Iterative projection methods for sparse nonlinear eigenvalue problems

Transcription

1 Iterative projection methods for sparse nonlinear eigenvalue problems Heinrich Voss Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

2 Outline 1 Introduction 2 Numerical methods for dense problems 3 Iterative projection methods 4 Variational characterization of eigenvalues 5 Numerical example TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

3 Outline Introduction 1 Introduction 2 Numerical methods for dense problems 3 Iterative projection methods 4 Variational characterization of eigenvalues 5 Numerical example TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

4 Introduction Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing positivity of solutions multiplicity of solution dependence of solutions on the parameter; bifurcation (change of ) stability of solutions TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

5 Introduction Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing positivity of solutions multiplicity of solution dependence of solutions on the parameter; bifurcation (change of ) stability of solutions TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

6 In this presentation Introduction For D C let T (λ) C n n, λ D be a family of matrices (more generally a family of closed linear operators on a Banach space). Find λ D and x 0 such that T (λ)x = 0. Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. In this talk we assume the matrices to be large and sparse. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

7 In this presentation Introduction For D C let T (λ) C n n, λ D be a family of matrices (more generally a family of closed linear operators on a Banach space). Find λ D and x 0 such that T (λ)x = 0. Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. In this talk we assume the matrices to be large and sparse. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

8 In this presentation Introduction For D C let T (λ) C n n, λ D be a family of matrices (more generally a family of closed linear operators on a Banach space). Find λ D and x 0 such that T (λ)x = 0. Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. In this talk we assume the matrices to be large and sparse. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

9 Introduction Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

10 Introduction Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

11 Introduction Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

12 Problem Introduction Determine relevant energy states (i.e. eigenvalues) and corresponding wave functions (i.e. eigenfunctions) of a three-dimensional quantum dot embedded in a matrix. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

13 Problem ct. Introduction Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinous across the heterojunction. Boundary and interface conditions Φ = 0 on outer boundary of matrix Ω m 1 Φ BenDaniel Duke condition m m n = 1 Φ Ωm m q n Ωq on interface TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

14 Problem ct. Introduction Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinous across the heterojunction. Boundary and interface conditions Φ = 0 on outer boundary of matrix Ω m 1 Φ BenDaniel Duke condition m m n = 1 Φ Ωm m q n Ωq on interface TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

15 Problem ct. Introduction Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinous across the heterojunction. Boundary and interface conditions Φ = 0 on outer boundary of matrix Ω m 1 Φ BenDaniel Duke condition m m n = 1 Φ Ωm m q n Ωq on interface TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

16 Introduction Electron effective mass The electron effective mass m = m j, j {q, m} depends an the energy level E. Most simulations in the literature consider constant electron effective masses (e.g. m q = 0.023m 0 and m m = 0.067m 0 for an InAs quantum dot and the GaAs matrix, respectively, where m 0 is the free electron mass). Numerical examples demonstrate that the electronic behavior is substantially different: For the pyramidal quantum dot in our examples there are only 3 confined electron states for the linear model whereas the nonlinear model exhibits 7 confined states (i.e. energy levels which are smaller than the confinement potential). The ground state of the nonlinear model is about 7% and the first exited state about 18% smaller than for the linear approximation. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

17 Introduction Electron effective mass The electron effective mass m = m j, j {q, m} depends an the energy level E. Most simulations in the literature consider constant electron effective masses (e.g. m q = 0.023m 0 and m m = 0.067m 0 for an InAs quantum dot and the GaAs matrix, respectively, where m 0 is the free electron mass). Numerical examples demonstrate that the electronic behavior is substantially different: For the pyramidal quantum dot in our examples there are only 3 confined electron states for the linear model whereas the nonlinear model exhibits 7 confined states (i.e. energy levels which are smaller than the confinement potential). The ground state of the nonlinear model is about 7% and the first exited state about 18% smaller than for the linear approximation. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

18 Introduction Electron effective mass The electron effective mass m = m j, j {q, m} depends an the energy level E. Most simulations in the literature consider constant electron effective masses (e.g. m q = 0.023m 0 and m m = 0.067m 0 for an InAs quantum dot and the GaAs matrix, respectively, where m 0 is the free electron mass). Numerical examples demonstrate that the electronic behavior is substantially different: For the pyramidal quantum dot in our examples there are only 3 confined electron states for the linear model whereas the nonlinear model exhibits 7 confined states (i.e. energy levels which are smaller than the confinement potential). The ground state of the nonlinear model is about 7% and the first exited state about 18% smaller than for the linear approximation. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

19 Introduction Electron effective mass The electron effective mass m = m j, j {q, m} depends an the energy level E. Most simulations in the literature consider constant electron effective masses (e.g. m q = 0.023m 0 and m m = 0.067m 0 for an InAs quantum dot and the GaAs matrix, respectively, where m 0 is the free electron mass). Numerical examples demonstrate that the electronic behavior is substantially different: For the pyramidal quantum dot in our examples there are only 3 confined electron states for the linear model whereas the nonlinear model exhibits 7 confined states (i.e. energy levels which are smaller than the confinement potential). The ground state of the nonlinear model is about 7% and the first exited state about 18% smaller than for the linear approximation. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

