Numerical Solution of Matrix Equations Arising in Control of Bilinear and Stochastic Systems

Transcription

1 MatTriad 2015 Coimbra September 7 11, 2015 Numerical Solution of Matrix Equations Arising in Control of Bilinear and Stochastic Systems Peter Benner Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg, Germany benner@mpi-magdeburg.mpg.de Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 1/38

2 Overview 1 Introduction 2 Applications 3 Solving Large-Scale Sylvester and Lyapunov Equations 4 Solving Large-Scale Lyapunov-plus-Positive Equations 5 References Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 2/38

3 Overview 1 Introduction Classification of Linear Matrix Equations Existence and Uniqueness of Solutions 2 Applications 3 Solving Large-Scale Sylvester and Lyapunov Equations 4 Solving Large-Scale Lyapunov-plus-Positive Equations 5 References Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 3/38

4 Introduction Linear Matrix Equations/Men with Beards Sylvester equation James Joseph Sylvester (September 3, 1814 March 15, 1897) AX + XB = C. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 4/38

5 Introduction Linear Matrix Equations/Men with Beards Sylvester equation Lyapunov equation James Joseph Sylvester (September 3, 1814 March 15, 1897) AX + XB = C. Alexander Michailowitsch Ljapunow (June 6, 1857 November 3, 1918) AX + XA T = C, C = C T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 4/38

6 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Generalized Sylvester equation: AXD + EXB = C. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

7 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Generalized Sylvester equation: Generalized Lyapunov equation: AXD + EXB = C. AXE T + EXA T = C, C = C T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

8 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Generalized Sylvester equation: Generalized Lyapunov equation: AXD + EXB = C. AXE T + EXA T = C, C = C T. Stein equation: X AXB = C. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

9 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Generalized Sylvester equation: Generalized Lyapunov equation: Stein equation: AXD + EXB = C. AXE T + EXA T = C, C = C T. X AXB = C. (Generalized) discrete Lyapunov/Stein equation: EXE T AXA T = C, C = C T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

10 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Generalized Sylvester equation: Generalized Lyapunov equation: Stein equation: AXD + EXB = C. AXE T + EXA T = C, C = C T. X AXB = C. (Generalized) discrete Lyapunov/Stein equation: Note: EXE T AXA T = C, C = C T. Consider only regular cases, having a unique solution! Solutions of symmetric cases are symmetric, X = X T R n n ; otherwise, X R n l with n l in general. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

11 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Bilinear Lyapunov equation/lyapunov-plus-positive equation: AX + XA T + m N k XNk T = C, C = C T. k=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

12 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Bilinear Lyapunov equation/lyapunov-plus-positive equation: AX + XA T + Bilinear Sylvester equation: m N k XNk T = C, C = C T. k=1 AX + XB + m N k XM k = C. k=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

13 Introduction Generalizations of Sylvester (AX + XB = C) and Lyapunov (AX + XA T = C) Equations Bilinear Lyapunov equation/lyapunov-plus-positive equation: AX + XA T + Bilinear Sylvester equation: m N k XNk T = C, C = C T. k=1 AX + XB + m N k XM k = C. k=1 (Generalized) discrete bilinear Lyapunov/Stein-minus-positive eq.: EXE T AXA T m N k XNk T = C, C = C T. k=1 Note: Again consider only regular cases, symmetric equations have symmetric solutions. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 5/38

14 Introduction Existence of Solutions of Linear Matrix Equations I Exemplarily, consider the generalized Sylvester equation AXD + EXB = C. (1) Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 6/38

15 Introduction Existence of Solutions of Linear Matrix Equations I Exemplarily, consider the generalized Sylvester equation AXD + EXB = C. (1) Vectorization (using Kronecker product) representation as linear system: ( D T A + B T E Ax = c. } {{ } =:A ) vec(x ) }{{} =:x = vec(c) }{{} =:c Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 6/38

16 Introduction Existence of Solutions of Linear Matrix Equations I Exemplarily, consider the generalized Sylvester equation AXD + EXB = C. (1) Vectorization (using Kronecker product) representation as linear system: ( D T A + B T E Ax = c. } {{ } =:A ) vec(x ) }{{} =:x = vec(c) }{{} =:c = (1) has a unique solution A is nonsingular Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 6/38

17 Introduction Existence of Solutions of Linear Matrix Equations I Exemplarily, consider the generalized Sylvester equation AXD + EXB = C. (1) Vectorization (using Kronecker product) representation as linear system: ( D T A + B T E Ax = c. } {{ } =:A ) vec(x ) }{{} =:x = vec(c) }{{} =:c = (1) has a unique solution A is nonsingular Lemma Λ (A) = {α j + β k α j Λ (A, E), β k Λ (B, D)}. Hence, (1) has unique solution Λ (A, E) Λ (B, D) =. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 6/38

18 Introduction Existence of Solutions of Linear Matrix Equations I Exemplarily, consider the generalized Sylvester equation AXD + EXB = C. (1) Vectorization (using Kronecker product) representation as linear system: ( D T A + B T E Ax = c. } {{ } =:A ) vec(x ) }{{} =:x = vec(c) }{{} =:c = (1) has a unique solution A is nonsingular Lemma Λ (A) = {α j + β k α j Λ (A, E), β k Λ (B, D)}. Hence, (1) has unique solution Λ (A, E) Λ (B, D) =. Example: Lyapunov equation AX + XA T = C has unique solution µ C : ±µ Λ (A). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 6/38

19 Introduction The Classical Lyapunov Theorem Theorem (Lyapunov 1892) Let A R n n and consider the Lyapunov operator L : X AX + XA T. Then the following are equivalent: (a) Y > 0: X > 0: L(X ) = Y, (b) Y > 0: X > 0: L(X ) = Y, (c) Λ (A) C := {z C Rz < 0}, i.e., A is (asymptotically) stable or Hurwitz. A. M. Lyapunov. The General Problem of the Stability of Motion (in Russian). Doctoral dissertation, Univ. Kharkov English translation: Stability of Motion, Academic Press, New-York & London, P. Lancaster, M. Tismenetsky. The Theory of Matrices (2nd edition). Academic Press, Orlando, FL, [Chapter 13] Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 7/38

