The waveguide eigenvalue problem and Giampaolo Tensor infinite - PowerPoint PPT Presentation

The waveguide eigenvalue problem and Tensor infinite Arnoldi The waveguide eigenvalue problem and Giampaolo Tensor infinite Arnoldi Mele Giampaolo Mele KTH Royal Institute of technology Dept. Math, Numerical analysis group WEP 27 August 2015 TIAR Combination Simulations Conclusions Joint work with Elias Jarlebring and Olof Runborg BIT Circus 2015 at Ume˚ a University

The waveguide Outline eigenvalue problem and Tensor infinite Arnoldi Giampaolo Mele ◮ WEP: Waveguide Eigenvalue Problem ◮ TIAR: Tensor infinite Arnoldi ◮ Specialization of TIAR to WEP and numerical WEP simulations TIAR Combination Simulations Conclusions

WEP: the waveguide eigenvalue problem

Helmholtz equation (single-periodic coefficients): The waveguide eigenvalue problem and Tensor infinite ∆ u ( x , z ) + κ ( x , z ) 2 u ( x , z ) Arnoldi = 0 when ( x , z ) ∈ R × R Giampaolo u ( x , · ) → 0 as x → ±∞ Mele ◮ κ ( x , z ) periodic z -direction. ◮ κ ( x , z ) constant for ( x , z ) �∈ [ x − , x + ] × R . . . . WEP TIAR Combination Simulations z Conclusions x . . . Some related computational works: [Tausch, Butler ’02], [Engstr¨ om, Hafner, Schmidt ’09, Engstr¨ om ’10], [Schmidt, Hiptmair ’13], [Spence, Poulton ’05], [Cox, Stevens ’99], . . .

We look for normal modes (Bloch solutions) The waveguide eigenvalue problem and Tensor infinite e λ z v ( x , z ) Arnoldi u ( x , z ) = Giampaolo v ( x , z ) = v ( x , z + 1) ⇒ Mele Periodic PDE-eigenvalue problem on a strip Find v ∈ C 1 ( R × [0 , 1] , R ) and λ such that: ∆ v + 2 λ v z + ( λ 2 + κ ( x , z ) 2 ) v = 0 v ( · , z ) → 0 as x → ±∞ WEP v ( x , z ) = v ( x , z + 1) TIAR Combination Simulations Conclusions Solutions of most interest: λ ∈ C − close to imaginary axis.

The waveguide eigenvalue problem DtN (Dirichlet to Neumann) equivalence and Tensor infinite Arnoldi Under generic conditions, equivalent in a weak sense Giampaolo ∆ v + 2 λ v z + ( λ 2 + κ ( x , z ) 2 ) v Mele = 0 , ( x , z ) ∈ [ x − , x + ] × [0 , 1] v ( x , z ) = v ( x , z + 1) v x ( x − , · ) = T − ,λ ( v ( x − , · )) v x ( x + , · ) = T + ,λ ( v ( x + , · )) T ± ,λ ( · ) has nonlinear dependence in λ . WEP Discretized problem TIAR Combination A particular type of FEM discretization leads to Simulations Conclusions � Q ( λ ) � C 1 ( λ ) M ( λ ) v = v = 0 C T R H P ( λ ) R 2 P ( λ ) nonlinear and non polynomial in λ .

The waveguide eigenvalue problem and Tensor infinite Arnoldi The nonlinear eigenvalue problem Giampaolo Find λ ∈ C , v � = 0 such that Mele M ( λ ) v = 0 where M analytic in a disk Ω ⊂ C . Selection of interesting works WEP [Ruhe ’73], [Mehrmann, Voss ’04], [Lancaster ’02], TIAR [Tisseur, et al. ’01], [Voss ’05], [Unger ’50], [Mackey, et al. ’09], Combination [Kressner ’09], [Bai, et al. ’05], [Meerbergen ’09], [Breda, et al. Simulations ’06], [Betcke, et al. ’04, ’10], [Asakura, et a. ’10], [Beyn ’12], Conclusions [Szyld, Xue ’13], [Hochstenbach, et al. ’08], [Neumaier ’85], [Gohberg, et al. ’82], [Effenberger ’13], [Van Beeumen, et al ’15] . . .

