Numerical algorithms for large-scale Hamiltonian eigenproblems - PowerPoint PPT Presentation

Numerical algorithms for large-scale Hamiltonian eigenproblems Peter Benner Professur Mathematik in Industrie und Technik Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz joint work with Heike Faßbender (TU Braunschweig), Martin Stoll (Oxford University) Workshop on Structured Linear Algebra Problems: Analysis, Algorithms, and Applications Cortona, Italy, September 15-19, 2008

Overview Large-Scale Hamiltonian 1 Introduction Eigenproblems Hamiltonian Eigenproblems Peter Benner Applications Introduction Symplectic 2 The Symplectic Lanczos Algorithm Lanczos The SR 3 The SR Algorithm Algorithm HKS 4 A Hamiltonian Krylov-Schur-Type Algorithm Numerical Examples Derivation Conclusions and Outlook 5 Numerical Examples References Quadratic Eigenvalue Problems Corner singularities Gyroscopic systems 6 Conclusions and Outlook 7 References

Introduction Hamiltonian Eigenproblems Large-Scale Hamiltonian Eigenproblems Peter Benner Definition Introduction Hamiltonian Eigenproblems � � 0 I n , then H ∈ R 2 n × 2 n is called Hamiltonian, if Applications Let J = − I n 0 Symplectic Lanczos ( HJ ) T = HJ . The SR Algorithm HKS Numerical Examples Explicit block form of Hamiltonian matrices Conclusions and Outlook � A References � G , where A , G , Q ∈ R n × n and G = G T , Q = Q T . − A T Q

Introduction Spectral Properties Large-Scale Hamiltonian Eigenproblems Hamiltonian Eigensymmetry Peter Benner Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: Introduction if λ is a finite eigenvalue of H , then ¯ λ, − λ, − ¯ λ are eigenvalues of H , Hamiltonian Eigenproblems too. Applications Symplectic Lanczos The SR Algorithm HKS Numerical Examples Conclusions and Outlook References

Introduction Spectral Properties Large-Scale Hamiltonian Eigenproblems Hamiltonian Eigensymmetry Peter Benner Hamiltonian matrices exhibit the Hamiltonian eigensymmetry: Introduction if λ is a finite eigenvalue of H , then ¯ λ, − λ, − ¯ λ are eigenvalues of H , Hamiltonian Eigenproblems too. Applications Symplectic Lanczos Typical Hamiltonian spectrum: The SR Algorithm HKS Numerical Examples Conclusions and Outlook References

Hamiltonian Eigenproblems Large-Scale Goal Hamiltonian Eigenproblems Structure-preserving algorithm, i.e., if ˜ λ is a computed eigenvalue of Peter Benner H , then ˜ λ, − ˜ λ, − ˜ λ should also be computed eigenvalues. Introduction Hamiltonian Eigenproblems Goal cannot be achieved by general methods for matrices or matrix Applications pencils like the QR, Lanczos, Arnoldi algorithms! Symplectic Lanczos For an algorithm based on similarity transformations, the goal is The SR Algorithm achieved if the Hamiltonian structure is preserved. HKS Numerical Examples Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian and S is symplectic, then S − 1 HS is Hamiltonian, too.

Hamiltonian Eigenproblems Large-Scale Goal Hamiltonian Eigenproblems Structure-preserving algorithm, i.e., if ˜ λ is a computed eigenvalue of Peter Benner H , then ˜ λ, − ˜ λ, − ˜ λ should also be computed eigenvalues. Introduction Hamiltonian Eigenproblems Goal cannot be achieved by general methods for matrices or matrix Applications pencils like the QR, Lanczos, Arnoldi algorithms! Symplectic Lanczos For an algorithm based on similarity transformations, the goal is The SR Algorithm achieved if the Hamiltonian structure is preserved. HKS Numerical Examples Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian and S is symplectic, then S − 1 HS is Hamiltonian, too.

Hamiltonian Eigenproblems Large-Scale Goal Hamiltonian Eigenproblems Structure-preserving algorithm, i.e., if ˜ λ is a computed eigenvalue of Peter Benner H , then ˜ λ, − ˜ λ, − ˜ λ should also be computed eigenvalues. Introduction Hamiltonian Eigenproblems Goal cannot be achieved by general methods for matrices or matrix Applications pencils like the QR, Lanczos, Arnoldi algorithms! Symplectic Lanczos For an algorithm based on similarity transformations, the goal is The SR Algorithm achieved if the Hamiltonian structure is preserved. HKS Numerical Examples Definition Conclusions and S ∈ R 2 n × 2 n is symplectic iff i.e., S − 1 = J T S T J . S T JS = J , Outlook References Lemma If H is Hamiltonian and S is symplectic, then S − 1 HS is Hamiltonian, too.

Introduction Applications Hamiltonian eigenproblems arise in many different applications, e.g.: Large-Scale Hamiltonian Eigenproblems Systems and control Peter Benner Model reduction Introduction Computational physics: Hamiltonian Eigenproblems exponential integrators for Hamiltonian dynamics. Applications Symplectic [ Eirola ’03, Lopez/Simoncini ’06 ] Lanczos Quantum chemistry: The SR Algorithm computing excitation energies in many-particle systems using HKS random phase approximation (RPA). Numerical Examples Quadratic eigenvalue problems with Hamiltonian symmetry: Conclusions and – computation of corner singularities in 3D anisotropic elastic Outlook References structures [ Apel/Mehrmann/Watkins ’01 ] ; – gyroscopic systems [ Lancaster ’99,. . . ] ; – vibro-acoustics [ Maess/Gaul ’05 ] ; – optical waveguide design [ Schmidt et al ’03 ] .