20 Effective mass ct. Introduction The dependence of m(x, E) on E can be derived from the eight-band k p analysis and effective mass theory. Projecting the 8 8 Hamiltonian onto the conduction band results in the single Hamiltonian eigenvalue problem with { mq (E), x Ω m(x, E) = q m m (E), x Ω m ( 1 m j (E) = P2 j E + g j V j 1 E + g j V j + δ j ), j {m, q} where m j is the electron effective mass, V j the confinement potential, P j the momentum, g j the main energy gap, and δ j the spin-orbit splitting in the jth region. Other types of effective mass (taking into account the effect of strain, e.g.) appear in the literature. They are all rational functions of E where 1/m(x, E) is monotonically decreasing with respect to E, and that s all we need. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

21 Effective mass ct. Introduction The dependence of m(x, E) on E can be derived from the eight-band k p analysis and effective mass theory. Projecting the 8 8 Hamiltonian onto the conduction band results in the single Hamiltonian eigenvalue problem with { mq (E), x Ω m(x, E) = q m m (E), x Ω m ( 1 m j (E) = P2 j E + g j V j 1 E + g j V j + δ j ), j {m, q} where m j is the electron effective mass, V j the confinement potential, P j the momentum, g j the main energy gap, and δ j the spin-orbit splitting in the jth region. Other types of effective mass (taking into account the effect of strain, e.g.) appear in the literature. They are all rational functions of E where 1/m(x, E) is monotonically decreasing with respect to E, and that s all we need. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

22 Variational form Introduction Find E R and Φ H 1 0 (Ω), Φ 0, Ω := Ω q Ω m, such that a(φ, Ψ; E) := Φ Ψ dx + m q (x, E) 2 Ω q Ω m 1 Φ Ψ dx m m (x, E) + V q (x)φψ dx + V m (x)φψ dx Ω q Ω m = E ΦΨ dx =: Eb(Φ, Ψ) for every Ψ H0 1 (Ω) Ω Discretizing by FEM yields a rational matrix eigenvalue problem TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

23 Variational form Introduction Find E R and Φ H 1 0 (Ω), Φ 0, Ω := Ω q Ω m, such that a(φ, Ψ; E) := Φ Ψ dx + m q (x, E) 2 Ω q Ω m 1 Φ Ψ dx m m (x, E) + V q (x)φψ dx + V m (x)φψ dx Ω q Ω m = E ΦΨ dx =: Eb(Φ, Ψ) for every Ψ H0 1 (Ω) Ω Discretizing by FEM yields a rational matrix eigenvalue problem TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

24 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

25 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Recent survey: Tisseur, Meerbergen 2001 T (λ) = λ 2 A + λb + C, A, B, C C n n Dynamic analysis of structures Stability analysis of flows in fluid mechanics Signal processing Vibration of spinning structures Vibration of fluid-solid structures (Conca et al. 1992) Lateral buckling analysis (Prandtl 1899) Corner singularities of anisotropic material (Wendland et al. 1992) Constrained least squares problems (Golub 1973) Regularization of total least squares problems (Sima et al. 2004) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

26 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

27 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems T (λ) = k λ j A j, A j C n n j=0 Optimal control problems (Mehrmann 1991) Corner singularities of anisotropic material (Kozlov et al. 2001) Dynamic element discretization of linear eigenproblems (V. 1987) Least squares element methods (Rothe 1989) System of coupled Euler-Bernoulli and Timoshenko beams (Balakrishnan et al. 2004) Nonlinear integrated optics (Botchev et al. 2008) Electronic states of quantum dots (Hwang et al. 2005) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

28 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

29 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems NMAT T (λ) = λ 2 1 M + K b m λ K m m=1 Vibration of structures with viscoelastic damping Vibration of sandwich plates (Soni 1981) Dynamics of plates with concentrated masses (Andreev et al. 1988) Vibration of fluid-solid structures (Conca et al. 1989) Exact condensation of linear eigenvalue problems (Wittrick, Williams 1971) Electronic states of semiconductor heterostructures (Luttinger, Kohn 1954) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

30 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

31 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems T (λ) = K λm + i p λ σj W j Exact dynamic element methods Nonlinear eigenproblems in accelerator design (Igarashi et al. 1995) Vibrations of poroelastic structures (Dazel et al. 2002) Vibro-acoustic behavior of piezoelectric/poroelastic structures (Batifol et al. 2007) Nonlinear integrated optics (Dirichlet-to-Neumann) (Botchev 2008) Stability of acoustic pressure levels in combustion chambers (van Leeuwen 2007) j=1 TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