20 Introduction The Classical Lyapunov Theorem Theorem (Lyapunov 1892) Let A R n n and consider the Lyapunov operator L : X AX + XA T. Then the following are equivalent: (a) Y > 0: X > 0: L(X ) = Y, (b) Y > 0: X > 0: L(X ) = Y, (c) Λ (A) C := {z C Rz < 0}, i.e., A is (asymptotically) stable or Hurwitz. The proof (c) (a) is trivial from the necessary and sufficient condition for existence and uniqueness, apart from the positive definiteness. The latter is shown by studying z H Yz for all eigenvectors z of A. A. M. Lyapunov. The General Problem of the Stability of Motion (in Russian). Doctoral dissertation, Univ. Kharkov English translation: Stability of Motion, Academic Press, New-York & London, P. Lancaster, M. Tismenetsky. The Theory of Matrices (2nd edition). Academic Press, Orlando, FL, [Chapter 13] Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 7/38

21 Introduction The Classical Lyapunov Theorem Theorem (Lyapunov 1892) Let A R n n and consider the Lyapunov operator L : X AX + XA T. Then the following are equivalent: (a) Y > 0: X > 0: L(X ) = Y, (b) Y > 0: X > 0: L(X ) = Y, (c) Λ (A) C := {z C Rz < 0}, i.e., A is (asymptotically) stable or Hurwitz. Important in applications: the nonnegative case: L(X ) = AX + XA T = WW T, where W R n n W, n W n. A Hurwitz unique solution X = ZZ T for Z R n n X with 1 n X n. A. M. Lyapunov. The General Problem of the Stability of Motion (in Russian). Doctoral dissertation, Univ. Kharkov English translation: Stability of Motion, Academic Press, New-York & London, P. Lancaster, M. Tismenetsky. The Theory of Matrices (2nd edition). Academic Press, Orlando, FL, [Chapter 13] Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 7/38

22 Introduction Existence of Solutions of Linear Matrix Equations II For Lyapunov-plus-positive-type equations, the solution theory is more involved. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 8/38

23 Introduction Existence of Solutions of Linear Matrix Equations II For Lyapunov-plus-positive-type equations, the solution theory is more involved. Focus on the Lyapunov-plus-positive case: m AX + XA }{{ T + N } k XNk T = C, C = C T 0. =:L(X ) k=1 }{{} =:P(X ) Note: The operator P(X ) m N k XNk T is nonnegative in the sense that P(X ) 0, whenever X 0. j=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 8/38

24 Introduction Existence of Solutions of Linear Matrix Equations II For Lyapunov-plus-positive-type equations, the solution theory is more involved. Focus on the Lyapunov-plus-positive case: m AX + XA }{{ T + N } k XNk T = C, C = C T 0. =:L(X ) k=1 }{{} =:P(X ) Note: The operator P(X ) m N k XNk T is nonnegative in the sense that P(X ) 0, whenever X 0. This nonnegative Lyapunov-plus-positive equation is the one occurring in applications like model order reduction. j=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 8/38

25 Introduction Existence of Solutions of Linear Matrix Equations II For Lyapunov-plus-positive-type equations, the solution theory is more involved. Focus on the Lyapunov-plus-positive case: m AX + XA }{{ T + N } k XNk T = C, C = C T 0. =:L(X ) k=1 }{{} =:P(X ) Note: The operator P(X ) m N k XNk T is nonnegative in the sense that P(X ) 0, whenever X 0. This nonnegative Lyapunov-plus-positive equation is the one occurring in applications like model order reduction. If A is Hurwitz and the N k are small enough, eigenvalue perturbation theory yields existence and uniqueness of solution. This is related to the concept of bounded-input bounded-output (BIBO) stability of dynamical systems. j=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 8/38

26 Introduction Existence of Solutions of Linear Matrix Equations II Theorem (Schneider 1965, Damm 2004) Let A R n n and consider the Lyapunov operator L : X AX + XA T and a nonnegative operator P (i.e., P(X ) 0 if X 0). The following are equivalent: (a) Y > 0: X > 0: L(X ) + P(X ) = Y, (b) Y > 0: X > 0: L(X ) + P(X ) = Y, (c) Y 0 with (A, Y ) controllable: X > 0: L(X ) + P(X ) = Y, (d) Λ (L + P) C := {z C Rz < 0}, (e) Λ (L) C and ρ(l 1 P) < 1, where ρ(t ) = max{ λ λ Λ (T )} = spectral radius of T. T. Damm. Rational Matrix Equations in Stochastic Control. Number 297 in Lecture Notes in Control and Information Sciences. Springer-Verlag, H. Schneider. Positive operators and an inertia theorem. Numerische Mathematik, 7:11 17, Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 8/38

27 Overview 1 Introduction 2 Applications Stability Theory Classical Control Applications Applications of Lyapunov-plus-Positive Equations 3 Solving Large-Scale Sylvester and Lyapunov Equations 4 Solving Large-Scale Lyapunov-plus-Positive Equations 5 References Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 9/38

28 Applications Stability Theory From Lyapunov s theorem, immediately obtain characterization of asymptotic stability of linear dynamical systems ẋ(t) = Ax(t). (2) Theorem (Lyapunov) The following are equivalent: For (2), the zero state is asymptotically stable. The Lyapunov equation AX + XA T = Y has a unique solution X = X T > 0 for all Y = Y T < 0. A is Hurwitz. A. M. Lyapunov. The General Problem of the Stability of Motion (In Russian). Doctoral dissertation, Univ. Kharkov English translation: Stability of Motion, Academic Press, New-York & London, Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 10/38

