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Symplectic Lanczos for Hamiltonian-Positive Matrices Encounters No Serious Breakdown!)

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Symplectic Lanczos for Hamiltonian-Positive Matrices Encounters No Serious Breakdown!) GAMM Annual Meeting 2013 Novi Sad, March 18{22, 2013 THE SYMPLECTIC LANCZOS PROCESS FOR HAMILTONIAN-POSITIVE MATRICES (Symplectic Lanczos for Hamiltonian-positive matrices encounters no serious breakdown!) Peter Benner Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg, Germany TU Chemnitz Mathematics in Industry and Technology Chemnitz, Germany Joint work with Ahmed Salam (ULCO, Calais) Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Hamiltonian Eigenproblems Definition 0 In Let J = , then H 2 R2n×2n is called Hamiltonian if −In 0 (HJ)T = HJ. Note: J−1 = JT = −J. Explicit block form of Hamiltonian matrices AB ; where A; B; C 2 n×n and B = BT ; C = C T : C −AT R Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 2/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Hamiltonian Eigenproblems Definition 0 In Let J = , then H 2 R2n×2n is called Hamiltonian if −In 0 (HJ)T = HJ. Note: J−1 = JT = −J. Explicit block form of Hamiltonian matrices AB ; where A; B; C 2 n×n and B = BT ; C = C T : C −AT R Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 2/13 Typical Hamiltonian spectrum Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Spectral Properties Hamiltonian Eigensymmetry Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: if λ is a finite eigenvalue of H, then λ,¯ −λ, −λ¯ are eigenvalues of H, too. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 3/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Spectral Properties Hamiltonian Eigensymmetry Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: if λ is a finite eigenvalue of H, then λ,¯ −λ, −λ¯ are eigenvalues of H, too. Typical Hamiltonian spectrum Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 3/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems Goal Structure-preserving algorithm, i.e., if λ~ is a computed eigenvalue of H, then λ,~ −λ,~ −λ~ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved. Definition V 2 R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk 2 R is symplectic or a J-isometry if Vk JnVk = Jk : Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems Goal Structure-preserving algorithm, i.e., if λ~ is a computed eigenvalue of H, then λ,~ −λ,~ −λ~ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved. Definition V 2 R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk 2 R is symplectic or a J-isometry if Vk JnVk = Jk : Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Hamiltonian Eigenproblems Goal Structure-preserving algorithm, i.e., if λ~ is a computed eigenvalue of H, then λ,~ −λ,~ −λ~ should also be computed eigenvalues. Goal cannot be achieved by general methods for matrices or matrix pencils like the QR/QZ, Lanczos, Arnoldi algorithms! For an algorithm based on similarity transformations, the goal is achieved if the Hamiltonian structure is preserved. Definition V 2 R2n×2n is symplectic if V T JV = J, i.e., V −1 = JT V T J. 2n×2k T Vk 2 R is symplectic or a J-isometry if Vk JnVk = Jk : Lemma If H is Hamiltonian and V is symplectic, then V −1HV is Hamiltonian, too. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 4/13 Introduction Symplectic Lanczos Hamiltonian-positiv References Introduction Applications Hamiltonian eigenproblems arise in many different applications, e.g.: Systems and control: Solution methods for algebraic and differential Riccati equations. Design of LQR/LQG/H2/H1 controllers and filters for continuous-time linear control systems. Stability radii and system norm computations; optimization of system norms. Passivity-preserving model reduction based on balancing. Reduced-order control for infinite-dimensional systems based on inertial manifolds. Computational physics: exponential integrators for Hamiltonian dynamics. [Eirola '03, Lopez/Simoncini '06, B./Meister '13] Quantum chemistry: computing excitation energies in many-particle systems using random phase approximation (RPA). Quadratic eigenvalue problems: in particular, gyroscopic systems. