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Recent Advances in Low Rank ADI for Lyapunov Equations

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Recent Advances in Low Rank ADI for Lyapunov Equations GAMM Anual Meeting Novi Sad, March 20th 2013 Recent advances in low rank ADI for Lyapunov equations Jens Saak joint work with Peter Benner, Patrick K¨urschner Computational Methods in Systems and Control Theory (CSC) Max Planck Institute for Dynamics of Complex Technical Systems MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 1/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Outline 1 Low Rank ADI Basics 2 Low Rank Factors of Residuals 3 Adaptive ADI Shift Computation 4 Numerical Experiments Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 2/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 3/20 Low-rank factor ADI iteration (LR-ADI) [Penzl '99, Benner/Li/Penzl '08] −1 −1 V1 = (F + p1I ) G; Vj = I − (pj + pj−1)(F + pj I ) Vj−1; p p Z1 := −2 Re (p1)V1; Zj := [Zj−1; −2 Re (pj )Vj ]: Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Low Rank Solution of Lyapunov Equations Lyapunov equation Consider FX + XF T = −GG T with F 2 Rn×n Hurwitz, G 2 Rn×m; m n. Goal: X = ZZ H Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 4/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Low Rank Solution of Lyapunov Equations Lyapunov equation Consider FX + XF T = −GG T with F 2 Rn×n Hurwitz, G 2 Rn×m; m n. Goal: X = ZZ H Low-rank factor ADI iteration (LR-ADI) [Penzl '99, Benner/Li/Penzl '08] −1 −1 V1 = (F + p1I ) G; Vj = I − (pj + pj−1)(F + pj I ) Vj−1; p p Z1 := −2 Re (p1)V1; Zj := [Zj−1; −2 Re (pj )Vj ]: Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 4/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Shift Parameters Optimal shift parameters For J ADI iterations, the optimal shift parameters solve the rational minmax problem 0 1 J Y pj − λ` min @ max A ; λ` 2 Λ(F ): p1;:::;pJ ⊂ − 1≤`≤n pj + λ` C j=1 (sub-)optimal shifts [Wachspress et al ∼'95] Optimal number J of shifts and their location computable for a given error tolerance for real spectra. suboptimal for Λ(F ) ⊂ C. Heuristic Penzl shifts [Penzl '99] Since λ` not easily available in large-scale setting, take small numbers of Ritz values of F and F −1 (generated with Arnoldi) instead. Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 5/20 L (Y ) := FY + YF T + GG T Residual Based Stopping T L(ZZ ) F computed via clever QR updates [Penzl'99] T T L(ZZ ) 2 = λmax(L(ZZ )) apply Arnoldi/Lanczos [S. et al '09-] Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Stopping Criteria Relative Change of Z: p −2 Re (pj )Vj F ≤ tol jjZj jjF Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 6/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank ADI Basics Stopping Criteria Relative Change of Z: p −2 Re (pj )Vj F ≤ tol jjZj jjF L (Y ) := FY + YF T + GG T Residual Based Stopping T L(ZZ ) F computed via clever QR updates [Penzl'99] T T L(ZZ ) 2 = λmax(L(ZZ )) apply Arnoldi/Lanczos [S. et al '09-] Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 6/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 7/20 And for the ADI iterates : X = F X F H − 2 Re (p )G G H j pj j−1 pj j pj pj T T Then: L(Xj ) = FXj + Xj F + GG T = F (Xj − X ) + (Xj − X ) F H j ! j ! Y T Y = ··· = Fpi GG Fpi i=1 i=1 H = Wj Wj : Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Equivalence of Lyapunov and Stein Equations e.g.[Hylla '10] T T p2C<0 H H FX + XF = −GG () X = FpXFp − 2 Re (p)GpGp −1 −1 Here: Fp := (F − pI )(F + pI ) and Gp := (F + pI ) G Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 8/20 T T Then: L(Xj ) = FXj + Xj F + GG T = F (Xj − X ) + (Xj − X ) F H j ! j ! Y T Y = ··· = Fpi GG Fpi i=1 i=1 H = Wj Wj : Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Equivalence of Lyapunov and Stein Equations e.g.