29 Classical Control Applications Algebraic Riccati Equations (ARE) Solving AREs by Newtons s Method Feedback control design often involves solution of A T X + XA XGX + H = 0, G = G T, H = H T. In each Newton step, solve Lyapunov equation (A GX j ) T X j+1 + X j+1 (A GX j ) = X j GX j H. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 11/38

30 Classical Control Applications Algebraic Riccati Equations (ARE) Solving AREs by Newtons s Method Feedback control design often involves solution of A T X + XA XGX + H = 0, G = G T, H = H T. In each Newton step, solve Lyapunov equation (A GX j ) T X j+1 + X j+1 (A GX j ) = X j GX j H. Decoupling of dynamical systems, e.g., in slow/fast modes, requires solution of nonsymmetric ARE AX + XF XGX + H = 0. In each Newton step, solve Sylvester equation (A X j G)X j+1 + X j+1 (F GX j ) = X j GX j H. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 11/38

31 Classical Control Applications Model Reduction Model Reduction via Balanced Truncation For linear dynamical system ẋ(t) = Ax(t) + Bu(t), y(t) = Cx r (t), x(t) R n find reduced-order system x r (t) = A r x r (t) + B r u(t), y r (t) = C r x r (t), x(t) R r, r n such that y(t) y r (t) < δ. The popular method balanced truncation requires the solution of the dual Lyapunov equations AX + XA T + BB T = 0, A T Y + YA + C T C = 0. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 11/38

32 Applications of Lyapunov-plus-Positive Equations Bilinear control systems: m ẋ(t) = Ax(t) + N i x(t)u i (t) + Bu(t), Σ : i=1 y(t) = Cx(t), x(0) = x 0, where A, N i R n n, B R n m, C R q n. Properties: Approximation of (weakly) nonlinear systems by Carleman linearization yields bilinear systems. Appear naturally in boundary control problems, control via coefficients of PDEs, Fokker-Planck equations,... Due to the close relation to linear systems, a lot of successful concepts can be extended, e.g. transfer functions, Gramians, Lyapunov equations,... Linear stochastic control systems possess an equivalent structure and can be treated alike [B./Damm 11]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 12/38

33 Applications of Lyapunov-plus-Positive Equations The concept of balanced truncation can be generalized to the case of bilinear systems, where we need the solutions of the Lyapunov-plus-positive equations: m AP + PA T + N i PA T i + BB T = 0, A T Q + QA T + i=1 m Ni T QA i + C T C = 0. i=1 Due to its approximation quality, balanced truncation is method of choice for model reduction of medium-size bilinear systems. For stationary iterative solvers, see [Damm 2008], extended to low-rank solutions recently by [Szyld/Shank/Simoncini 2014]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 13/38

34 Applications of Lyapunov-plus-Positive Equations The concept of balanced truncation can be generalized to the case of bilinear systems, where we need the solutions of the Lyapunov-plus-positive equations: m AP + PA T + N i PA T i + BB T = 0, Further applications: A T Q + QA T + i=1 m Ni T QA i + C T C = 0. i=1 Analysis and model reduction for linear stochastic control systems driven by Wiener noise [B./Damm 2011], Lévy processes [B./Redmann 2011/15]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 13/38

35 Applications of Lyapunov-plus-Positive Equations The concept of balanced truncation can be generalized to the case of bilinear systems, where we need the solutions of the Lyapunov-plus-positive equations: m AP + PA T + N i PA T i + BB T = 0, Further applications: A T Q + QA T + i=1 m Ni T QA i + C T C = 0. i=1 Analysis and model reduction for linear stochastic control systems driven by Wiener noise [B./Damm 2011], Lévy processes [B./Redmann 2011/15]. Model reduction of linear parameter-varying (LPV) systems using bilinearization approach [B./Breiten 2011, B./Bruns 2015]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 13/38

36 Applications of Lyapunov-plus-Positive Equations The concept of balanced truncation can be generalized to the case of bilinear systems, where we need the solutions of the Lyapunov-plus-positive equations: m AP + PA T + N i PA T i + BB T = 0, Further applications: A T Q + QA T + i=1 m Ni T QA i + C T C = 0. i=1 Analysis and model reduction for linear stochastic control systems driven by Wiener noise [B./Damm 2011], Lévy processes [B./Redmann 2011/15]. Model reduction of linear parameter-varying (LPV) systems using bilinearization approach [B./Breiten 2011, B./Bruns 2015]. Model reduction for Fokker-Planck equations [Hartmann et al. 2013]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 13/38

37 Applications of Lyapunov-plus-Positive Equations The concept of balanced truncation can be generalized to the case of bilinear systems, where we need the solutions of the Lyapunov-plus-positive equations: m AP + PA T + N i PA T i + BB T = 0 Further applications: i=1 Analysis and model reduction for linear stochastic control systems driven by Wiener noise [B./Damm 2011], Lévy processes [B./Redmann 2011/15]. Model reduction of linear parameter-varying (LPV) systems using bilinearization approach [B./Breiten 2011, B./Bruns 2015]. Model reduction for Fokker-Planck equations [Hartmann et al. 2013]. Linear-quadratic regulators for stochastic systems require solution of AREs of the form m AP + PA T XC T CX + N i PA T i + BB T = 0, application of Newton s method 1 L-p-P equation/iteration. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 13/38 i=1

38 Overview This part: joint work with Patrick Kürschner and Jens Saak (MPI Magdeburg) 1 Introduction 2 Applications 3 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics LR-ADI Derivation The New LR-ADI Applied to Lyapunov Equations 4 Solving Large-Scale Lyapunov-plus-Positive Equations 5 References Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 14/38