TIAR: tensor infinite Arnoldi

The waveguide eigenvalue problem and Tensor infinite Arnoldi Properties / features of infinite Arnoldi method Giampaolo Mele ◮ Equivalent to Arnoldi’s method on a companion matrix, for any truncation parameter N with N > k ◮ Equivalent to Arnoldi’s method on an operator B ◮ Convergence theory (?) ◮ Requires adaption of computation of y 0 . For Taylor WEP version: TIAR y 0 = M (ˆ λ ) − 1 ( M ′ (ˆ λ ) x 1 + · · · + M ( k ) (ˆ λ ) x k ) Combination Simulations ◮ Complexity of orthogonalization at step k : O ( k 2 n ) Conclusions Described in: [Jarlebring, et al. ’11, ’12, ’15]

Observation: The basis matrix has a structure The waveguide eigenvalue problem and Tensor infinite Arnoldi Giampaolo v 00 v 01 v 02 v 03 Mele v 11 v 12 v 13 v 22 v 23 v 33 Theorem (Implicit representation of the basis matrix WEP [Jarlebring, M., Runborg ’15]) TIAR Combination There exists Z = [ z 1 , . . . , z k ] ∈ C n × k and tensor [ a i , j ,ℓ ] k i , j ,ℓ =1 , Simulations such that the blocks in the basis matrix generated by k steps Conclusions of infinite Arnoldi method can factorized as k � q i , j = a i , j , k z k . ℓ =1

Observation: The basis matrix has a structure The waveguide eigenvalue problem and Tensor infinite Arnoldi Giampaolo v 00 v 01 v 02 v 03 y 0 Mele y 1 v 11 v 12 v 13 v 22 v 23 y 2 v 33 y 3 y 4 Theorem (Implicit representation of the basis matrix WEP [Jarlebring, M., Runborg ’15]) TIAR Combination There exists Z = [ z 1 , . . . , z k ] ∈ C n × k and tensor [ a i , j ,ℓ ] k i , j ,ℓ =1 , Simulations such that the blocks in the basis matrix generated by k steps Conclusions of infinite Arnoldi method can factorized as k � q i , j = a i , j , k z k . ℓ =1

Observation: The basis matrix has a structure The waveguide eigenvalue problem and Tensor infinite Arnoldi Giampaolo v 00 v 01 v 02 v 03 y 0 v 04 Mele y 1 v 11 v 12 v 13 v 14 v 22 v 23 y 2 v 24 v 33 y 3 v 34 y 4 v 44 Theorem (Implicit representation of the basis matrix WEP [Jarlebring, M., Runborg ’15]) TIAR Combination There exists Z = [ z 1 , . . . , z k ] ∈ C n × k and tensor [ a i , j ,ℓ ] k i , j ,ℓ =1 , Simulations such that the blocks in the basis matrix generated by k steps Conclusions of infinite Arnoldi method can factorized as k � q i , j = a i , j , k z k . ℓ =1

The waveguide Key ideas of TIAR eigenvalue problem and Tensor infinite Arnoldi ◮ Rephrase IAR using implicit representation of basis Giampaolo matrix as a Z ∈ C n × k and [ a i , j ,ℓ ] k i , j ,ℓ =1 . Mele ◮ Maintain orthogonality of Z for numerical stability TIAR vs IAR ◮ TIAR involves less memory O ( nm 2 ) vs. O ( nm ), WEP TIAR ◮ Complexity for m steps: O ( nm 3 ) for both, Combination ◮ TIAR involves less data and is much faster due to Simulations modern CPU-caching issues Conclusions Other literature with compact representations ◮ TOAR: [Zhang, Su, ’13], [Kressner, Roman ’14] ◮ CORK: [V. Beeumen, et al ’15]