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 5/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Symplectic Lanczos Algorithm for Hamiltonian operators H is based on transpose-free unsymmetric Lanczos process [Freund '94]; computes partial J-tridiagonalization; 2n×2k provides a symplectic (J-orthogonal) Lanczos basis Vk 2 R , T i.e., Vk JnVk = Jk ; was derived in several variants: [Freund/Mehrmann '94, Ferng/Lin/Wang '97, B./Faßbender '97, Watkins '04]; requires re-J-orthogonalization using, e.g., modified symplectic Gram-Schmidt; can be restarted implicitly using implicit SR steps [B./Faßbender '97] or Krylov-Schur restarting [B./Faßbender/Stoll '11] which allows easy locking & purging procedure; does not provide an orthogonal Lanczos basis and is prone to serious breakdown. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 6/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Hamiltonian J-Tridiagonal Form or Hamiltonian J-Hessenberg Form 2 δ1 β1 ζ2 3 6 δ2 ζ2 β2 ζ3 7 6 7 6 .. .. 7 6 δ ζ . 7 6 3 3 7 6 . 7 6 . 7 6 . ζn 7 6 7 2n×2n Tn = 6 δn ζn βn 7 2 R ; 6 7 6 ν1 −δ1 7 6 7 6 ν2 −δ2 7 6 7 6 ν3 −δ3 7 6 7 6 . 7 4 .. .. 5 νn −δn −1 can be computed by symplectic similarity Tn = V HV for almost (serious breakdown!) any V1 = V (:; 1), is computed partially by symplectic Lanczos process, based on symplectic Lanczos recursion T HVk = Vk Tk + ζk+1vk+1e2k ; Vk = [V (:; 1 : k); V (:; n + 1 : n + k)]: Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 7/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm T Algorithm based on symplectic Lanczos recursion HVk = Vk Tk + ζk+1vk+1e2k 2n×2n 2n INPUT: H 2 R , k 2 N, and start vectorv ~1 6= 0 2 R . 2k×2k 2n×2k OUTPUT: Tk 2 R , Vk 2 R , ζk+1, and vk+1. 1 ζ1 = kv~1k2 1 2 v1 = v~1 ζ1 3 FOR m = 1; 2;:::; k (a) t = Hvm (b) δm = ht; vmi % B./Faßbender '97: δm = 1, Watkins '04: δm = 0. (c)~wm = t − δmvm (d) νm = ht; vmiJ 1 (e) wm = w~m νm (f) u = Hwm (g) βm = −hu; wmiJ (h)~vm+1 = u − ζmvm−1 − βmvm + δmwm (i) ζm+1 = kv~m+1k2 1 (j) vm+1 = v~m+1 ζm+1 ENDFOR Note: 3(b) yields orthogonality of vk ; wk [Ferng/Lin/Wang '97] and optimal conditioning of Lanczos basis [B. '03] if kvk2 = 1 is forced. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 8/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm T Algorithm based on symplectic Lanczos recursion HVk = Vk Tk + ζk+1vk+1e2k 2n×2n 2n INPUT: H 2 R , k 2 N, and start vectorv ~1 6= 0 2 R . 2k×2k 2n×2k OUTPUT: Tk 2 R , Vk 2 R , ζk+1, and vk+1. 1 ζ1 = kv~1k2 1 2 v1 = v~1 ζ1 3 FOR m = 1; 2;:::; k (a) t = Hvm (b) δm = ht; vmi % B./Faßbender '97: δm = 1, Watkins '04: δm = 0. (c)~wm = t − δmvm (d) νm = ht; vmiJ 1 (e) wm = w~m νm (f) u = Hwm (g) βm = −hu; wmiJ (h)~vm+1 = u − ζmvm−1 − βmvm + δmwm (i) ζm+1 = kv~m+1k2 1 (j) vm+1 = v~m+1 ζm+1 ENDFOR Note: 3(b) yields orthogonality of vk ; wk [Ferng/Lin/Wang '97] and optimal conditioning of Lanczos basis [B. '03] if kvk2 = 1 is forced. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 8/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Numerical example: rolling tire Results from [B./Faßbender/Stoll '08/'11] Modeling the noise of rolling tires requires to determine the transient vibrations, [Nackenhorst/von Estorff '01]. FEM model of a deformable wheel rolling on a rigid plane surface results in a gyroscopic system of order n = 124; 992 [Nackenhorst '04]. Here, use reduced-order model of size n = 2; 635 computed by AMLS [Elssel/Voß '06]. Max-Planck-Institut Magdeburg Peter Benner, Symplectic Lanczos for Hamiltonian-positive matrices 9/13 Introduction Symplectic Lanczos Hamiltonian-positiv References The Symplectic Lanczos Algorithm Numerical example: rolling tire Results from [B./Faßbender/Stoll '08/'11] Compare eigs and symplectic Lanczos with Krylov-Schur restarting (HKS) applied to H−1 to compute the 12 smallest eigenvalues. eigs needs 8, HKS 6 iterations. max(cond (SR)) = 331. Eigenvalues scaled by 1,000. eigs HKS
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