32 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

33 Introduction Classes of nonlinear eigenproblems Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues d dt s(t) = A 0s(t) + p A j s(t τ j ), s(t) = xe λt j=1 T (λ) = λi + A 0 + p e τ j λ A j j=1 Stability of time-delay systems TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

34 Outline Numerical methods for dense problems 1 Introduction 2 Numerical methods for dense problems 3 Iterative projection methods 4 Variational characterization of eigenvalues 5 Numerical example TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

35 Numerical methods for dense problems Inverse iteration Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair 1: for k = 0, 1, 2,... until convergence do 2: Solve T (λ k )u k+1 = T (λ k )x k for u k+1 3: λ k+1 = λ k (v H u k+1 )/(v H x k ) 4: Normalize x k+1 = u k+1 /v H u k+1 5: end for Inverse iteration (being a variant of Newton s method) converges locally and quadratically for isolated eigenpairs. Replacing the update of λ by the Rayleigh functional inverse iteration becomes cubically convergent for real symmetric problems (Rothe, V. 1989); in the general case cubic convergence was proved for a two-sided version of Rayleigh functional iteration (Schreiber 2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

36 Numerical methods for dense problems Inverse iteration Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair 1: for k = 0, 1, 2,... until convergence do 2: Solve T (λ k )u k+1 = T (λ k )x k for u k+1 3: λ k+1 = λ k (v H u k+1 )/(v H x k ) 4: Normalize x k+1 = u k+1 /v H u k+1 5: end for Inverse iteration (being a variant of Newton s method) converges locally and quadratically for isolated eigenpairs. Replacing the update of λ by the Rayleigh functional inverse iteration becomes cubically convergent for real symmetric problems (Rothe, V. 1989); in the general case cubic convergence was proved for a two-sided version of Rayleigh functional iteration (Schreiber 2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

37 Numerical methods for dense problems Inverse iteration Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair 1: for k = 0, 1, 2,... until convergence do 2: Solve T (λ k )u k+1 = T (λ k )x k for u k+1 3: λ k+1 = λ k (v H u k+1 )/(v H x k ) 4: Normalize x k+1 = u k+1 /v H u k+1 5: end for Inverse iteration (being a variant of Newton s method) converges locally and quadratically for isolated eigenpairs. Replacing the update of λ by the Rayleigh functional inverse iteration becomes cubically convergent for real symmetric problems (Rothe, V. 1989); in the general case cubic convergence was proved for a two-sided version of Rayleigh functional iteration (Schreiber 2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

38 Numerical methods for dense problems Residual inverse iteration (Neumaier 1985) Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair and shift σ 1: for k = 0, 1, 2,... until convergence do 2: Solve v H T (σ) 1 T (λ k+1 )x k = 0 for λ k+1 or x H k T (λ k+1)x k = 0 if T (λ) is Hermitian and λ k+1 is real 3: Solve T (σ)d k = T (λ k+1 )x k for d k 4: Set u k+1 = x k d k, and normalize x k+1 = u k+1 /v H u k+1 5: end for If T (λ) is twice continuously differentiable, ˆλ is a simple zero of det T (λ), and if ˆx is an eigenvector normalized by v H ˆx = 1, then residual inverse iteration convergence for all σ sufficiently close to ˆλ and it holds that x k+1 ˆx x k ˆx = O( σ ˆλ ) and λ k+1 ˆλ = ( x k ˆx q ) where q = 2 if T (λ) is Hermitian, ˆλ is real, and λ k+1 solves x H k T (λ k+1)x k = 0 in Step 3, and q = 1 otherwise. The domains of attraction of inverse iteration and residual inverse iteration is often very small, and both methods are very sensitive with respect to inexact solves of linear systems. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

39 Numerical methods for dense problems Residual inverse iteration (Neumaier 1985) Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair and shift σ 1: for k = 0, 1, 2,... until convergence do 2: Solve v H T (σ) 1 T (λ k+1 )x k = 0 for λ k+1 or x H k T (λ k+1)x k = 0 if T (λ) is Hermitian and λ k+1 is real 3: Solve T (σ)d k = T (λ k+1 )x k for d k 4: Set u k+1 = x k d k, and normalize x k+1 = u k+1 /v H u k+1 5: end for If T (λ) is twice continuously differentiable, ˆλ is a simple zero of det T (λ), and if ˆx is an eigenvector normalized by v H ˆx = 1, then residual inverse iteration convergence for all σ sufficiently close to ˆλ and it holds that x k+1 ˆx x k ˆx = O( σ ˆλ ) and λ k+1 ˆλ = ( x k ˆx q ) where q = 2 if T (λ) is Hermitian, ˆλ is real, and λ k+1 solves x H k T (λ k+1)x k = 0 in Step 3, and q = 1 otherwise. The domains of attraction of inverse iteration and residual inverse iteration is often very small, and both methods are very sensitive with respect to inexact solves of linear systems. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