[Hylla '10] T T p2C<0 H H FX + XF = −GG () X = FpXFp − 2 Re (p)GpGp −1 −1 Here: Fp := (F − pI )(F + pI ) and Gp := (F + pI ) G And for the ADI iterates : X = F X F H − 2 Re (p )G G H j pj j−1 pj j pj pj Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 8/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Equivalence of Lyapunov and Stein Equations e.g.[Hylla '10] T T p2C<0 H H FX + XF = −GG () X = FpXFp − 2 Re (p)GpGp −1 −1 Here: Fp := (F − pI )(F + pI ) and Gp := (F + pI ) G And for the ADI iterates : X = F X F H − 2 Re (p )G G H j pj j−1 pj j pj pj T T Then: L(Xj ) = FXj + Xj F + GG T = F (Xj − X ) + (Xj − X ) F H j ! j ! Y T Y = ··· = Fpi GG Fpi i=1 i=1 H = Wj Wj : Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 8/20 −1 Vj = I − (pj−1 + pj )(F + pj I ) Vj−1 j−1 ! −1 Y = ··· =( F + pj I ) Fpi G: i=1 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ j ! H Y L(Xj ) = Wj Wj for Wj = Fpi G i=1 Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 9/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ j ! H Y L(Xj ) = Wj Wj for Wj = Fpi G i=1 −1 Vj = I − (pj−1 + pj )(F + pj I ) Vj−1 −1 = (F − pj−1I )(F + pj I ) Vj−1 −1 −1 = (F − pj−2I )( F + pj−1I ) (F − pj−1I )( F + pj I ) Vj−2 −1 = (F − pj−2I ) Fpj−1 (F + pj I ) Vj−2 j−1 ! −1 Y = ··· =( F + pj I ) Fpi G: i=1 Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 9/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ j ! H Y L(Xj ) = Wj Wj for Wj = Fpi G i=1 −1 Vj = I − (pj−1 + pj )(F + pj I ) Vj−1 j−1 ! −1 Y = ··· =( F + pj I ) Fpi G: i=1 j ! j−1 ! Y −1 Y Wj = Fpi G = (F − pj I )( F + pj I ) Fpi G i=1 i=1 Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 9/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ j ! H Y L(Xj ) = Wj Wj for Wj = Fpi G i=1 −1 Vj = I − (pj−1 + pj )(F + pj I ) Vj−1 j−1 ! −1 Y = ··· =( F + pj I ) Fpi G: i=1 Wj = (F − pj I ) Vj Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 9/20 −1 Wj = (F − pj I ) Vj = (F − pj I )(F + pj I ) Wj−1 −1 = I − (pj + pj )(F + pj I ) Wj−1 = Wj−1 − 2 Re (pj )Vj The latter result was independently derived from a Krylov subspace recurrence perspective in [Wolf/Panzer/Lohmann '13] Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ Wj = (F − pj I ) Vj Moreover: −1 Vj = (F + pj I ) (F − pj−1I ) Vj−1 −1 = (F + pj I ) Wj−1 Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 10/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals Rank and Computability of the Residual Factor [Benner/Kurschner/S.'13]¨ Wj = (F − pj I ) Vj Moreover: −1 Vj = (F + pj I ) (F − pj−1I ) Vj−1 −1 = (F + pj I ) Wj−1 −1 Wj = (F − pj I ) Vj = (F − pj I )(F + pj I ) Wj−1 −1 = I − (pj + pj )(F + pj I ) Wj−1 = Wj−1 − 2 Re (pj )Vj The latter result was independently derived from a Krylov subspace recurrence perspective in [Wolf/Panzer/Lohmann '13] Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 10/20 Note: H H W`−1W`−1 = W`−1W`−1 2=F 2=F H and W`−1W`−1 inner product m × m ≡ small Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals A New LR-ADI Formulation [Benner/Kurschner/S.'13]¨ Algorithm 1 LR-ADI Re-Formulation T T Input: F ,G in FX + XF = −GG , and proper shifts pj ,(j = 1;:::; J) Output: Z with X ≈ ZZ H 1: Z = [] 2: W0 = G 3: ` = 1 H 4: while ( W`−1W`−1 > tol and ` ≤ J) do −1 5: V` = (F + p`I ) W`−1 6: W` = W`−1 − 2 Re (p`)V` h p i 7: Z = Z; −2 Re (p`)V` 8: ` + + 9: end while Max Planck Institute Magdeburg Jens Saak, Recent advances in LR-ADI 11/20 Low Rank ADI Basics Low Rank Factors of Residuals Adaptive ADI Shift Computation Numerical Experiments Low Rank Factors of Residuals A New LR-ADI Formulation [Benner/Kurschner/S.'13]¨ Algorithm 1 LR-ADI Re-Formulation T T Input: F ,G in FX + XF = −GG , and proper shifts pj ,(j = 1;:::; J) Output: Z with X ≈ ZZ H 1: Z = [] 2: W0 = G 3: ` = 1 H 4: while ( W`−1W`−1 > tol and ` ≤ J) do −1
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