iterative projection methods for sparse nonlinear ... · introduction outline 1 introduction 2...
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Iterative projection methods for sparse nonlineareigenvalue problems
Heinrich [email protected]
Hamburg University of TechnologyInstitute of Mathematics
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 1 / 61
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, 2012 2 / 61
Introduction
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, 2012 3 / 61
Introduction
Nonlinear eigenvalue problem
The term nonlinear eigenvalue problem is not used in a unique way in theliterature
On the one hand: Parameter dependent nonlinear (with respect to the statevariable) 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, 2012 4 / 61
Introduction
Nonlinear eigenvalue problem
The term nonlinear eigenvalue problem is not used in a unique way in theliterature
On the one hand: Parameter dependent nonlinear (with respect to the statevariable) 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, 2012 4 / 61
Introduction
In this presentation
For D ⊂ C let T (λ) ∈ Cn×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 6= 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, 2012 5 / 61
Introduction
In this presentation
For D ⊂ C let T (λ) ∈ Cn×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 6= 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, 2012 5 / 61
Introduction
In this presentation
For D ⊂ C let T (λ) ∈ Cn×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 6= 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, 2012 5 / 61
Introduction
Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the pastfew years because of their special physical properties and their potential forapplications in micro– and optoelectronic devices.
In such nanostructures, the free carriers are confined to a small region ofspace by potential barriers, and if the size of this region is less than theelectron wavelength, the electronic states become quantized at discreteenergy levels.
The ultimate limit of low dimensional structures is the quantum dot, in whichthe carriers are confined in all three directions, thus reducing the degrees offreedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 6 / 61
Introduction
Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the pastfew years because of their special physical properties and their potential forapplications in micro– and optoelectronic devices.
In such nanostructures, the free carriers are confined to a small region ofspace by potential barriers, and if the size of this region is less than theelectron wavelength, the electronic states become quantized at discreteenergy levels.
The ultimate limit of low dimensional structures is the quantum dot, in whichthe carriers are confined in all three directions, thus reducing the degrees offreedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 6 / 61
Introduction
Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the pastfew years because of their special physical properties and their potential forapplications in micro– and optoelectronic devices.
In such nanostructures, the free carriers are confined to a small region ofspace by potential barriers, and if the size of this region is less than theelectron wavelength, the electronic states become quantized at discreteenergy levels.
The ultimate limit of low dimensional structures is the quantum dot, in whichthe carriers are confined in all three directions, thus reducing the degrees offreedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 6 / 61
Introduction
ProblemDetermine relevant energy states (i.e. eigenvalues) and corresponding wavefunctions (i.e. eigenfunctions) of a three-dimensional quantum dot embeddedin a matrix.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 7 / 61
Introduction
Problem ct.
Governing equation: Schrodinger equation
−∇ ·(
~2
2m(x ,E)∇Φ
)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm
where ~ is the reduced Planck constant, m(x ,E) is the electron effectivemass, 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 Ωm
BenDaniel–Duke condition1
mm
∂Φ
∂n
∣∣∣∣∂Ωm
=1
mq
∂Φ
∂n
∣∣∣∣∂Ωq
on interface
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 8 / 61
Introduction
Problem ct.
Governing equation: Schrodinger equation
−∇ ·(
~2
2m(x ,E)∇Φ
)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm
where ~ is the reduced Planck constant, m(x ,E) is the electron effectivemass, 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 Ωm
BenDaniel–Duke condition1
mm
∂Φ
∂n
∣∣∣∣∂Ωm
=1
mq
∂Φ
∂n
∣∣∣∣∂Ωq
on interface
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 8 / 61
Introduction
Problem ct.
Governing equation: Schrodinger equation
−∇ ·(
~2
2m(x ,E)∇Φ
)+ V (x)Φ = EΦ, x ∈ Ωq ∪ Ωm
where ~ is the reduced Planck constant, m(x ,E) is the electron effectivemass, 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 Ωm
BenDaniel–Duke condition1
mm
∂Φ
∂n
∣∣∣∣∂Ωm
=1
mq
∂Φ
∂n
∣∣∣∣∂Ωq
on interface
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 8 / 61
Introduction
Electron effective mass
The electron effective mass m = mj , j ∈ q,m depends an the energy levelE .
Most simulations in the literature consider constant electron effective masses(e.g. mq = 0.023m0 and mm = 0.067m0 for an InAs quantum dot and theGaAs matrix, respectively, where m0 is the free electron mass).
Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 9 / 61
Introduction
Electron effective mass
The electron effective mass m = mj , j ∈ q,m depends an the energy levelE .
Most simulations in the literature consider constant electron effective masses(e.g. mq = 0.023m0 and mm = 0.067m0 for an InAs quantum dot and theGaAs matrix, respectively, where m0 is the free electron mass).
Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 9 / 61
Introduction
Electron effective mass
The electron effective mass m = mj , j ∈ q,m depends an the energy levelE .
Most simulations in the literature consider constant electron effective masses(e.g. mq = 0.023m0 and mm = 0.067m0 for an InAs quantum dot and theGaAs matrix, respectively, where m0 is the free electron mass).
Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 9 / 61
Introduction
Electron effective mass
The electron effective mass m = mj , j ∈ q,m depends an the energy levelE .
Most simulations in the literature consider constant electron effective masses(e.g. mq = 0.023m0 and mm = 0.067m0 for an InAs quantum dot and theGaAs matrix, respectively, where m0 is the free electron mass).
Numerical examples demonstrate that the electronic behavior is substantiallydifferent: For the pyramidal quantum dot in our examples there are only 3confined electron states for the linear model whereas the nonlinear modelexhibits 7 confined states (i.e. energy levels which are smaller than theconfinement potential).The ground state of the nonlinear model is about 7% and the first exited stateabout 18% smaller than for the linear approximation.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 9 / 61
Introduction
Effective mass ct.
The dependence of m(x ,E) on E can be derived from the eight-band k · panalysis and effective mass theory. Projecting the 8× 8 Hamiltonian onto theconduction band results in the single Hamiltonian eigenvalue problem with
m(x ,E) =
mq(E), x ∈ Ωqmm(E), x ∈ Ωm
1mj (E)
=P2
j
~2
(2
E + gj − Vj+
1E + gj − Vj + δj
), j ∈ m,q
where mj is the electron effective mass, Vj the confinement potential, Pj themomentum, gj the main energy gap, and δj the spin-orbit splitting in the j thregion.
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) ismonotonically decreasing with respect to E , and that’s all we need.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 10 / 61
Introduction
Effective mass ct.
The dependence of m(x ,E) on E can be derived from the eight-band k · panalysis and effective mass theory. Projecting the 8× 8 Hamiltonian onto theconduction band results in the single Hamiltonian eigenvalue problem with
m(x ,E) =
mq(E), x ∈ Ωqmm(E), x ∈ Ωm
1mj (E)
=P2
j
~2
(2
E + gj − Vj+
1E + gj − Vj + δj
), j ∈ m,q
where mj is the electron effective mass, Vj the confinement potential, Pj themomentum, gj the main energy gap, and δj the spin-orbit splitting in the j thregion.