39 Solving Large-Scale Sylvester and Lyapunov Equations The Low-Rank Structure Sylvester Equations Find X R n m solving AX X B = F G T, where A R n n, B R m m, F R n r, G R m r. If n, m large, but r n, m X has a small numerical rank. [Penzl 1999, Grasedyck 2004, Antoulas/Sorensen/Zhou 2002] rank(x, τ) = f min(n, m) singular values of example 10 0 σ(x ) σ 155 u u Compute low-rank solution factors Z R n f, Y R m f, D R f f, such that X ZDY T with f min(n, m). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 15/38

40 Solving Large-Scale Sylvester and Lyapunov Equations The Low-Rank Structure Lyapunov Equations Find X R n n solving where A R n n, F R n r. If n large, but r n X has a small numerical rank. [Penzl 1999, Grasedyck 2004, Antoulas/Sorensen/Zhou 2002] rank(x, τ) = f n AX +X A T = F F T, u Compute low-rank solution factors Z R n f, D R f f, such that X ZDZ T with f n. singular values of example σ 155 u σ(x ) Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 15/38

41 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

42 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). This cannot be used for numerical solutions unless nm 1, 000 (or so), as it requires O(n 2 m 2 ) of storage; Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

43 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). This cannot be used for numerical solutions unless nm 1, 000 (or so), as it requires O(n 2 m 2 ) of storage; direct solver needs O(n 3 m 3 ) flops; Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

44 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). This cannot be used for numerical solutions unless nm 1, 000 (or so), as it requires O(n 2 m 2 ) of storage; direct solver needs O(n 3 m 3 ) flops; low (tensor-)rank of right-hand side is ignored; Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

45 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). This cannot be used for numerical solutions unless nm 1, 000 (or so), as it requires O(n 2 m 2 ) of storage; direct solver needs O(n 3 m 3 ) flops; low (tensor-)rank of right-hand side is ignored; in Lyapunov case, symmetry and possible definiteness are not respected. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

46 Solving Large-Scale Sylvester and Lyapunov Equations Some Basics Sylvester equation AX XB = FG T is equivalent to linear system of equations ( Im A B T I n ) vec(x ) = vec(fg T ). This cannot be used for numerical solutions unless nm 1, 000 (or so), as it requires O(n 2 m 2 ) of storage; direct solver needs O(n 3 m 3 ) flops; low (tensor-)rank of right-hand side is ignored; in Lyapunov case, symmetry and possible definiteness are not respected. Possible solvers: Standard Krylov subspace solvers in operator from [Hochbruck, Starke, Reichel, Bao,... ]. Block-Tensor-Krylov subspace methods with truncation [Kressner/Tobler, Bollhöfer/Eppler, B./Breiten,... ]. Galerkin-type methods based on (extended, rational) Krylov subspace methods [Jaimoukha, Kasenally, Jbilou, Simoncini, Druskin, Knizhermann,... ] Doubling-type methods [Smith, Chu et al., B./Sadkane/El Khoury,... ]. ADI methods [Wachspress, Reichel et al., Li, Penzl, B., Saak, Kürschner,... ]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 16/38

47 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Sylvester and Stein equations Let α β with α / Λ(B), β / Λ(A), then AX XB = FG T }{{} Sylvester equation X = A X B + (β α )F G H }{{} Stein equation with the Cayley like transformations A := (A β I n ) 1 (A α I n ), B := (B α I m ) 1 (B β I m ), F := (A β I n ) 1 F, G := (B α I m ) H G. fix point iteration X k = A X k 1 B + (β α )F G H for k 1, X 0 R n m. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 17/38

48 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Sylvester and Stein equations Let α k β k with α k / Λ(B), β k / Λ(A), then AX XB = FG T }{{} Sylvester equation X = A k X B k + (β k α k )F k G k H }{{} Stein equation with the Cayley like transformations A k := (A β k I n ) 1 (A α k I n ), B k := (B α k I m ) 1 (B β k I m ), F k := (A β k I n ) 1 F, G k := (B α k I m ) H G. alternating directions implicit (ADI) iteration X k = A k X k 1 B k + (β k α k )F k G H k for k 1, X 0 R n m. [Wachspress 1988] Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 17/38

49 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Sylvester ADI iteration [Wachspress 1988] X k = A k X k 1 B k + (β k α k )F k G H k, A k := (A β k I n ) 1 (A α k I n ), B k := (B α k I m ) 1 (B β k I m ), F k := (A β k I n ) 1 F R n r, G k := (B α k I m ) H G C m r. Now set X 0 = 0 and find factorization X k = Z k D k Y H k X 1 = A 1 X 0 B 1 + (β 1 α 1 )F 1 G H 1, Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 18/38

50 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Sylvester ADI iteration [Wachspress 1988] X k = A k X k 1 B k + (β k α k )F k G H k, A k := (A β k I n ) 1 (A α k I n ), B k := (B α k I m ) 1 (B β k I m ), F k := (A β k I n ) 1 F R n r, G k := (B α k I m ) H G C m r. Now set X 0 = 0 and find factorization X k = Z k D k Yk H X 1 = (β 1 α 1 )(A β 1 I n ) 1 F G T (B α 1 I m ) 1 V 1 := Z 1 = (A β 1 I n ) 1 F R n r, D 1 = (β 1 α 1 )I r R r r, W 1 := Y 1 = (B α 1 I m ) H G C m r. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 18/38

51 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Sylvester ADI iteration [Wachspress 1988] X k = A k X k 1 B k + (β k α k )F k G H k, A k := (A β k I n ) 1 (A α k I n ), B k := (B α k I m ) 1 (B β k I m ), F k := (A β k I n ) 1 F R n r, G k := (B α k I m ) H G C m r. Now set X 0 = 0 and find factorization X k = Z k D k Yk H X 2 = A 2 X 1 B 2 + (β 2 α 2 )F 2 G2 H =... = V 2 = V 1 + (β 2 α 1 )(A + β 2 I ) 1 V 1 R n r, W 2 = W 1 + (α 2 β 1 )(B + α 2 I ) H W 1 R m r, Z 2 = [Z 1, V 2 ], D 2 = diag (D 1, (β 2 α 2 )I r ), Y 2 = [Y 1, W 2 ]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 18/38