40 Numerical methods for dense problems Residual inverse iteration (Neumaier 1985) Require: Initial pair (λ 0, x 0 ) close to wanted eigenpair and shift σ 1: for k = 0, 1, 2,... until convergence do 2: Solve v H T (σ) 1 T (λ k+1 )x k = 0 for λ k+1 or x H k T (λ k+1)x k = 0 if T (λ) is Hermitian and λ k+1 is real 3: Solve T (σ)d k = T (λ k+1 )x k for d k 4: Set u k+1 = x k d k, and normalize x k+1 = u k+1 /v H u k+1 5: end for If T (λ) is twice continuously differentiable, ˆλ is a simple zero of det T (λ), and if ˆx is an eigenvector normalized by v H ˆx = 1, then residual inverse iteration convergence for all σ sufficiently close to ˆλ and it holds that x k+1 ˆx x k ˆx = O( σ ˆλ ) and λ k+1 ˆλ = ( x k ˆx q ) where q = 2 if T (λ) is Hermitian, ˆλ is real, and λ k+1 solves x H k T (λ k+1)x k = 0 in Step 3, and q = 1 otherwise. The domains of attraction of inverse iteration and residual inverse iteration is often very small, and both methods are very sensitive with respect to inexact solves of linear systems. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

41 Outline Iterative projection methods 1 Introduction 2 Numerical methods for dense problems 3 Iterative projection methods 4 Variational characterization of eigenvalues 5 Numerical example TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

42 Iterative projection methods Iterative projection methods For linear sparse eigenproblems T (λ) = λb A very efficient methods are iterative projection methods (Lanczos, Arnoldi, Jacobi Davidson method, e.g.), where approximations to the wanted eigenvalues and eigenvectors are obtained from projections of the eigenproblem to subspaces of small dimension which are expanded in the course of the algorithm. Generalizations to nonlinear sparse eigenproblems nonlinear rational Krylov: Ruhe(2000,2005), Jarlebring, V. (2005)) Arnoldi method: quadratic probl.: Meerbergen (2001) general problems: V. (2003,2004); Liao, Bai, Lee, Ko (2006), Liao (2007) Jacobi-Davidson: polynomial problems: Sleijpen, Boten, Fokkema, van der Vorst (1996) Hwang, Lin, Wang, Wang (2004,2005) general problems:t. Betcke, V. (2004), V. (2004,2007), Schwetlick, Schreiber (2006), Schreiber (2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

43 Iterative projection methods Iterative projection methods For linear sparse eigenproblems T (λ) = λb A very efficient methods are iterative projection methods (Lanczos, Arnoldi, Jacobi Davidson method, e.g.), where approximations to the wanted eigenvalues and eigenvectors are obtained from projections of the eigenproblem to subspaces of small dimension which are expanded in the course of the algorithm. Generalizations to nonlinear sparse eigenproblems nonlinear rational Krylov: Ruhe(2000,2005), Jarlebring, V. (2005)) Arnoldi method: quadratic probl.: Meerbergen (2001) general problems: V. (2003,2004); Liao, Bai, Lee, Ko (2006), Liao (2007) Jacobi-Davidson: polynomial problems: Sleijpen, Boten, Fokkema, van der Vorst (1996) Hwang, Lin, Wang, Wang (2004,2005) general problems:t. Betcke, V. (2004), V. (2004,2007), Schwetlick, Schreiber (2006), Schreiber (2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

44 Iterative projection methods Iterative projection methods For linear sparse eigenproblems T (λ) = λb A very efficient methods are iterative projection methods (Lanczos, Arnoldi, Jacobi Davidson method, e.g.), where approximations to the wanted eigenvalues and eigenvectors are obtained from projections of the eigenproblem to subspaces of small dimension which are expanded in the course of the algorithm. Generalizations to nonlinear sparse eigenproblems nonlinear rational Krylov: Ruhe(2000,2005), Jarlebring, V. (2005)) Arnoldi method: quadratic probl.: Meerbergen (2001) general problems: V. (2003,2004); Liao, Bai, Lee, Ko (2006), Liao (2007) Jacobi-Davidson: polynomial problems: Sleijpen, Boten, Fokkema, van der Vorst (1996) Hwang, Lin, Wang, Wang (2004,2005) general problems:t. Betcke, V. (2004), V. (2004,2007), Schwetlick, Schreiber (2006), Schreiber (2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