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) ismonotonically decreasing with respect to E , and that’s all we need.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 10 / 61
Introduction
Variational form
Find E ∈ R and Φ ∈ H10 (Ω), Φ 6= 0, Ω := Ωq ∪ Ωm, such that
a(Φ,Ψ; E) :=~2
2
∫Ωq
1mq(x ,E)
∇Φ · ∇Ψ dx +~2
2
∫Ωm
1mm(x ,E)
∇Φ · ∇Ψ dx
+
∫Ωq
Vq(x)ΦΨ dx +
∫Ωm
Vm(x)ΦΨ dx
= E∫Ω
ΦΨ dx =: Eb(Φ,Ψ) for every Ψ ∈ H10 (Ω)
Discretizing by FEM yields a rational matrix eigenvalue problem
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 11 / 61
Introduction
Variational form
Find E ∈ R and Φ ∈ H10 (Ω), Φ 6= 0, Ω := Ωq ∪ Ωm, such that
a(Φ,Ψ; E) :=~2
2
∫Ωq
1mq(x ,E)
∇Φ · ∇Ψ dx +~2
2
∫Ωm
1mm(x ,E)
∇Φ · ∇Ψ dx
+
∫Ωq
Vq(x)ΦΨ dx +
∫Ωm
Vm(x)ΦΨ dx
= E∫Ω
ΦΨ dx =: Eb(Φ,Ψ) for every Ψ ∈ H10 (Ω)
Discretizing by FEM yields a rational matrix eigenvalue problem
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 11 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problems
Polynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Recent survey: Tisseur, Meerbergen 2001
T (λ) = λ2A + λB + C, A,B,C ∈ Cn×n
Dynamic analysis of structuresStability analysis of flows in fluid mechanicsSignal processingVibration of spinning structuresVibration 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)
Polynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problems
Rational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problems
T (λ) =k∑
j=0
λjAj , Aj ∈ Cn×n
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)
Rational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problems
General nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problems
T (λ) = λ2M + K +NMAT∑m=1
11 + bmλ
∆Km
Vibration of structures with viscoelastic dampingVibration 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)
General nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblems
Exponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblems
T (λ) = K − λM + ip∑
j=1
√λ− σjWj
Exact dynamic element methodsNonlinear 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)
Exponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Introduction
Classes of nonlinear eigenproblems
Quadratic eigenvalue problemsPolynomial eigenvalue problemsRational eigenvalue problemsGeneral nonlinear eigenproblemsExponential dependence on eigenvalues
ddt
s(t) = A0s(t) +
p∑j=1
Ajs(t − τj ), s(t) = xeλt
⇒ T (λ) = −λI + A0 +
p∑j=1
e−τjλAj
Stability of time-delay systems
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 12 / 61
Numerical methods for dense problems
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, 2012 13 / 61
Numerical methods for dense problems
Inverse iteration
Require: Initial pair (λ0, x0) close to wanted eigenpair1: for k = 0,1,2, . . . until convergence do2: Solve T (λk )uk+1 = T ′(λk )xk for uk+13: λk+1 = λk − (vHuk+1)/(vHxk )4: Normalize xk+1 = uk+1/vHuk+15: end for
Inverse iteration (being a variant of Newton’s method) converges locally andquadratically for isolated eigenpairs.
Replacing the update of λ by the Rayleigh functional inverse iterationbecomes cubically convergent for real symmetric problems (Rothe, V. 1989);in the general case cubic convergence was proved for a two-sided version ofRayleigh functional iteration (Schreiber 2008)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 14 / 61
Numerical methods for dense problems
Inverse iteration
Require: Initial pair (λ0, x0) close to wanted eigenpair1: for k = 0,1,2, . . . until convergence do2: Solve T (λk )uk+1 = T ′(λk )xk for uk+13: λk+1 = λk − (vHuk+1)/(vHxk )4: Normalize xk+1 = uk+1/vHuk+15: end for
Inverse iteration (being a variant of Newton’s method) converges locally andquadratically for isolated eigenpairs.
Replacing the update of λ by the Rayleigh functional inverse iterationbecomes cubically convergent for real symmetric problems (Rothe, V. 1989);in the general case cubic convergence was proved for a two-sided version ofRayleigh functional iteration (Schreiber 2008)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 14 / 61
Numerical methods for dense problems
Inverse iteration
Require: Initial pair (λ0, x0) close to wanted eigenpair1: for k = 0,1,2, . . . until convergence do2: Solve T (λk )uk+1 = T ′(λk )xk for uk+13: λk+1 = λk − (vHuk+1)/(vHxk )4: Normalize xk+1 = uk+1/vHuk+15: end for
Inverse iteration (being a variant of Newton’s method) converges locally andquadratically for isolated eigenpairs.