52 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Algorithm [B. 2005, Li/Truhar 2008, B./Li/Truhar 2009] Algorithm 1: Low-rank Sylvester ADI / factored ADI (fadi) Input : Matrices defining AX XB = FG T and shift parameters {α 1,..., α kmax }, {β 1,..., β kmax }. Output: Z, D, Y such that ZDY H X. 1 Z 1 = V 1 = (A β 1 I n ) 1 F, 2 Y 1 = W 1 = (B α 1 I m ) H G. 3 D 1 = (β 1 α 1 )I r 4 for k = 2,..., k max do 5 V k = V k 1 + (β k α k 1 )(A β k I n ) 1 V k 1. 6 W k = W k 1 + (α k β k 1 )(B α k I n ) H W k 1. 7 Update solution factors Z k = [Z k 1, V k ], Y k = [Y k 1, W k ], D k = diag (D k 1, (β k α k )I r ). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 19/38

53 Solving Large-Scale Sylvester and Lyapunov Equations ADI Shifts Optimal Shifts Solution of rational optimization problem min max α j C λ Λ(A) j=1 β j C µ Λ(B) k (λ α j )(µ β j ) (λ β j )(µ α j ), for which no analytic solution is known in general. Some shift generation approaches: generalized Bagby points, [Levenberg/Reichel 1993] adaption of Penzl s cheap heuristic approach available [Penzl 1999, Li/Truhar 2008] approximate Λ(A), Λ(B) by small number of Ritz values w.r.t. A, A 1, B, B 1 via Arnoldi, just taking these Ritz values alone also works well quite often. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 20/38

54 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Disadvantages of Low-Rank ADI as of 2012: 1 No efficient stopping criteria: Difference in iterates norm of added columns/step: not reliable, stops often too late. Residual is a full dense matrix, can not be calculated as such. 2 Requires complex arithmetic for real coefficients when complex shifts are used. 3 Expensive (only semi-automatic) set-up phase to precompute ADI shifts. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 21/38

55 Solving Large-Scale Sylvester and Lyapunov Equations LR-ADI Derivation Disadvantages of Low-Rank ADI as of 2012: 1 No efficient stopping criteria: Difference in iterates norm of added columns/step: not reliable, stops often too late. Residual is a full dense matrix, can not be calculated as such. 2 Requires complex arithmetic for real coefficients when complex shifts are used. 3 Expensive (only semi-automatic) set-up phase to precompute ADI shifts. None of these disadvantages exists as of today = speed-ups old vs. new LR-ADI can be up to 20! Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 21/38

56 Projection-Based Lyapunov Solvers for Lyapunov equation 0 = AX + XA T + BB T Projection-based methods for Lyapunov equations with A + A T < 0: 1 Compute orthonormal basis range (Z), Z R n r, for subspace Z R n, dim Z = r. 2 Set  := Z T AZ, ˆB := Z T B. 3 Solve small-size Lyapunov equation  ˆX + ˆX  T + ˆB ˆB T = 0. 4 Use X Z ˆX Z T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 22/38

57 Projection-Based Lyapunov Solvers for Lyapunov equation 0 = AX + XA T + BB T Projection-based methods for Lyapunov equations with A + A T < 0: 1 Compute orthonormal basis range (Z), Z R n r, for subspace Z R n, dim Z = r. 2 Set  := Z T AZ, ˆB := Z T B. 3 Solve small-size Lyapunov equation  ˆX + ˆX  T + ˆB ˆB T = 0. 4 Use X Z ˆX Z T. Examples: Krylov subspace methods, i.e., for m = 1: Z = K(A, B, r) = span{b, AB, A 2 B,..., A r 1 B} [Saad 1990, Jaimoukha/Kasenally 1994, Jbilou ]. Extended Krylov subspace method (EKSM) [Simoncini 2007], Z = K(A, B, r) K(A 1, B, r). Rational Krylov subspace methods (RKSM) [Druskin/Simoncini 2011]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 22/38

58 The New LR-ADI Applied to Lyapunov Equations Example: an ocean circulation problem [Van Gijzen et al. 1998] FEM discretization of a simple 3D ocean circulation model (barotropic, constant depth) stiffness matrix A with n = 42, 249, choose artificial constant term B = rand(n,5). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 23/38

59 The New LR-ADI Applied to Lyapunov Equations Example: an ocean circulation problem [Van Gijzen et al. 1998] FEM discretization of a simple 3D ocean circulation model (barotropic, constant depth) stiffness matrix A with n = 42, 249, choose artificial constant term B = rand(n,5). Convergence history: R / B T B TOL LR-ADI with adaptive shifts vs. EKSM coldim(z) LR-ADI EKSM Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 23/38

60 The New LR-ADI Applied to Lyapunov Equations Example: an ocean circulation problem [Van Gijzen et al. 1998] FEM discretization of a simple 3D ocean circulation model (barotropic, constant depth) stiffness matrix A with n = 42, 249, choose artificial constant term B = rand(n,5). Convergence history: R / B T B TOL LR-ADI with adaptive shifts vs. EKSM coldim(z) CPU times: LR-ADI 110 sec, EKSM 135 sec. LR-ADI EKSM Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 23/38

