residual norm space dimension space dimension Parameter computation: Left: adaptive RKS on A Right: adaptive RKS on A BB X k (Lin & Simoncini 2015) 25

26 On the residual matrix and adaptive RKS R k := AX k +X k A X k BB X k +C C theorem. Let T k = T k B k B k Y k. Then R k = R k V k +V k R k, with Rk = AV k Y k +V k Y k T k +C (CV k ) so that R k F = 2 R k F At least formally: V k Y k V k is a solution to the Riccati equation (R k = 0) if and only if Z k = V k Y k is the solution to the Sylvester equation ( R k = 0) 26

27 On the residual matrix and adaptive RKS R k = R k V k +V k R k Expression for the semi-residual Rk : theorem. Assume C R n, Range(V k )= RK k (A,C,s). Assume that T k = T k B k B k Y k is diagonalizable. Then R k = ψ k,tk (A)C CV k (ψ k,tk ( T k )) 1. where ψ k,tk (z) = det(zi T k) k j=1 (z s j) (see also Beckermann 2011 for the linear case) 27

28 On the choice of the next parameters s k+1 with ψ k,tk (z) = det(zi T k) k j=1 (z s j) R k = ψ k,tk (A)C CV k (ψ k,tk ( T k )) 1. Greedy strategy: Next shift should make (ψ k,tk ( T k )) 1 smaller Determine for which s in the spectral region of T k the quantity (ψ k,tk ( s)) 1 is large, and add a root there s k+1 = arg max s S k 1 ψ k,tk (s) S k region enclosing the eigenvalues of T k = (T k B k B k Y k) (This argument is new also for linear equations) 28

29 Selection of s k+1 in RKS. An example A: D Laplacian, B = t1 with t j = 5 10 j, C = [1, 2,1, 2,1, 2,...] absolute residual norm t 2 t t 1 t 1 t 2 t space dimension RKS convergence with and without modified shift selection as t varies Solid curves: use of T k Dashed curves: use of T k 29

30 Further results not presented but relevant Stabilization properties of the approx solution X k Accuracy tracking as the approximation space grows Interpretation via invariant subspace approximation (V.Simoncini, 2016) Wrap-up Projection-type methods fill the gap between MOR and Riccati equation Clearer role of the non-linear term during the projection 30

31 Further results not presented but relevant Stabilization properties of the approx solution X k Accuracy tracking as the approximation space grows Interpretation via invariant subspace approximation (V.Simoncini, 2016) Wrap-up Projection-type methods fill the gap between MOR and Riccati equation Clearer role of the non-linear term during the projection 31

32 Outlook Petrov-Galerkin projection (à la Balanced truncation) (work in progress, with A. Alla, Florida State Univ.) Projected Differential Riccati equations (see, e.g., Koskela & Mena, tr 2017) Parameterized Algebraic Riccati equations (see, e.g., Schmidt & Haasdonk, tr 2017) References - V. Simoncini, Daniel B. Szyld and Marlliny Monsalve, On two numerical methods for the solution of large-scale algebraic Riccati equations IMA Journal of Numerical Analysis, (2014) - Yiding Lin and V. Simoncini, A new subspace iteration method for the algebraic Riccati equation Numerical Linear Algebra w/appl., (2015) - V.Simoncini, Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations SIAM J. Matrix Analysis Appl, (2016). - V. Simoncini, Computational methods for linear matrix equations (Survey article) SIAM Review, (2016). 32

33 Outlook Petrov-Galerkin projection (à la Balanced truncation) (work in progress, with A. Alla, Florida State Univ.) Projected Differential Riccati equations (see, e.g., Koskela & Mena, tr 2017) Parameterized Algebraic Riccati equations (see, e.g., Schmidt & Haasdonk, tr 2017) References - V. Simoncini, Daniel B. Szyld and Marlliny Monsalve, On two numerical methods for the solution of large-scale algebraic Riccati equations IMA Journal of Numerical Analysis, (2014) - Yiding Lin and V. Simoncini, A new subspace iteration method for the algebraic Riccati equation Numerical Linear Algebra w/appl., (2015) - V.Simoncini, Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations SIAM J. Matrix Anal.Appl, (2016) - V. Simoncini, Computational methods for linear matrix equations SIAM Review, (2016) 33

On the numerical solution of large-scale linear matrix equations. V. Simoncini

On the numerical solution of large-scale linear matrix equations V. Simoncini Dipartimento di Matematica, Università di Bologna (Italy) valeria.simoncini@unibo.it 1 Some matrix equations Sylvester matrix