45 Iterative projection methods Iterative projection methods For linear sparse eigenproblems T (λ) = λb A very efficient methods are iterative projection methods (Lanczos, Arnoldi, Jacobi Davidson method, e.g.), where approximations to the wanted eigenvalues and eigenvectors are obtained from projections of the eigenproblem to subspaces of small dimension which are expanded in the course of the algorithm. Generalizations to nonlinear sparse eigenproblems nonlinear rational Krylov: Ruhe(2000,2005), Jarlebring, V. (2005)) Arnoldi method: quadratic probl.: Meerbergen (2001) general problems: V. (2003,2004); Liao, Bai, Lee, Ko (2006), Liao (2007) Jacobi-Davidson: polynomial problems: Sleijpen, Boten, Fokkema, van der Vorst (1996) Hwang, Lin, Wang, Wang (2004,2005) general problems:t. Betcke, V. (2004), V. (2004,2007), Schwetlick, Schreiber (2006), Schreiber (2008) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

46 Iterative projection methods Iterative projection method Require: Initial basis V with V H V = I; set m = 1 1: while m number of wanted eigenvalues do 2: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 3: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 4: if r < ε then 5: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 6: reduce search space V if necessary 7: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m )u 8: end if 9: expand search space V = [V, vnew] 10: update projected problem 11: end while Main tasks expand search space choose eigenpair of projected problem (locking, purging) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

47 Iterative projection methods Iterative projection method Require: Initial basis V with V H V = I; set m = 1 1: while m number of wanted eigenvalues do 2: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 3: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 4: if r < ε then 5: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 6: reduce search space V if necessary 7: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m )u 8: end if 9: expand search space V = [V, vnew] 10: update projected problem 11: end while Main tasks expand search space choose eigenpair of projected problem (locking, purging) TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

48 Iterative projection methods Expanding the subspace Given subspace V C n. Expand V by a direction with high approximation potential for the next wanted eigenvector. Let θ be an eigenvalue of the projected problem V H T (λ)vy = 0 and x = Vy corresponding Ritz vector, then inverse iteration yields suitable candidate v := T (θ) 1 T (θ)x BUT: In each step have to solve large linear system with varying matrix and in a truly large problem the vector v will not be accessible but only an inexact solution ṽ := v + e of T (θ)v = T (θ)x, and the next iterate will be a solution of the projection of T (λ)x = 0 upon the expanded space Ṽ := span{v, ṽ}. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

49 Iterative projection methods Expanding the subspace Given subspace V C n. Expand V by a direction with high approximation potential for the next wanted eigenvector. Let θ be an eigenvalue of the projected problem V H T (λ)vy = 0 and x = Vy corresponding Ritz vector, then inverse iteration yields suitable candidate v := T (θ) 1 T (θ)x BUT: In each step have to solve large linear system with varying matrix and in a truly large problem the vector v will not be accessible but only an inexact solution ṽ := v + e of T (θ)v = T (θ)x, and the next iterate will be a solution of the projection of T (λ)x = 0 upon the expanded space Ṽ := span{v, ṽ}. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

50 Iterative projection methods Expanding the subspace Given subspace V C n. Expand V by a direction with high approximation potential for the next wanted eigenvector. Let θ be an eigenvalue of the projected problem V H T (λ)vy = 0 and x = Vy corresponding Ritz vector, then inverse iteration yields suitable candidate v := T (θ) 1 T (θ)x BUT: In each step have to solve large linear system with varying matrix and in a truly large problem the vector v will not be accessible but only an inexact solution ṽ := v + e of T (θ)v = T (θ)x, and the next iterate will be a solution of the projection of T (λ)x = 0 upon the expanded space Ṽ := span{v, ṽ}. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

51 Iterative projection methods Expanding the subspace Given subspace V C n. Expand V by a direction with high approximation potential for the next wanted eigenvector. Let θ be an eigenvalue of the projected problem V H T (λ)vy = 0 and x = Vy corresponding Ritz vector, then inverse iteration yields suitable candidate v := T (θ) 1 T (θ)x BUT: In each step have to solve large linear system with varying matrix and in a truly large problem the vector v will not be accessible but only an inexact solution ṽ := v + e of T (θ)v = T (θ)x, and the next iterate will be a solution of the projection of T (λ)x = 0 upon the expanded space Ṽ := span{v, ṽ}. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

52 Iterative projection methods Expansion of search space ct. We assume that x is already a good approximation to an eigenvector of T ( ). Then v will be an even better approximation, and therefore the eigenvector we are looking for will be very close to the plane E := span{x, v}. We therefore neglect the influence of the orthogonal complement of x in V on the next iterate and discuss the nearness of the planes E and Ẽ := span{x, ṽ}. If the angle between these two planes is small, then the projection of T (λ) upon Ṽ should be similar to the one upon span{v, v}, and the approximation properties of inverse iteration should be maintained. If this angle can become large, then it is not surprising that the convergence properties of inverse iteration are not reflected by the projection method. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