61 Solving Large-Scale Sylvester and Lyapunov Equations Summary & Outlook Numerical enhancements of low-rank ADI for large Sylvester/Lyapunov equations: 1 low-rank residuals, reformulated implementation, 2 compute real low-rank factors in the presence of complex shifts, 3 self-generating shift strategies (quantification in progress). For diffusion-convection-reaction example: sec. down to sec. acceleration by factor almost 20. Generalized version enables derivation of low-rank solvers for various generalized Sylvester equations. Ongoing work: Apply LR-ADI in Newton methods for algebraic Riccati equations R(X ) = AX + XA T + GG T XSS T X = 0, D(X ) = AXA T EXE T + GG T + A T XF (I r + F T XF ) 1 F T XA = 0. For nonlinear AREs see P. Benner, P. Kürschner, J. Saak. Low-rank Newton-ADI methods for large nonsymmetric algebraic Riccati equations. J. Franklin Inst., Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 24/38

62 Overview This part: joint work with Tobias Breiten (KFU Graz, Austria) 1 Introduction 2 Applications 3 Solving Large-Scale Sylvester and Lyapunov Equations 4 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Generalized ADI Iteration Bilinear EKSM Tensorized Krylov Subspace Methods Comparison of Methods 5 References Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 25/38

63 Solving Large-Scale Lyapunov-plus-Positive Equations Some basic facts and assumptions AX + XA T + m N j XNj T + BB T = 0. (3) j=1 Need a positive semi-definite symmetric solution X. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 26/38

64 Solving Large-Scale Lyapunov-plus-Positive Equations Some basic facts and assumptions AX + XA T + m N j XNj T + BB T = 0. (3) j=1 Need a positive semi-definite symmetric solution X. As discussed before, solution theory for Lyapuonv-plus-positive equation is more involved then in standard Lyapuonv case. Here, existence and uniqueness of positive semi-definite solution X = X T is assumed. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 26/38

65 Solving Large-Scale Lyapunov-plus-Positive Equations Some basic facts and assumptions AX + XA T + m N j XNj T + BB T = 0. (3) j=1 Need a positive semi-definite symmetric solution X. As discussed before, solution theory for Lyapuonv-plus-positive equation is more involved then in standard Lyapuonv case. Here, existence and uniqueness of positive semi-definite solution X = X T is assumed. Want: solution methods for large scale problems, i.e., only matrix-matrix multiplication with A, N j, solves with (shifted) A allowed! Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 26/38

66 Solving Large-Scale Lyapunov-plus-Positive Equations Some basic facts and assumptions AX + XA T + m N j XNj T + BB T = 0. (3) j=1 Need a positive semi-definite symmetric solution X. As discussed before, solution theory for Lyapuonv-plus-positive equation is more involved then in standard Lyapuonv case. Here, existence and uniqueness of positive semi-definite solution X = X T is assumed. Want: solution methods for large scale problems, i.e., only matrix-matrix multiplication with A, N j, solves with (shifted) A allowed! Requires to compute data-sparse approximation to generally dense X ; here: X ZZ T with Z R n n Z, n Z n! Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 26/38

67 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

68 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Standard Lyapunov case: [Grasedyck 04] AX + XA T + BB T = 0 (I n A + A I n) vec(x ) = vec(bb T ). }{{} =:A Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

69 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Standard Lyapunov case: [Grasedyck 04] Apply AX + XA T + BB T = 0 (I n A + A I n) vec(x ) = vec(bb T ). }{{} =:A M 1 = 0 exp(tm)dt to A and approximate the integral via (sinc) quadrature A 1 k ω i exp(t k A), i= k with error exp( k) (exp( k) if A = A T ), then an approximate Lyapunov solution is given by k vec(x ) vec(x k ) = ω i exp(t i A) vec(bb T ). i= k Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

70 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Standard Lyapunov case: [Grasedyck 04] AX + XA T + BB T = 0 (I n A + A I n) vec(x ) = vec(bb T ). }{{} =:A Now observe that vec(x ) vec(x k ) = k ω i exp(t i A) vec(bb T ). i= k exp(t i A) = exp (t i (I n A + A I n)) exp(t i A) exp(t i A). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

71 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Standard Lyapunov case: [Grasedyck 04] AX + XA T + BB T = 0 (I n A + A I n) vec(x ) = vec(bb T ). }{{} =:A Now observe that Hence, vec(x ) vec(x k ) = k ω i exp(t i A) vec(bb T ). i= k exp(t i A) = exp (t i (I n A + A I n)) exp(t i A) exp(t i A). vec(x k ) = k ω i (exp(t i A) exp(t i A)) vec(bb T ) i= k Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

72 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Standard Lyapunov case: [Grasedyck 04] Hence, AX + XA T + BB T = 0 (I n A + A I n) vec(x ) = vec(bb T ). }{{} =:A vec(x k ) = = X k = k ω i (exp(t i A) exp(t i A)) vec(bb T ) i= k k k ω i exp(t i A)BB T exp(t i A T ) ω i B i Bi T, i= k i= k so that rank (X k ) (2k + 1)m with X X k 2 exp( k) ( exp( k) for A = A T )! Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

73 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Problem: in general, m m exp t i (I A + A I + N j N j ) (exp (t i A) exp (t i A)) exp t i ( N j N j ). j=1 j=1 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

74 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Assume that m = 1 and N 1 = UV T with U, V R n r and consider ( I n A + A I }{{} n +N 1 N 1 ) vec(x ) = vec(bb T ). }{{} =A =:y Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

75 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Assume that m = 1 and N 1 = UV T with U, V R n r and consider ( I n A + A I }{{} n +N 1 N 1 ) vec(x ) = vec(bb T ). }{{} =A =:y Sherman-Morrison-Woodbury = ( Ir I r + (V T V T )A 1 (U U) ) w = (V T V T )A 1 y, A vec(x ) = y (U U)w. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

76 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Question Can we expect low-rank approximations ZZ T X to the solution of m AX + XA T + N j XNj T + BB T = 0? j=1 Assume that m = 1 and N 1 = UV T with U, V R n r and consider ( I n A + A I }{{} n +N 1 N 1 ) vec(x ) = vec(bb T ). }{{} =A =:y Sherman-Morrison-Woodbury = ( Ir I r + (V T V T )A 1 (U U) ) w = (V T V T )A 1 y, A vec(x ) = y (U U)w. Matrix rank of RHS BB T U vec 1 (w) U T is r + 1! Apply results for linear Lyapunov equations with r.h.s of rank r + 1. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 27/38