Replacing the update of λ by the Rayleigh functional inverse iterationbecomes cubically convergent for real symmetric problems (Rothe, V. 1989);in the general case cubic convergence was proved for a two-sided version ofRayleigh functional iteration (Schreiber 2008)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 14 / 61
Numerical methods for dense problems
Residual inverse iteration (Neumaier 1985)
Require: Initial pair (λ0, x0) close to wanted eigenpair and shift σ1: for k = 0,1,2, . . . until convergence do2: Solve vHT (σ)−1T (λk+1)xk = 0 for λk+1
or xHk T (λk+1)xk = 0 if T (λ) is Hermitian and λk+1 is real
3: Solve T (σ)dk = T (λk+1)xk for dk4: Set uk+1 = xk − dk , and normalize xk+1 = uk+1/vHuk+15: end for
If T (λ) is twice continuously differentiable, λ is a simple zero of det T (λ), andif x is an eigenvector normalized by vH x = 1, then residual inverse iterationconvergence for all σ sufficiently close to λ and it holds that
‖xk+1 − x‖‖xk − x‖
= O(|σ − λ|) and |λk+1 − λ| = (‖xk − x‖q)
where q = 2 if T (λ) is Hermitian, λ is real, and λk+1 solves xHk T (λk+1)xk = 0
in Step 3, and q = 1 otherwise.The domains of attraction of inverse iteration and residual inverse iteration isoften very small, and both methods are very sensitive with respect to inexactsolves of linear systems.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 15 / 61
Numerical methods for dense problems
Residual inverse iteration (Neumaier 1985)
Require: Initial pair (λ0, x0) close to wanted eigenpair and shift σ1: for k = 0,1,2, . . . until convergence do2: Solve vHT (σ)−1T (λk+1)xk = 0 for λk+1
or xHk T (λk+1)xk = 0 if T (λ) is Hermitian and λk+1 is real
3: Solve T (σ)dk = T (λk+1)xk for dk4: Set uk+1 = xk − dk , and normalize xk+1 = uk+1/vHuk+15: end for
If T (λ) is twice continuously differentiable, λ is a simple zero of det T (λ), andif x is an eigenvector normalized by vH x = 1, then residual inverse iterationconvergence for all σ sufficiently close to λ and it holds that
‖xk+1 − x‖‖xk − x‖
= O(|σ − λ|) and |λk+1 − λ| = (‖xk − x‖q)
where q = 2 if T (λ) is Hermitian, λ is real, and λk+1 solves xHk T (λk+1)xk = 0
in Step 3, and q = 1 otherwise.The domains of attraction of inverse iteration and residual inverse iteration isoften very small, and both methods are very sensitive with respect to inexactsolves of linear systems.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 15 / 61
Numerical methods for dense problems
Residual inverse iteration (Neumaier 1985)
Require: Initial pair (λ0, x0) close to wanted eigenpair and shift σ1: for k = 0,1,2, . . . until convergence do2: Solve vHT (σ)−1T (λk+1)xk = 0 for λk+1
or xHk T (λk+1)xk = 0 if T (λ) is Hermitian and λk+1 is real
3: Solve T (σ)dk = T (λk+1)xk for dk4: Set uk+1 = xk − dk , and normalize xk+1 = uk+1/vHuk+15: end for
If T (λ) is twice continuously differentiable, λ is a simple zero of det T (λ), andif x is an eigenvector normalized by vH x = 1, then residual inverse iterationconvergence for all σ sufficiently close to λ and it holds that
‖xk+1 − x‖‖xk − x‖
= O(|σ − λ|) and |λk+1 − λ| = (‖xk − x‖q)
where q = 2 if T (λ) is Hermitian, λ is real, and λk+1 solves xHk T (λk+1)xk = 0
in Step 3, and q = 1 otherwise.The domains of attraction of inverse iteration and residual inverse iteration isoften very small, and both methods are very sensitive with respect to inexactsolves of linear systems.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 15 / 61
Iterative projection methods
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, 2012 16 / 61
Iterative projection methods
Iterative projection methodsFor 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 wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.
Generalizations to nonlinear sparse eigenproblemsnonlinear 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, 2012 17 / 61
Iterative projection methods
Iterative projection methodsFor 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 wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.
Generalizations to nonlinear sparse eigenproblemsnonlinear 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, 2012 17 / 61
Iterative projection methods
Iterative projection methodsFor 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 wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.
Generalizations to nonlinear sparse eigenproblemsnonlinear 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, 2012 17 / 61
Iterative projection methods
Iterative projection methodsFor 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 wantedeigenvalues and eigenvectors are obtained from projections of theeigenproblem to subspaces of small dimension which are expanded in thecourse of the algorithm.
Generalizations to nonlinear sparse eigenproblemsnonlinear 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, 2012 17 / 61
Iterative projection methods
Iterative projection method
Require: Initial basis V with V HV = I; set m = 11: while m ≤ number of wanted eigenvalues do2: 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)u4: if ‖r‖ < ε then5: accept approximate eigenpair λm = µ, xm = u; increase m← m + 16: reduce search space V if necessary7: choose approximation (λm,u) to next eigenpair, and compute
r = T (λm)u8: end if9: expand search space V = [V , vnew]
10: update projected problem11: end while
Main tasksexpand search spacechoose eigenpair of projected problem (locking, purging)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 18 / 61
Iterative projection methods
Iterative projection method
Require: Initial basis V with V HV = I; set m = 11: while m ≤ number of wanted eigenvalues do2: 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)u4: if ‖r‖ < ε then5: accept approximate eigenpair λm = µ, xm = u; increase m← m + 16: reduce search space V if necessary7: choose approximation (λm,u) to next eigenpair, and compute
r = T (λm)u8: end if9: expand search space V = [V , vnew]
10: update projected problem11: end while
Main tasksexpand search spacechoose eigenpair of projected problem (locking, purging)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 18 / 61
Iterative projection methods
Expanding the subspace
Given subspace V ⊂ Cn . Expand V by a direction with high approximationpotential for the next wanted eigenvector.
Let θ be an eigenvalue of the projected problem
V HT (λ)Vy = 0
and x = Vy corresponding Ritz vector, then inverse iteration yieldssuitable candidate
v := T (θ)−1T ′(θ)x
BUT: In each step have to solve large linear system with varying matrixand in a truly large problem the vector v will not be accessible but only aninexact solution v := v + e of T (θ)v = T ′(θ)x , and the next iterate will be asolution of the projection of T (λ)x = 0 upon the expanded spaceV := spanV, v.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 19 / 61
Iterative projection methods
Expanding the subspace
Given subspace V ⊂ Cn . Expand V by a direction with high approximationpotential for the next wanted eigenvector.
Let θ be an eigenvalue of the projected problem
V HT (λ)Vy = 0
and x = Vy corresponding Ritz vector, then inverse iteration yieldssuitable candidate
v := T (θ)−1T ′(θ)x
BUT: In each step have to solve large linear system with varying matrixand in a truly large problem the vector v will not be accessible but only aninexact solution v := v + e of T (θ)v = T ′(θ)x , and the next iterate will be asolution of the projection of T (λ)x = 0 upon the expanded spaceV := spanV, v.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 19 / 61
Iterative projection methods
Expanding the subspace
Given subspace V ⊂ Cn . Expand V by a direction with high approximationpotential for the next wanted eigenvector.
Let θ be an eigenvalue of the projected problem
V HT (λ)Vy = 0
and x = Vy corresponding Ritz vector, then inverse iteration yieldssuitable candidate
v := T (θ)−1T ′(θ)x
BUT: In each step have to solve large linear system with varying matrixand in a truly large problem the vector v will not be accessible but only aninexact solution v := v + e of T (θ)v = T ′(θ)x , and the next iterate will be asolution of the projection of T (λ)x = 0 upon the expanded spaceV := spanV, v.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 19 / 61
Iterative projection methods
Expanding the subspace
Given subspace V ⊂ Cn . Expand V by a direction with high approximationpotential for the next wanted eigenvector.
Let θ be an eigenvalue of the projected problem
V HT (λ)Vy = 0
and x = Vy corresponding Ritz vector, then inverse iteration yieldssuitable candidate
v := T (θ)−1T ′(θ)x
BUT: In each step have to solve large linear system with varying matrixand in a truly large problem the vector v will not be accessible but only aninexact solution v := v + e of T (θ)v = T ′(θ)x , and the next iterate will be asolution of the projection of T (λ)x = 0 upon the expanded spaceV := spanV, v.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 19 / 61
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 weare looking for will be very close to the plane E := spanx , v.