53 Iterative projection methods Expansion of search space ct. We assume that x is already a good approximation to an eigenvector of T ( ). Then v will be an even better approximation, and therefore the eigenvector we are looking for will be very close to the plane E := span{x, v}. We therefore neglect the influence of the orthogonal complement of x in V on the next iterate and discuss the nearness of the planes E and Ẽ := span{x, ṽ}. If the angle between these two planes is small, then the projection of T (λ) upon Ṽ should be similar to the one upon span{v, v}, and the approximation properties of inverse iteration should be maintained. If this angle can become large, then it is not surprising that the convergence properties of inverse iteration are not reflected by the projection method. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

54 Iterative projection methods Expansion of search space ct. We assume that x is already a good approximation to an eigenvector of T ( ). Then v will be an even better approximation, and therefore the eigenvector we are looking for will be very close to the plane E := span{x, v}. We therefore neglect the influence of the orthogonal complement of x in V on the next iterate and discuss the nearness of the planes E and Ẽ := span{x, ṽ}. If the angle between these two planes is small, then the projection of T (λ) upon Ṽ should be similar to the one upon span{v, v}, and the approximation properties of inverse iteration should be maintained. If this angle can become large, then it is not surprising that the convergence properties of inverse iteration are not reflected by the projection method. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

55 Iterative projection methods Expansion of search space ct. We assume that x is already a good approximation to an eigenvector of T ( ). Then v will be an even better approximation, and therefore the eigenvector we are looking for will be very close to the plane E := span{x, v}. We therefore neglect the influence of the orthogonal complement of x in V on the next iterate and discuss the nearness of the planes E and Ẽ := span{x, ṽ}. If the angle between these two planes is small, then the projection of T (λ) upon Ṽ should be similar to the one upon span{v, v}, and the approximation properties of inverse iteration should be maintained. If this angle can become large, then it is not surprising that the convergence properties of inverse iteration are not reflected by the projection method. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

56 Theorem Iterative projection methods Let φ 0 = arccos(x T v) denote the angle between x and v, and the relative error of ṽ by ε := e. Then the maximal possible acute angle between the planes E and Ẽ is { β(ε) = arccos 1 ε 2 / sin 2 φ 0 if ε sin φ 0 π 2 if ε sin φ 0 TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

57 Theorem Iterative projection methods Let φ 0 = arccos(x T v) denote the angle between x and v, and the relative error of ṽ by ε := e. Then the maximal possible acute angle between the planes E and Ẽ is { β(ε) = arccos 1 ε 2 / sin 2 φ 0 if ε sin φ 0 π 2 if ε sin φ 0 TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

58 Proof Iterative projection methods TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

59 Iterative projection methods Expansion by inexact inverse iteration Obviously for every α R, α 0 the plane E is also spanned by x and x + αv. If Ẽ(α) is the plane which is spanned by x and a perturbed realization x + αv + e of x + αv then by the same arguments as in the proof of the Theorem the maximum angle between E and Ẽ(α) is { γ(α, ε) = arccos 1 ε 2 / sin 2 φ(α) if ε sin φ(α) π 2 if ε sin φ(α) where φ(α) denotes the angle between x and x + αv. Since the mapping φ arccos 1 ε 2 / sin 2 φ decreases monotonically the expansion of the search space by an inexact realization of t := x + αv is most robust with respect to small perturbations, if α is chosen such that x and x + αv are orthogonal TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

60 Iterative projection methods Expansion by inexact inverse iteration Obviously for every α R, α 0 the plane E is also spanned by x and x + αv. If Ẽ(α) is the plane which is spanned by x and a perturbed realization x + αv + e of x + αv then by the same arguments as in the proof of the Theorem the maximum angle between E and Ẽ(α) is { γ(α, ε) = arccos 1 ε 2 / sin 2 φ(α) if ε sin φ(α) π 2 if ε sin φ(α) where φ(α) denotes the angle between x and x + αv. Since the mapping φ arccos 1 ε 2 / sin 2 φ decreases monotonically the expansion of the search space by an inexact realization of t := x + αv is most robust with respect to small perturbations, if α is chosen such that x and x + αv are orthogonal TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

61 Iterative projection methods Expansion by inexact inverse iteration Obviously for every α R, α 0 the plane E is also spanned by x and x + αv. If Ẽ(α) is the plane which is spanned by x and a perturbed realization x + αv + e of x + αv then by the same arguments as in the proof of the Theorem the maximum angle between E and Ẽ(α) is { γ(α, ε) = arccos 1 ε 2 / sin 2 φ(α) if ε sin φ(α) π 2 if ε sin φ(α) where φ(α) denotes the angle between x and x + αv. Since the mapping φ arccos 1 ε 2 / sin 2 φ decreases monotonically the expansion of the search space by an inexact realization of t := x + αv is most robust with respect to small perturbations, if α is chosen such that x and x + αv are orthogonal TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