77 Solving Large-Scale Lyapunov-plus-Positive Equations Existence of Low-Rank Approximations Theorem [B./Breiten 2012] Assume existence and uniqueness with stable A and N j = U j Vj T, with U j, V j R n r j. Set r = m j=1 r j. Then the solution X of AX + XA T + m N j XNj T + BB T = 0 j=1 can be approximated by X k of rank (2k + 1)(m + r), with an error satisfying X X k 2 exp( k). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 28/38

78 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Let us again consider the Lyapunov-plus-positive equation AP + PA T + NPN T + BB T = 0. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

79 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Let us again consider the Lyapunov-plus-positive equation AP + PA T + NPN T + BB T = 0. For a fixed parameter p, we can rewrite the linear Lyapunov operator as AP + PA T = 1 ( (A + pi )P(A + pi ) T (A pi )P(A pi ) T ) 2p Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

80 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Let us again consider the Lyapunov-plus-positive equation AP + PA T + NPN T + BB T = 0. For a fixed parameter p, we can rewrite the linear Lyapunov operator as AP + PA T = 1 ( (A + pi )P(A + pi ) T (A pi )P(A pi ) T ) 2p leading to the fix point iteration [Damm 2008] P j = (A pi ) 1 (A + pi )P j 1 (A + pi ) T (A pi ) T + 2p(A pi ) 1 (NP j 1 N T + BB T )(A pi ) T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

81 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Let us again consider the Lyapunov-plus-positive equation AP + PA T + NPN T + BB T = 0. For a fixed parameter p, we can rewrite the linear Lyapunov operator as AP + PA T = 1 ( (A + pi )P(A + pi ) T (A pi )P(A pi ) T ) 2p leading to the fix point iteration [Damm 2008] P j = (A pi ) 1 (A + pi )P j 1 (A + pi ) T (A pi ) T + 2p(A pi ) 1 (NP j 1 N T + BB T )(A pi ) T. P j Z j Z T j Z j Z T j (rank (Z j ) n) factored iteration = (A pi ) 1 (A + pi )Z j 1 Z T j 1(A + pi ) T (A pi ) T + 2p(A pi ) 1 (NZ j 1 Z T j 1N T + BB T )(A pi ) T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

82 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Hence, for a given sequence of shift parameters {p 1,..., p q }, we can extend the linear ADI iteration as follows: Z 1 = 2p 1 (A p 1 I ) 1 B, Z j = (A p j I ) 1 [ (A + p j I ) Z j 1 2pj B 2p j NZ j 1 ], j q. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

83 Solving Large-Scale Lyapunov-plus-Positive Equations Generalized ADI Iteration Hence, for a given sequence of shift parameters {p 1,..., p q }, we can extend the linear ADI iteration as follows: Z 1 = 2p 1 (A p 1 I ) 1 B, Z j = (A p j I ) 1 [ (A + p j I ) Z j 1 2pj B 2p j NZ j 1 ], j q. Problems: A and N in general do not commute we have to operate on full preceding subspace Z j 1 in each step. Rapid increase of rank (Z j ) perform some kind of column compression. Choice of shift parameters? No obvious generalization of minimax problem. Here, we will use shifts minimizing a certain H 2 -optimization problem, see [B./Breiten 2011/14]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 29/38

84 Generalized ADI Iteration Numerical Example: A Heat Transfer Model with Uncertainty 2-dimensional heat distribution motivated by [Benner/Saak 05] boundary control by a cooling fluid with an uncertain spraying intensity x 10 Γ 2 x 01 x 02 x 03 x 11 x 12 x 13 x 14 Ω = (0, 1) (0, 1) x t = x in Ω Γ 1 x 20 x 21 x 22 x 23 x 24 Γ 4 n x = (0.5 + dω 1 )x on Γ 1 x = u on Γ 2 x 30 x 31 x 32 x 33 x 34 x = 0 on Γ 3, Γ 4 spatial discretization k k-grid dx Axdt + Nxdω i + Budt output: C = 1 [ ] k 2 x 41 x 42 x 43 Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 30/38 Γ 3

85 Generalized ADI Iteration Numerical Example: A Heat Transfer Model with Uncertainty Conv. history for bilinear low-rank ADI method (n = 40, 000) 10 5 Relative Residual Absolute Residual Residual Iteration Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 30/38

86 Solving Large-Scale Lyapunov-plus-Positive Equations Generalizing the Extended Krylov Subspace Method (EKSM) [Simoncini 07] Low-rank solutions of the Lyapunov-plus-positive equation may be obtained by projecting the original equation onto a suitable smaller subspace V = span(v ), V R n k, with V T V = I. In more detail, solve ( V T AV ) ˆX + ˆX ( V T A T V ) + ( V T NV ) ˆX ( V T N T V ) + ( V T B ) ( V T B ) T = 0 and prolongate X V ˆX V T. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 31/38

87 Solving Large-Scale Lyapunov-plus-Positive Equations Generalizing the Extended Krylov Subspace Method (EKSM) [Simoncini 07] Low-rank solutions of the Lyapunov-plus-positive equation may be obtained by projecting the original equation onto a suitable smaller subspace V = span(v ), V R n k, with V T V = I. In more detail, solve ( V T AV ) ˆX + ˆX ( V T A T V ) + ( V T NV ) ˆX ( V T N T V ) + ( V T B ) ( V T B ) T = 0 and prolongate X V ˆX V T. For this, one might use the extended Krylov subspace method (EKSM) algorithm in the following way: Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 31/38