We therefore neglect the influence of the orthogonal complement of x in V onthe next iterate and discuss the nearness of the planes E andE := spanx , v.
If the angle between these two planes is small, then the projection of T (λ)upon V should be similar to the one upon spanV, v, and the approximationproperties of inverse iteration should be maintained.
If this angle can become large, then it is not surprising that the convergenceproperties of inverse iteration are not reflected by the projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 20 / 61
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 weare looking for will be very close to the plane E := spanx , v.
We therefore neglect the influence of the orthogonal complement of x in V onthe next iterate and discuss the nearness of the planes E andE := spanx , v.
If the angle between these two planes is small, then the projection of T (λ)upon V should be similar to the one upon spanV, v, and the approximationproperties of inverse iteration should be maintained.
If this angle can become large, then it is not surprising that the convergenceproperties of inverse iteration are not reflected by the projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 20 / 61
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 weare looking for will be very close to the plane E := spanx , v.
We therefore neglect the influence of the orthogonal complement of x in V onthe next iterate and discuss the nearness of the planes E andE := spanx , v.
If the angle between these two planes is small, then the projection of T (λ)upon V should be similar to the one upon spanV, v, and the approximationproperties of inverse iteration should be maintained.
If this angle can become large, then it is not surprising that the convergenceproperties of inverse iteration are not reflected by the projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 20 / 61
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 weare looking for will be very close to the plane E := spanx , v.
We therefore neglect the influence of the orthogonal complement of x in V onthe next iterate and discuss the nearness of the planes E andE := spanx , v.
If the angle between these two planes is small, then the projection of T (λ)upon V should be similar to the one upon spanV, v, and the approximationproperties of inverse iteration should be maintained.
If this angle can become large, then it is not surprising that the convergenceproperties of inverse iteration are not reflected by the projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 20 / 61
Iterative projection methods
Theorem
Let φ0 = arccos(xT v) denote the angle between x and v , and the relativeerror of v by ε := ‖e‖.
Then the maximal possible acute angle between the planes E and E is
β(ε) =
arccos
√1− ε2/ sin2 φ0 if ε ≤ | sinφ0|π2 if ε ≥ | sinφ0|
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 21 / 61
Iterative projection methods
Theorem
Let φ0 = arccos(xT v) denote the angle between x and v , and the relativeerror of v by ε := ‖e‖.
Then the maximal possible acute angle between the planes E and E is
β(ε) =
arccos
√1− ε2/ sin2 φ0 if ε ≤ | sinφ0|π2 if ε ≥ | sinφ0|
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 21 / 61
Iterative projection methods
“Proof”
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 22 / 61
Iterative projection methods
Expansion by inexact inverse iteration
Obviously for every α ∈ R, α 6= 0 the plane E is also spanned by x and x +αv .
If E(α) is the plane which is spanned by x and a perturbed realizationx + αv + e of x + αv then by the same arguments as in the proof of theTheorem the maximum angle between E and E(α) is
γ(α, ε) =
arccos
√1− ε2/ sin2 φ(α) if ε ≤ | sinφ(α)|
π2 if ε ≥ | sinφ(α)|
where φ(α) denotes the angle between x and x + αv .
Since the mapping
φ 7→ arccos√
1− ε2/ sin2 φ
decreases monotonically the expansion of the search space by an inexactrealization of t := x + αv is most robust with respect to small perturbations, ifα is chosen such that x and x + αv are orthogonal
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 23 / 61
Iterative projection methods
Expansion by inexact inverse iteration
Obviously for every α ∈ R, α 6= 0 the plane E is also spanned by x and x +αv .
If E(α) is the plane which is spanned by x and a perturbed realizationx + αv + e of x + αv then by the same arguments as in the proof of theTheorem the maximum angle between E and E(α) is
γ(α, ε) =
arccos
√1− ε2/ sin2 φ(α) if ε ≤ | sinφ(α)|
π2 if ε ≥ | sinφ(α)|
where φ(α) denotes the angle between x and x + αv .
Since the mapping
φ 7→ arccos√
1− ε2/ sin2 φ
decreases monotonically the expansion of the search space by an inexactrealization of t := x + αv is most robust with respect to small perturbations, ifα is chosen such that x and x + αv are orthogonal
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 23 / 61
Iterative projection methods
Expansion by inexact inverse iteration
Obviously for every α ∈ R, α 6= 0 the plane E is also spanned by x and x +αv .
If E(α) is the plane which is spanned by x and a perturbed realizationx + αv + e of x + αv then by the same arguments as in the proof of theTheorem the maximum angle between E and E(α) is
γ(α, ε) =
arccos
√1− ε2/ sin2 φ(α) if ε ≤ | sinφ(α)|
π2 if ε ≥ | sinφ(α)|
where φ(α) denotes the angle between x and x + αv .
Since the mapping
φ 7→ arccos√
1− ε2/ sin2 φ
decreases monotonically the expansion of the search space by an inexactrealization of t := x + αv is most robust with respect to small perturbations, ifα is chosen such that x and x + αv are orthogonal
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 23 / 61
Iterative projection methods
Expansion by inexact inverse iteration ct.
t := x + αv is orthogonal to x iff
v = x − xHxxHT (θ)−1T ′(θ)x
T (θ)−1T (θ)x .. (∗)
which yields a maximum acute angle between E and E(α)
γ(α, ε) =
arccos
√1− ε2 if ε ≤ 1
π2 if ε ≥ 1
.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 24 / 61
Iterative projection methods
Expansion by inexact inverse iteration ct.
t := x + αv is orthogonal to x iff
v = x − xHxxHT (θ)−1T ′(θ)x
T (θ)−1T (θ)x .. (∗)
which yields a maximum acute angle between E and E(α)
γ(α, ε) =
arccos
√1− ε2 if ε ≤ 1
π2 if ε ≥ 1
.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 24 / 61
Iterative projection methods
Expansion by inexact inverse iteration
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 25 / 61
Iterative projection methods
Expansion by inexact inverse iteration ct.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 26 / 61
Iterative projection methods
Expansion by inexact inverse iteration ct.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 27 / 61
Iterative projection methods
Jacobi–Davidson method
The expansion
v = x − xHxxHT (θ)−1T ′(θ)x
T (θ)−1T (θ)x .. (∗)
of the current search space V is the solution of the equation
(I − T ′(θ)xxH
xHT ′(θ)x)T (θ)(I − xxH)t = −r , t⊥x
This is the so called correction equation of the Jacobi–Davidson method whichwas derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpenand van der Vorst (1996) for linear and polynomial eigenvalue problems.
Hence, the Jacobi–Davidson method is the most robust realization of anexpansion of a search space such that the direction of inverse iteration iscontained in the expanded space in the sense that it is least sensitive toinexact solves of linear systems T (θ)v = T ′(θ)x .