62 Iterative projection methods Expansion by inexact inverse iteration ct. t := x + αv is orthogonal to x iff v = x x H x x H T (θ) 1 T (θ)x T (θ) 1 T (θ)x.. ( ) which yields a maximum acute angle between E and Ẽ(α) { arccos 1 ε 2 if ε 1 γ(α, ε) = π 2 if ε 1. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

63 Iterative projection methods Expansion by inexact inverse iteration ct. t := x + αv is orthogonal to x iff v = x x H x x H T (θ) 1 T (θ)x T (θ) 1 T (θ)x.. ( ) which yields a maximum acute angle between E and Ẽ(α) { arccos 1 ε 2 if ε 1 γ(α, ε) = π 2 if ε 1. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

64 Iterative projection methods Expansion by inexact inverse iteration TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

65 Iterative projection methods Expansion by inexact inverse iteration ct. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

66 Iterative projection methods Expansion by inexact inverse iteration ct. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

67 Iterative projection methods Jacobi Davidson method The expansion v = x x H x x H T (θ) 1 T (θ)x T (θ) 1 T (θ)x.. ( ) of the current search space V is the solution of the equation (I T (θ)xx H x H T (θ)x )T (θ)(i xx H )t = r, t x This is the so called correction equation of the Jacobi Davidson method which was derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpen and van der Vorst (1996) for linear and polynomial eigenvalue problems. Hence, the Jacobi Davidson method is the most robust realization of an expansion of a search space such that the direction of inverse iteration is contained in the expanded space in the sense that it is least sensitive to inexact solves of linear systems T (θ)v = T (θ)x. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

68 Iterative projection methods Jacobi Davidson method The expansion v = x x H x x H T (θ) 1 T (θ)x T (θ) 1 T (θ)x.. ( ) of the current search space V is the solution of the equation (I T (θ)xx H x H T (θ)x )T (θ)(i xx H )t = r, t x This is the so called correction equation of the Jacobi Davidson method which was derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpen and van der Vorst (1996) for linear and polynomial eigenvalue problems. Hence, the Jacobi Davidson method is the most robust realization of an expansion of a search space such that the direction of inverse iteration is contained in the expanded space in the sense that it is least sensitive to inexact solves of linear systems T (θ)v = T (θ)x. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

69 Iterative projection methods Jacobi Davidson method The expansion v = x x H x x H T (θ) 1 T (θ)x T (θ) 1 T (θ)x.. ( ) of the current search space V is the solution of the equation (I T (θ)xx H x H T (θ)x )T (θ)(i xx H )t = r, t x This is the so called correction equation of the Jacobi Davidson method which was derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpen and van der Vorst (1996) for linear and polynomial eigenvalue problems. Hence, the Jacobi Davidson method is the most robust realization of an expansion of a search space such that the direction of inverse iteration is contained in the expanded space in the sense that it is least sensitive to inexact solves of linear systems T (θ)v = T (θ)x. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

70 Iterative projection methods Nonlinear Jacobi-Davidson 1: Start with orthonormal basis V ; set m = 1 2: determine preconditioner M T (σ) 1 ; σ close to first wanted eigenvalue 3: while m number of wanted eigenvalues do 4: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 5: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 6: if r < ε then 7: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 8: reduce search space V if necessary 9: choose new preconditioner M T (µ) if indicated 10: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m)u 11: end if 12: solve approximately correction equation (I T (µ)uu H u H T (µ)u ) T (µ) (I uuh u H u ) t = r, 13: t = t VV T t,ṽ = t/ t, reorthogonalize if necessary 14: expand search space V = [V, ṽ] 15: update projected problem 16: end while t u TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

71 Comments Iterative projection methods 1: Start with orthonormal basis V ; set m = 1 2: determine preconditioner M T (σ) 1 ; σ close to first wanted eigenvalue 3: while m number of wanted eigenvalues do 4: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 5: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 6: if r < ε then 7: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 8: reduce search space V if necessary 9: choose new preconditioner M T (µ) if indicated 10: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m)u 11: end if 12: solve approximately correction equation (I T (µ)uu H u H T (µ)u ) T (µ) (I uuh u H u ) t = r, 13: t = t VV T t,ṽ = t/ t, reorthogonalize if necessary 14: expand search space V = [V, ṽ] 15: update projected problem 16: end while t u TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

72 Comments 1: Iterative projection methods Here preinformation on eigenvectors can be introduced into the algorithm (approximate eigenvectors of contigous problems in reanalysis, e.g.) If no information on eigenvectors is at hand, and if we are interested in eigenvalues close to the parameter σ D: choose initial vector at random, and execute a few Arnoldi steps for the linear eigenproblem T (σ)u = θu or T (σ)u = θt (σ)u, and choose the eigenvector corresponding to the smallest eigenvalue in modulus or a small number of Schur vectors as initial basis of the search space. Starting with a random vector without this preprocessing usually will yield a value µ in step 3 which is far away from σ and will avert convergence. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