88 Solving Large-Scale Lyapunov-plus-Positive Equations Generalizing the Extended Krylov Subspace Method (EKSM) [Simoncini 07] Low-rank solutions of the Lyapunov-plus-positive equation may be obtained by projecting the original equation onto a suitable smaller subspace V = span(v ), V R n k, with V T V = I. In more detail, solve ( V T AV ) ˆX + ˆX ( V T A T V ) + ( V T NV ) ˆX ( V T N T V ) + ( V T B ) ( V T B ) T = 0 and prolongate X V ˆX V T. For this, one might use the extended Krylov subspace method (EKSM) algorithm in the following way: V 1 = [ B A 1 B ], V r = [ AV r 1 A 1 V r 1 NV r 1 ], r = 2, 3,... Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 31/38

89 Solving Large-Scale Lyapunov-plus-Positive Equations Generalizing the Extended Krylov Subspace Method (EKSM) [Simoncini 07] Low-rank solutions of the Lyapunov-plus-positive equation may be obtained by projecting the original equation onto a suitable smaller subspace V = span(v ), V R n k, with V T V = I. In more detail, solve ( V T AV ) ˆX + ˆX ( V T A T V ) + ( V T NV ) ˆX ( V T N T V ) + ( V T B ) ( V T B ) T = 0 and prolongate X V ˆX V T. For this, one might use the extended Krylov subspace method (EKSM) algorithm in the following way: V 1 = [ B A 1 B ], V r = [ AV r 1 A 1 V r 1 NV r 1 ], r = 2, 3,... However, criteria like dissipativity of A for the linear case which ensure solvability of the projected equation have to be further investigated. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 31/38

90 Bilinear EKSM Residual Computation in O(k 3 ) Theorem (B./Breiten 2012) Let V i R n k i be the extend Krylov matrix after i generalized EKSM steps. Denote the residual associated with the approximate solution X i = V i ˆX i Vi T by R i := AX i + X i A T + NX i N T + BB T, where ˆX i is the solution of the reduced Lyapunov-plus-positive equation Then: V T i AV i ˆXi + ˆX i V T i range (R i ) range (V i+1 ), A T V i + V T i NV i ˆXi Vi T N T V i + Vi T BB T V i = 0. R i = V T i+1r i V i+1 for the Frobenius and spectral norms. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 32/38

91 Bilinear EKSM Residual Computation in O(k 3 ) Theorem (B./Breiten 2012) Let V i R n k i be the extend Krylov matrix after i generalized EKSM steps. Denote the residual associated with the approximate solution X i = V i ˆX i Vi T by R i := AX i + X i A T + NX i N T + BB T, where ˆX i is the solution of the reduced Lyapunov-plus-positive equation Then: V T i AV i ˆXi + ˆX i V T i range (R i ) range (V i+1 ), A T V i + V T i NV i ˆXi Vi T N T V i + Vi T BB T V i = 0. R i = V T i+1r i V i+1 for the Frobenius and spectral norms. Remarks: Residual evaluation only requires quantities needed in i + 1st projection step plus O(k 3 i+1) operations. No Hessenberg structure of reduced system matrix that allows to simplify residual expression as in standard Lyapunov case! Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 32/38

92 Bilinear EKSM Numerical Example: A Heat Transfer Model with Uncertainty Convergence history for bilinear EKSM variant (n = 6, 400) 10 4 Relative Residual Absolute Residual 10 0 Residual Iteration Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 33/38

93 Solving Large-Scale Lyapunov-plus-Positive Equations Tensorized Krylov Subspace Methods Another possibility is to iteratively solve the linear system (I n A + A I n + N N) vec(x ) = vec(bb T ), with a fixed number of ADI iteration steps used as a preconditioner M M 1 (I n A + A I n + N N) vec(x ) = M 1 vec(bb T ). We implemented this approach for PCG and BiCGstab. Updates like X k+1 X k + ω k P k require truncation operator to preserve low-order structure. Note, that the low-rank factorization X ZZ T has to be replaced by X ZDZ T, D possibly indefinite. Similar to more general tensorized Krylov solvers, see [Kressner/Tobler 2010/12]. Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 34/38

94 Tensorized Krylov Subspace Methods Vanilla Implementation of Tensor-PCG for Matrix Equations Algorithm 2: Preconditioned CG method for A(X ) = B Input : Matrix functions A, M : R n n R n n, low rank factor B of right-hand side B = BB T. Truncation operator T w.r.t. relative accuracy ɛ rel. Output: Low rank approximation X = LDL T with A(X ) B F tol. 1 X 0 = 0, R 0 = B, Z 0 = M 1 (R 0 ), P 0 = Z 0, Q 0 = A(P 0 ), ξ 0 = P 0, Q 0, k = 0 2 while R k F > tol do 3 ω k = R k,p k ξ k 4 X k+1 = X k + ω k P k, X k+1 T (X k+1 ) 5 R k+1 = B A(X k+1 ), Optionally: R k+1 T (R k+1 ) 6 Z k+1 = M 1 (R k+1 ) 7 β k = Z k+1,q k ξ k 8 P k+1 = Z k+1 + β k P k, P k+1 T (P k+1 ) 9 Q k+1 = A(P k+1 ), Optionally: Q k+1 T (Q k+1 ) 10 ξ k+1 = P k+1, Q k+1 11 k = k X = X k Here, A : X AX + XA T + NXN T, M: l steps of (bilinear) ADI, both in low-rank ( ZDZ T format). Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 35/38

95 Comparison of Methods Heat Equation with Boundary Control Comparison of low rank solution methods for n = 562, 500. Bilinear ADI (6 H 2 -optimal shifts) Bilinear ADI (10 H 2 -optimal shifts) CG (Bilinear ADI Precond.) Bilinear ADI (8 H 2 -optimal shifts) Bilinear ADI (4 Wachspress shifts) Bilinear EKSM Relative residual Iteration number Rank of X k Max Planck Institute Magdeburg P. Benner, Numerical Solution of Matrix Equations 36/38