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 28 / 61
Iterative projection methods
Jacobi–Davidson method
The expansion
v = x − xHxxHT (θ)−1T ′(θ)x
T (θ)−1T (θ)x .. (∗)
of the current search space V is the solution of the equation
(I − T ′(θ)xxH
xHT ′(θ)x)T (θ)(I − xxH)t = −r , t⊥x
This is the so called correction equation of the Jacobi–Davidson method whichwas derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpenand van der Vorst (1996) for linear and polynomial eigenvalue problems.
Hence, the Jacobi–Davidson method is the most robust realization of anexpansion of a search space such that the direction of inverse iteration iscontained in the expanded space in the sense that it is least sensitive toinexact solves of linear systems T (θ)v = T ′(θ)x .
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 28 / 61
Iterative projection methods
Jacobi–Davidson method
The expansion
v = x − xHxxHT (θ)−1T ′(θ)x
T (θ)−1T (θ)x .. (∗)
of the current search space V is the solution of the equation
(I − T ′(θ)xxH
xHT ′(θ)x)T (θ)(I − xxH)t = −r , t⊥x
This is the so called correction equation of the Jacobi–Davidson method whichwas derived in T. Betcke & Voss (2004) generalizing the approach of Sleijpenand van der Vorst (1996) for linear and polynomial eigenvalue problems.
Hence, the Jacobi–Davidson method is the most robust realization of anexpansion of a search space such that the direction of inverse iteration iscontained in the expanded space in the sense that it is least sensitive toinexact solves of linear systems T (θ)v = T ′(θ)x .
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 28 / 61
Iterative projection methods
Nonlinear Jacobi-Davidson
1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ T (σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: 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)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ T (µ) if indicated
10: choose approximation (λm, u) to next eigenpair, and compute r = T (λm)u11: end if12: solve approximately correction equation(
I − T ′(µ)uuH
uHT ′(µ)u
)T (µ)
(I − uuH
uHu
)t = −r , t ⊥ u
13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 29 / 61
Iterative projection methods
Comments
1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ T (σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: 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)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ T (µ) if indicated
10: choose approximation (λm, u) to next eigenpair, and compute r = T (λm)u11: end if12: solve approximately correction equation(
I − T ′(µ)uuH
uHT ′(µ)u
)T (µ)
(I − uuH
uHu
)t = −r , t ⊥ u
13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 30 / 61
Iterative projection methods
Comments 1:
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 ineigenvalues close to the parameter σ ∈ D:
choose initial vector at random, and execute a few Arnoldi steps for the lineareigenproblem T (σ)u = θu or T (σ)u = θT ′(σ)u, and choose the eigenvectorcorresponding to the smallest eigenvalue in modulus or a small number ofSchur vectors as initial basis of the search space.
Starting with a random vector without this preprocessing usually will yield avalue µ in step 3 which is far away from σ and will avert convergence.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 31 / 61
Iterative projection methods
Comments 1:
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 ineigenvalues close to the parameter σ ∈ D:
choose initial vector at random, and execute a few Arnoldi steps for the lineareigenproblem T (σ)u = θu or T (σ)u = θT ′(σ)u, and choose the eigenvectorcorresponding to the smallest eigenvalue in modulus or a small number ofSchur vectors as initial basis of the search space.
Starting with a random vector without this preprocessing usually will yield avalue µ in step 3 which is far away from σ and will avert convergence.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 31 / 61
Iterative projection methods
Comments 1:
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 ineigenvalues close to the parameter σ ∈ D:
choose initial vector at random, and execute a few Arnoldi steps for the lineareigenproblem T (σ)u = θu or T (σ)u = θT ′(σ)u, and choose the eigenvectorcorresponding to the smallest eigenvalue in modulus or a small number ofSchur vectors as initial basis of the search space.
Starting with a random vector without this preprocessing usually will yield avalue µ in step 3 which is far away from σ and will avert convergence.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 31 / 61
Iterative projection methods
Comments
1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ T (σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: 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)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ T (µ) if indicated
10: choose approximation (λm, u) to next eigenpair, and compute r = T (λm)u11: end if12: solve approximately correction equation(
I − T ′(µ)uuH
uHT ′(µ)u
)T (µ)
(I − uuH
uHu
)t = −r , t ⊥ u
13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 32 / 61
Iterative projection methods
Comments 4:
Since the dimension of the projected problem is small it can be solved by anydense solver like
linearization if T (λ) is polynomialmethods based on characteristic function det T (λ) = 0inverse iterationresidual inverse iterationsuccessive linear problems
Notice that symmetry properties that the original problem may have areinherited by the projected problem, and one can take advantage of theseproperties when solving the projected problem (safeguarded iteration,structure preserving methods).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 33 / 61
Iterative projection methods
Comments 4:
Since the dimension of the projected problem is small it can be solved by anydense solver like
linearization if T (λ) is polynomialmethods based on characteristic function det T (λ) = 0inverse iterationresidual inverse iterationsuccessive linear problems
Notice that symmetry properties that the original problem may have areinherited by the projected problem, and one can take advantage of theseproperties when solving the projected problem (safeguarded iteration,structure preserving methods).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 33 / 61
Iterative projection methods
Comments
1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ T (σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: 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)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ T (µ) if indicated
10: choose approximation (λm, u) to next eigenpair, and compute r = T (λm)u11: end if12: solve approximately correction equation(
I − T ′(µ)uuH
uHT ′(µ)u
)T (µ)
(I − uuH
uHu
)t = −r , t ⊥ u
13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 34 / 61
Iterative projection methods
Comments 8:
As the subspaces expand in the course of the algorithm the increasingstorage or the computational cost for solving the projected eigenvalueproblems may make it necessary to restart the algorithm and purge some ofthe basis vectors.
Since a restart destroys information on the eigenvectors and particularly onthe one the method is just aiming at, we restart only if an eigenvector has justconverged.
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, 2012 35 / 61
Iterative projection methods
Comments 8:
As the subspaces expand in the course of the algorithm the increasingstorage or the computational cost for solving the projected eigenvalueproblems may make it necessary to restart the algorithm and purge some ofthe basis vectors.
Since a restart destroys information on the eigenvectors and particularly onthe one the method is just aiming at, we restart only if an eigenvector has justconverged.
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, 2012 35 / 61
Iterative projection methods
Comments 8:
As the subspaces expand in the course of the algorithm the increasingstorage or the computational cost for solving the projected eigenvalueproblems may make it necessary to restart the algorithm and purge some ofthe basis vectors.
Since a restart destroys information on the eigenvectors and particularly onthe one the method is just aiming at, we restart only if an eigenvector has justconverged.