The Symplectic Lanczos Algorithm Large-Scale Hamiltonian Eigenproblems Symplectic Lanczos Algorithm for Hamiltonian operators H Peter Benner is based on transpose-free unsymmetric Lanczos process Introduction [ Freund ’94 ] ; Symplectic Lanczos computes partial J -tridiagonalization; The SR Algorithm provides a symplectic ( J -orthogonal) Lanczos basis V k ∈ R 2 n × 2 k , HKS i.e., V T k J n V k = J k ; Numerical Examples was derived in several variants: [ Freund/Mehrmann ’94, Conclusions and Ferng/Lin/Wang ’97, B./Faßbender ’97, Watkins ’04 ] ; Outlook requires re- J -orthogonalization using, e.g., modified symplectic References Gram-Schmidt; can be restarted implicitly using implicit SR steps [ B./Faßbender ’97 ] ; exhibits convergence problems without locking & purging.

The Hamiltonian J -Tridiagonal Form or Hamiltonian J -Hessenberg Form Large-Scale Hamiltonian 2 3 Eigenproblems δ 1 β 1 ζ 2 Peter Benner 6 δ 2 ζ 2 β 2 ζ 3 7 6 7 6 ... ... 7 Introduction 6 7 δ 3 ζ 3 6 7 Symplectic 6 7 ... ... ... Lanczos 6 7 ζ n 6 7 The SR 6 7 Algorithm δ n ζ n β n ∈ R 2 n × 2 n , T n = 6 7 6 7 HKS ν 1 − δ 1 6 7 6 7 Numerical ν 2 − δ 2 6 7 Examples 6 7 ν 3 − δ 3 6 7 Conclusions and 6 7 ... ... Outlook 4 5 References ν n − δ n can be computed by symplectic similarity T n = S − 1 HS almost always, is computed partially by symplectic Lanczos process, based on symplectic Lanczos recursion HV k = V k T k + ζ k +1 v k +1 e T 2 k , V k = [ S (: , 1 : k ) , S (: , n + 1 : n + k )] .

The Symplectic Lanczos Algorithm Derivation using Partial J -Tridiagonalization Large-Scale Theorem Hamiltonian Eigenproblems If T n = S − 1 HS is in Hamiltonian J -tridiagonal form, then Peter Benner K ( H , 2 n − 1 , v ) = SR with s 1 = v Introduction Symplectic Lanczos is an SR decomposition of the Krylov matrix The SR Algorithm K ( H , 2 n − 1 , v ) := [ v , Hv , . . . , H 2 n − 1 v ] . HKS Numerical If R is nonsingular, then T is unreduced, i.e., ζ j � = 0 for all j . Examples Conclusions and Column-wise evaluation of HS = ST n yields ( S := [ v 1 , . . . , v n , w 1 , . . . , w n ]) Outlook References Hv k = δ k v k + ν k w k ⇐ ⇒ ν k w k = Hv k − δ k v k =: e w k , Hw k = ζ m v k − 1 + β k v k − δ k w k + ζ k +1 v k +1 ⇐ ⇒ ζ k +1 v k +1 = Hw k − ζ k v k − 1 − β k v k + δ k w k =: e v k +1 . = ⇒ Choose parameters δ k , β k , ν k , ζ k such that resulting algorithm computes symplectic ( J -orthogonal) basis of Krylov subspace K ( H , v 1 , 2 m ) = span { v 1 , Hv 1 , . . . , H 2 m − 1 v 1 } .

The Symplectic Lanczos Algorithm Derivation using Partial J -Tridiagonalization Large-Scale Theorem Hamiltonian Eigenproblems If T n = S − 1 HS is in Hamiltonian J -tridiagonal form, then Peter Benner K ( H , 2 n − 1 , v ) = SR with s 1 = v Introduction Symplectic Lanczos is an SR decomposition of the Krylov matrix The SR Algorithm K ( H , 2 n − 1 , v ) := [ v , Hv , . . . , H 2 n − 1 v ] . HKS Numerical If R is nonsingular, then T is unreduced, i.e., ζ j � = 0 for all j . Examples Conclusions and Column-wise evaluation of HS = ST n yields ( S := [ v 1 , . . . , v n , w 1 , . . . , w n ]) Outlook References Hv k = δ k v k + ν k w k ⇐ ⇒ ν k w k = Hv k − δ k v k =: e w k , Hw k = ζ m v k − 1 + β k v k − δ k w k + ζ k +1 v k +1 ⇐ ⇒ ζ k +1 v k +1 = Hw k − ζ k v k − 1 − β k v k + δ k w k =: e v k +1 . = ⇒ Choose parameters δ k , β k , ν k , ζ k such that resulting algorithm computes symplectic ( J -orthogonal) basis of Krylov subspace K ( H , v 1 , 2 m ) = span { v 1 , Hv 1 , . . . , H 2 m − 1 v 1 } .