73 Comments 1: Iterative projection methods Here preinformation on eigenvectors can be introduced into the algorithm (approximate eigenvectors of contigous problems in reanalysis, e.g.) If no information on eigenvectors is at hand, and if we are interested in eigenvalues close to the parameter σ D: choose initial vector at random, and execute a few Arnoldi steps for the linear eigenproblem T (σ)u = θu or T (σ)u = θt (σ)u, and choose the eigenvector corresponding to the smallest eigenvalue in modulus or a small number of Schur vectors as initial basis of the search space. Starting with a random vector without this preprocessing usually will yield a value µ in step 3 which is far away from σ and will avert convergence. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

74 Comments 1: Iterative projection methods Here preinformation on eigenvectors can be introduced into the algorithm (approximate eigenvectors of contigous problems in reanalysis, e.g.) If no information on eigenvectors is at hand, and if we are interested in eigenvalues close to the parameter σ D: choose initial vector at random, and execute a few Arnoldi steps for the linear eigenproblem T (σ)u = θu or T (σ)u = θt (σ)u, and choose the eigenvector corresponding to the smallest eigenvalue in modulus or a small number of Schur vectors as initial basis of the search space. Starting with a random vector without this preprocessing usually will yield a value µ in step 3 which is far away from σ and will avert convergence. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

75 Comments Iterative projection methods 1: Start with orthonormal basis V ; set m = 1 2: determine preconditioner M T (σ) 1 ; σ close to first wanted eigenvalue 3: while m number of wanted eigenvalues do 4: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 5: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 6: if r < ε then 7: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 8: reduce search space V if necessary 9: choose new preconditioner M T (µ) if indicated 10: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m)u 11: end if 12: solve approximately correction equation (I T (µ)uu H u H T (µ)u ) T (µ) (I uuh u H u ) t = r, 13: t = t VV T t,ṽ = t/ t, reorthogonalize if necessary 14: expand search space V = [V, ṽ] 15: update projected problem 16: end while t u TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

76 Comments 4: Iterative projection methods Since the dimension of the projected problem is small it can be solved by any dense solver like linearization if T (λ) is polynomial methods based on characteristic function det T (λ) = 0 inverse iteration residual inverse iteration successive linear problems Notice that symmetry properties that the original problem may have are inherited by the projected problem, and one can take advantage of these properties when solving the projected problem (safeguarded iteration, structure preserving methods). TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

77 Comments 4: Iterative projection methods Since the dimension of the projected problem is small it can be solved by any dense solver like linearization if T (λ) is polynomial methods based on characteristic function det T (λ) = 0 inverse iteration residual inverse iteration successive linear problems Notice that symmetry properties that the original problem may have are inherited by the projected problem, and one can take advantage of these properties when solving the projected problem (safeguarded iteration, structure preserving methods). TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

78 Comments Iterative projection methods 1: Start with orthonormal basis V ; set m = 1 2: determine preconditioner M T (σ) 1 ; σ close to first wanted eigenvalue 3: while m number of wanted eigenvalues do 4: compute eigenpair (µ, y) of projected problem V T T (λ)vy = 0. 5: determine Ritz vector u = Vy, u = 1,and residual r = T (µ)u 6: if r < ε then 7: accept approximate eigenpair λ m = µ, x m = u; increase m m + 1 8: reduce search space V if necessary 9: choose new preconditioner M T (µ) if indicated 10: choose approximation (λ m, u) to next eigenpair, and compute r = T (λ m)u 11: end if 12: solve approximately correction equation (I T (µ)uu H u H T (µ)u ) T (µ) (I uuh u H u ) t = r, 13: t = t VV T t,ṽ = t/ t, reorthogonalize if necessary 14: expand search space V = [V, ṽ] 15: update projected problem 16: end while t u TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

79 Comments 8: Iterative projection methods As the subspaces expand in the course of the algorithm the increasing storage or the computational cost for solving the projected eigenvalue problems may make it necessary to restart the algorithm and purge some of the basis vectors. Since a restart destroys information on the eigenvectors and particularly on the one the method is just aiming at, we restart only if an eigenvector has just converged. Resonable search spaces after restart are the space spanned by the already converged eigenvectors (or a space slightly larger) an invariant space of T (σ) or eigenspace of T (σ)y = µt (σ)y corresponding to small eigenvalues in modulus. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61

80 Comments 8: Iterative projection methods As the subspaces expand in the course of the algorithm the increasing storage or the computational cost for solving the projected eigenvalue problems may make it necessary to restart the algorithm and purge some of the basis vectors. Since a restart destroys information on the eigenvectors and particularly on the one the method is just aiming at, we restart only if an eigenvector has just converged. Resonable search spaces after restart are the space spanned by the already converged eigenvectors (or a space slightly larger) an invariant space of T (σ) or eigenspace of T (σ)y = µt (σ)y corresponding to small eigenvalues in modulus. TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, / 61