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, 2012 35 / 61
Iterative projection methods
Restarts
0 20 40 60 80 100 1200
50
100
150
200
250
300
350
400
450
500
iteration
time
[s]
CPU time
nonlinear eigensolver
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 36 / 61
Iterative projection methods
Restarts
0 20 40 60 80 100 1200
50
100
150
200
250
iteration
time
[s]
CPU time
LU update
nonlin. solver
restart
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 37 / 61
Iterative projection methods
Comments
1: Start with orthonormal basis V ; set m = 12: determine preconditioner M ≈ T (σ)−1; σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: 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)u6: if ‖r‖ < ε then7: accept approximate eigenpair λm = µ, xm = u; increase m← m + 18: reduce search space V if necessary9: choose new preconditioner M ≈ T (µ) if indicated
10: choose approximation (λm, u) to next eigenpair, and compute r = T (λm)u11: end if12: solve approximately correction equation(
I − T ′(µ)uuH
uHT ′(µ)u
)T (µ)
(I − uuH
uHu
)t = −r , t ⊥ u
13: t = t − VV T t ,v = t/‖t‖, reorthogonalize if necessary14: expand search space V = [V , v ]15: update projected problem16: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 38 / 61
Iterative projection methods
Comments 12
The correction equation is solved approximately by a few steps of an iterativesolver (GMRES or BiCGStab).
The operator T (σ) is restricted to map the subspace x⊥ into itself. Hence, ifM ≈ T (σ) is a preconditioner of T (σ), σ ≈ µ, then a preconditioner for aniterativ solver of the correction equation should be modified correspondingly to
M := (I − T ′(µ)xxH
xHT (µ)x)M(I − xxH
xHx).
Taking into account the projectors in the preconditioner, i.e. using M instead ofM, raises the cost of the preconditioned Krylov solver only slightly (cf.Sleijpen, van der Vorst). Only one additional linear solve with system matrix Mis required.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 39 / 61
Iterative projection methods
Comments 12
The correction equation is solved approximately by a few steps of an iterativesolver (GMRES or BiCGStab).
The operator T (σ) is restricted to map the subspace x⊥ into itself. Hence, ifM ≈ T (σ) is a preconditioner of T (σ), σ ≈ µ, then a preconditioner for aniterativ solver of the correction equation should be modified correspondingly to
M := (I − T ′(µ)xxH
xHT (µ)x)M(I − xxH
xHx).
Taking into account the projectors in the preconditioner, i.e. using M instead ofM, raises the cost of the preconditioned Krylov solver only slightly (cf.Sleijpen, van der Vorst). Only one additional linear solve with system matrix Mis required.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 39 / 61
Iterative projection methods
Comments 12
The correction equation is solved approximately by a few steps of an iterativesolver (GMRES or BiCGStab).
The operator T (σ) is restricted to map the subspace x⊥ into itself. Hence, ifM ≈ T (σ) is a preconditioner of T (σ), σ ≈ µ, then a preconditioner for aniterativ solver of the correction equation should be modified correspondingly to
M := (I − T ′(µ)xxH
xHT (µ)x)M(I − xxH
xHx).
Taking into account the projectors in the preconditioner, i.e. using M instead ofM, raises the cost of the preconditioned Krylov solver only slightly (cf.Sleijpen, van der Vorst). Only one additional linear solve with system matrix Mis required.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 39 / 61
Iterative projection methods
Non-iterative approximate solution
Hwang, Lin, Wang, Wang (2004) considered a non-iterative solution (whichwas already contained in the paper of Sleijpen et al. (1996), but was notconsidered in subsequent papers):Solve (
I − T ′(θ)uuH
uHT ′(θ)u
)T (θ)
(I − uuH
uHu
)t = −r , t ⊥ u
approximately
M
by computing
t = M−1r + αM−1T ′(θ)u with α :=uHM−1r
uHM−1T ′(θ)u
where M is a preconditioner of T (θ).
Method combines preconditioned Arnoldi method (M−1r ) and simplifiedinverse iteration (M−1T ′(θ)u)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 40 / 61
Iterative projection methods
Non-iterative approximate solution
Hwang, Lin, Wang, Wang (2004) considered a non-iterative solution (whichwas already contained in the paper of Sleijpen et al. (1996), but was notconsidered in subsequent papers):Solve (
I − T ′(θ)uuH
uHT ′(θ)u
)T (θ)
(I − uuH
uHu
)t = −r , t ⊥ u
approximately
M
by computing
t = M−1r + αM−1T ′(θ)u with α :=uHM−1r
uHM−1T ′(θ)u
where M is a preconditioner of T (θ).
Method combines preconditioned Arnoldi method (M−1r ) and simplifiedinverse iteration (M−1T ′(θ)u)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 40 / 61
Iterative projection methods
Non-iterative approximate solution
Hwang, Lin, Wang, Wang (2004) considered a non-iterative solution (whichwas already contained in the paper of Sleijpen et al. (1996), but was notconsidered in subsequent papers):Solve (
I − T ′(θ)uuH
uHT ′(θ)u
)T (θ)
(I − uuH
uHu
)t = −r , t ⊥ u
approximately
M
by computing
t = M−1r + αM−1T ′(θ)u with α :=uHM−1r
uHM−1T ′(θ)u
where M is a preconditioner of T (θ).
Method combines preconditioned Arnoldi method (M−1r ) and simplifiedinverse iteration (M−1T ′(θ)u)
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 40 / 61
Iterative projection methods
Alternative expansion: Residual inverse It.
1: start with an approximation x1 to an eigenvector of T (λ)x = 02: for ` = 1,2, . . . until convergence do3: solve xH
` T (µ`+1)x` = 0 for µ`+14: compute the residual r` = T (µ`+1)x`5: solve T (σ)d` = r`6: set x`+1 = x` − d`, x`+1 = x`+1/‖x`+1‖7: end for
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 41 / 61
Iterative projection methods
Arnoldi methodThis suggests: If θ is an eigenvalue of the projected problem V HT (λ)Vy = 0and x = Vy is a corresponding Ritz vector, then expand V by new direction
vRI = x − T (σ)−1T (θ)x
Since in iterative projection methods the new search direction isorthogonalized against the basis of the current search space and since x isalready contained in V, the expansion is chosen to be
vA := T (σ)−1T (θ)x .
The choice of vA is more robust than vRI since
|xHvRI |/‖vRI‖ → 1 and xHvA/‖vA‖ → 0
For the linear problem T (λ) = λI − A this is exactly the Cayley transformationor shifted-and-inverted Arnoldi method. Therefore the resulting iterativeprojection method is called nonlinear Arnoldi method although no Krylovspace is constructed and no Arnoldi recursion holds
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 42 / 61
Iterative projection methods
Arnoldi methodThis suggests: If θ is an eigenvalue of the projected problem V HT (λ)Vy = 0and x = Vy is a corresponding Ritz vector, then expand V by new direction
vRI = x − T (σ)−1T (θ)x
Since in iterative projection methods the new search direction isorthogonalized against the basis of the current search space and since x isalready contained in V, the expansion is chosen to be
vA := T (σ)−1T (θ)x .