Specialization of TIAR to WEP and numerical simulations

Recall WEP: The waveguide eigenvalue problem and Tensor infinite � Q ( λ ) � C 1 ( λ ) Arnoldi M ( λ ) = C T R H P ( λ ) R Giampaolo 2 Mele and Q ( λ ) = A 0 + A 1 λ + A 2 λ 2 and C 1 ( λ ) = C 1 , 0 + C 1 , 1 λ + C 1 , 2 λ 2 P ( λ ) = diag( s − , − p ( λ ) , . . . , s − , p ( λ ) , s + , − p ( λ ) , . . . , s + , p ( λ )) where WEP � s ± , k ( λ ) = ρ k (( λ + 2 i π k ) + i κ ± )(( λ + 2 i π k ) − i κ ± ) . TIAR Combination Bad news: O ( √ n ) branch-point singularities Simulations Good news: All singularities are on i R Conclusions Solution Cayley transformation brings all singularities to unit circle. Apply algorithm to Cayley transformed problem.

The waveguide In order to implement IAR or TIAR: We need an efficient eigenvalue problem and Tensor infinite way to compute Arnoldi Giampaolo y 0 = M (0) − 1 ( M ′ (0) x 1 + · · · + M ( k ) (0) x k ) Mele Compute by exploiting structure √ a λ 2 + b λ + c after Cayley ◮ Derivatives of transformation computable with Gegenbauer WEP polynomials (inspired by [Tausch, Butler 02’]) TIAR Combination ◮ Use FFT-for dense (2,2)-block ◮ Higher order derivatives have O ( √ n ) non-zero elements Simulations (reduces dominant O ( n )-term to O ( √ n )) Conclusions ◮ Use Schur complement and LU-factorization of (1 , 1)-block

The waveguide eigenvalue problem and Tensor infinite Arnoldi Simulations for a (more difficult) variant of the waveguide in Giampaolo [Tausch, Butler ’02] Mele One of the eigenfunctions of interest WEP TIAR 1 1 Combination 0.5 0.5 z Simulations 0 0 Conclusions -3 -2 -1 0 1 2 3 x Largest problem with our approach: n ≈ 10 7 .

0 10 0 Ritz values in Ω Ritz values outside Ω Eigenvalues −1 Relative residual norm E(w, γ ) Singularitues −5 10 γ 0 −2 −10 −3 10 −4 −15 10 −5 −6 −20 10 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0 20 40 60 80 100 Iteration k CPU time storage of Q m n n x n z IAR WTIAR IAR TIAR 462 20 21 8.35 secs 2.58 secs 35.24 MB 7.98 MB 1,722 40 41 28.90 secs 2.83 secs 131.38 MB 8.94 MB 6,642 80 81 1 min and 59 secs 4.81 secs 506.74 MB 12.70 MB 26,082 160 161 8 mins and 13.37 secs 13.9 secs 1.94 GB 27.52 MB 103,362 320 321 out of memory 45.50 secs out of memory 86.48 MB 411,522 640 641 out of memory 3 mins and 30.29 secs out of memory 321.60 MB 1,642,242 1280 1281 out of memory 15 mins and 20.61 secs out of memory 1.23 GB Using different computer: n = 9 , 009 , 002, several hours CPU-time.

The waveguide eigenvalue problem and Tensor infinite Arnoldi CONCLUSIONS Giampaolo Mele New contributions ◮ A structured discretization of a waveguide eigenvalue problem (WEP) ◮ A new algorithm: TIAR WEP ◮ Specialization of TIAR to WEP TIAR Combination Online material: Simulations Conclusions ◮ Preprint: http://arxiv.org/abs/1503.02096 ◮ Software: http://www.math.kth.se/~gmele/waveguide