The choice of vA is more robust than vRI since
|xHvRI |/‖vRI‖ → 1 and xHvA/‖vA‖ → 0
For the linear problem T (λ) = λI − A this is exactly the Cayley transformationor shifted-and-inverted Arnoldi method. Therefore the resulting iterativeprojection method is called nonlinear Arnoldi method although no Krylovspace is constructed and no Arnoldi recursion holds
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 42 / 61
Iterative projection methods
Arnoldi methodThis suggests: If θ is an eigenvalue of the projected problem V HT (λ)Vy = 0and x = Vy is a corresponding Ritz vector, then expand V by new direction
vRI = x − T (σ)−1T (θ)x
Since in iterative projection methods the new search direction isorthogonalized against the basis of the current search space and since x isalready contained in V, the expansion is chosen to be
vA := T (σ)−1T (θ)x .
The choice of vA is more robust than vRI since
|xHvRI |/‖vRI‖ → 1 and xHvA/‖vA‖ → 0
For the linear problem T (λ) = λI − A this is exactly the Cayley transformationor shifted-and-inverted Arnoldi method. Therefore the resulting iterativeprojection method is called nonlinear Arnoldi method although no Krylovspace is constructed and no Arnoldi recursion holds
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 42 / 61
Iterative projection methods
Arnoldi methodThis suggests: If θ is an eigenvalue of the projected problem V HT (λ)Vy = 0and x = Vy is a corresponding Ritz vector, then expand V by new direction
vRI = x − T (σ)−1T (θ)x
Since in iterative projection methods the new search direction isorthogonalized against the basis of the current search space and since x isalready contained in V, the expansion is chosen to be
vA := T (σ)−1T (θ)x .
The choice of vA is more robust than vRI since
|xHvRI |/‖vRI‖ → 1 and xHvA/‖vA‖ → 0
For the linear problem T (λ) = λI − A this is exactly the Cayley transformationor shifted-and-inverted Arnoldi method. Therefore the resulting iterativeprojection method is called nonlinear Arnoldi method although no Krylovspace is constructed and no Arnoldi recursion holds
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 42 / 61
Iterative projection methods
Nonlinear Arnoldi Method
1: start with initial basis V , V HV = I; set k = m = 12: determine preconditioner M ≈ T (σ)−1, σ close to first wanted eigenvalue3: while m ≤ number of wanted eigenvalues do4: solve V HT (µ)Vy = 0 for (µ, y) and set u = Vy , rk = T (µ)u5: if ‖rk‖/‖u‖ < ε then6: Accept eigenpair λm = µ, xm = u,7: if m == number of wanted eigenvalues then STOP end if8: m = m + 19: if ‖rk−1‖/‖rk‖ > tol then
10: choose new pole σ, determine preconditioner M ≈ T (σ)−1
11: end if12: restart if necessary13: Choose approximations µ and u to next eigenvalue and eigenvector14: determine r = T (µ)u and set k = 015: end if16: v = Mr , k = k + 117: v = v − VV Hv ,v = v/‖v‖, V = [V , v ] and reorthogonalize if necessary18: end while
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 43 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Iterative projection methods
Locking of converged eigenvectors
A major problem with iterative projection methods for nonlinear eigenproblemswhen approximating more than one eigenvalue is to inhibit the method fromconverging to an eigenpair which was detected already previously.
linear problems: (incomplete) Schur decompositionquadratic problems: Meerbergen (2001) based on linearization andSchur form of linearized problem (lock 2 vectors in each step)cubic problem: Hwang, Lin, Liu, Wang (2005) based on linearization andknowledge of all eigenvalues of the linearized problemApproach of Hwang et al. can be generalized directly to general problemsif all eigenvalues of projected problems are determinedfor Hermitian problems one can often take advantage of a variationalcharacterization of eigenvaluesA new method for locking known eigenpairs based on invariant pairs wasintroduced by Kressner (2009) for Newton’s method. However, it is not yetclear how to combine it with iterative projection method.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 44 / 61
Variational characterization of eigenvalues
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, 2012 45 / 61
Variational characterization of eigenvalues
Nonlinear minmax theory
Let T (λ) ∈ Cn×n, T (λ) = T (λ)H , λ ∈ J ⊂ R an open interval, and assume thatthe entries of T depend continuously on λ.
Assume that for fixed x ∈ Cn, x 6= 0 the real equation
f (λ, x) := xHT (λ)x = 0
has at most one solution λ =: p(x) in J.
Then equation f (λ, x) = 0 implicitly defines a functional p on some subset Dof Cn which we call the Rayleigh functional.
Let(λ− p(x))f (λ, x) > 0 for every λ 6= p(x) and every x ∈ D.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 46 / 61
Variational characterization of eigenvalues
Nonlinear minmax theory
Let T (λ) ∈ Cn×n, T (λ) = T (λ)H , λ ∈ J ⊂ R an open interval, and assume thatthe entries of T depend continuously on λ.
Assume that for fixed x ∈ Cn, x 6= 0 the real equation
f (λ, x) := xHT (λ)x = 0
has at most one solution λ =: p(x) in J.
Then equation f (λ, x) = 0 implicitly defines a functional p on some subset Dof Cn which we call the Rayleigh functional.
Let(λ− p(x))f (λ, x) > 0 for every λ 6= p(x) and every x ∈ D.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 46 / 61
Variational characterization of eigenvalues
Nonlinear minmax theory
Let T (λ) ∈ Cn×n, T (λ) = T (λ)H , λ ∈ J ⊂ R an open interval, and assume thatthe entries of T depend continuously on λ.
Assume that for fixed x ∈ Cn, x 6= 0 the real equation
f (λ, x) := xHT (λ)x = 0
has at most one solution λ =: p(x) in J.
Then equation f (λ, x) = 0 implicitly defines a functional p on some subset Dof Cn which we call the Rayleigh functional.
Let(λ− p(x))f (λ, x) > 0 for every λ 6= p(x) and every x ∈ D.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 46 / 61
Variational characterization of eigenvalues
Nonlinear minmax theory
Let T (λ) ∈ Cn×n, T (λ) = T (λ)H , λ ∈ J ⊂ R an open interval, and assume thatthe entries of T depend continuously on λ.
Assume that for fixed x ∈ Cn, x 6= 0 the real equation
f (λ, x) := xHT (λ)x = 0
has at most one solution λ =: p(x) in J.
Then equation f (λ, x) = 0 implicitly defines a functional p on some subset Dof Cn which we call the Rayleigh functional.
Let(λ− p(x))f (λ, x) > 0 for every λ 6= p(x) and every x ∈ D.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 46 / 61
Variational characterization of eigenvalues
Overdamped problems
If the Rayleigh functional p is defined on the entire space H \ 0 then theeigenproblem T (λ)x = 0 is called overdamped.
This notation is motivated by the finite dimensional quadratic eigenvalueproblem
T (λ)x = λ2Mx + λαCx + Kx = 0
where M, C and K are symmetric and positive definite matrices.
α = 0 all eigenvalues on imaginary axis
increase α eigenvalues go into left half plane as conjugate complex pairs
increase α complex pairs reach real axis, run in opposite directions
increase α all eigenvalues on the negative real axis
increase α all eigenvalues going to the left are smallerthan all eigenvalues going to the rightsystem is overdamped
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 47 / 61
Variational characterization of eigenvalues
Overdamped problems
If the Rayleigh functional p is defined on the entire space H \ 0 then theeigenproblem T (λ)x = 0 is called overdamped.
This notation is motivated by the finite dimensional quadratic eigenvalueproblem
T (λ)x = λ2Mx + λαCx + Kx = 0
where M, C and K are symmetric and positive definite matrices.
α = 0 all eigenvalues on imaginary axis
increase α eigenvalues go into left half plane as conjugate complex pairs
increase α complex pairs reach real axis, run in opposite directions
increase α all eigenvalues on the negative real axis
increase α all eigenvalues going to the left are smallerthan all eigenvalues going to the rightsystem is overdamped
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 47 / 61
Variational characterization of eigenvalues
Overdamped problems
If the Rayleigh functional p is defined on the entire space H \ 0 then theeigenproblem T (λ)x = 0 is called overdamped.
This notation is motivated by the finite dimensional quadratic eigenvalueproblem
T (λ)x = λ2Mx + λαCx + Kx = 0
where M, C and K are symmetric and positive definite matrices.
α = 0 all eigenvalues on imaginary axis
increase α eigenvalues go into left half plane as conjugate complex pairs
increase α complex pairs reach real axis, run in opposite directions
increase α all eigenvalues on the negative real axis
increase α all eigenvalues going to the left are smallerthan all eigenvalues going to the rightsystem is overdamped
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 47 / 61
Variational characterization of eigenvalues
Quadratic overdamped problems
For quadratic overdamped systems the two solutions
p±(x) =12
(−α〈Cx , x〉 ±
√α2〈Cx , x〉2 − 4〈Mx , x〉〈Kx , x〉
)/〈Mx , x〉.
of the quadratic equation
〈T (λ)x , x〉 = λ2〈Mx , x〉+ λα〈Cx , x〉+ 〈Kx , x〉 = 0 (1)
are real, and they satisfy supx 6=0 p−(x) < infx 6=<0 p+(x).
Hence, equation (1) defines two Rayleigh functionals p− and p+
corresponding to the intervals
J− := (−∞, infx 6=0
p+(x)) and J+ := (supx 6=0
p−(x),∞).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 48 / 61
Variational characterization of eigenvalues
Quadratic overdamped problems
For quadratic overdamped systems the two solutions
p±(x) =12
(−α〈Cx , x〉 ±
√α2〈Cx , x〉2 − 4〈Mx , x〉〈Kx , x〉
)/〈Mx , x〉.
of the quadratic equation
〈T (λ)x , x〉 = λ2〈Mx , x〉+ λα〈Cx , x〉+ 〈Kx , x〉 = 0 (1)
are real, and they satisfy supx 6=0 p−(x) < infx 6=<0 p+(x).
Hence, equation (1) defines two Rayleigh functionals p− and p+
corresponding to the intervals
J− := (−∞, infx 6=0
p+(x)) and J+ := (supx 6=0
p−(x),∞).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 48 / 61
Variational characterization of eigenvalues
Minmax principle
Poincare’s maxmin characterization was first generalized by Duffin (1955) tooverdamped quadratic eigenproblems of finite dimension, and for moregeneral overdamped problems of finite dimension it was proved by Rogers(1964).
Infinite dimensional eigenvalue problems were studied by Turner (1967),Langer (1968), and Weinberger (1969) who proved generalizations of both,the maxmin characterization of Poincare and of the minmax characterizationof Courant, Fischer and Weyl for quadratic (and by Turner (1968) forpolynomial) overdamped problems.
The corresponding generalizations for general overdamped problems ofinfinite dimension were derived by Hadeler (1968). Similar results (weakeningthe compactness or smoothness requirements) are contained in Rogers(1968), Werner (1971), Abramov (1973), Hadeler (1975), Markus (1985),Maksudov & Gasanov (1992), and Hasanov (2002).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 49 / 61
Variational characterization of eigenvalues
Minmax principle
Poincare’s maxmin characterization was first generalized by Duffin (1955) tooverdamped quadratic eigenproblems of finite dimension, and for moregeneral overdamped problems of finite dimension it was proved by Rogers(1964).
Infinite dimensional eigenvalue problems were studied by Turner (1967),Langer (1968), and Weinberger (1969) who proved generalizations of both,the maxmin characterization of Poincare and of the minmax characterizationof Courant, Fischer and Weyl for quadratic (and by Turner (1968) forpolynomial) overdamped problems.
The corresponding generalizations for general overdamped problems ofinfinite dimension were derived by Hadeler (1968). Similar results (weakeningthe compactness or smoothness requirements) are contained in Rogers(1968), Werner (1971), Abramov (1973), Hadeler (1975), Markus (1985),Maksudov & Gasanov (1992), and Hasanov (2002).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 49 / 61
Variational characterization of eigenvalues
Minmax principle
Poincare’s maxmin characterization was first generalized by Duffin (1955) tooverdamped quadratic eigenproblems of finite dimension, and for moregeneral overdamped problems of finite dimension it was proved by Rogers(1964).
Infinite dimensional eigenvalue problems were studied by Turner (1967),Langer (1968), and Weinberger (1969) who proved generalizations of both,the maxmin characterization of Poincare and of the minmax characterizationof Courant, Fischer and Weyl for quadratic (and by Turner (1968) forpolynomial) overdamped problems.
The corresponding generalizations for general overdamped problems ofinfinite dimension were derived by Hadeler (1968). Similar results (weakeningthe compactness or smoothness requirements) are contained in Rogers(1968), Werner (1971), Abramov (1973), Hadeler (1975), Markus (1985),Maksudov & Gasanov (1992), and Hasanov (2002).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 49 / 61
Variational characterization of eigenvalues
Nonoverdamped problems
For nonoverdamped eigenproblems the natural ordering to call the smallesteigenvalue the first one, the second smallest the second one, etc., is notappropriate.
This is obvious if we make a linear eigenvalue
T (λ)x := (λI − A)x = 0
nonlinear by restricting it to an interval J which does not contain the smallesteigenvalue of A.
Then all conditions are satisfied, p is the restriction of the Rayleigh quotientRA to
D := x 6= 0 : RA(x) ∈ J,
and infx∈D p(x) will in general not be an eigenvalue.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 50 / 61
Variational characterization of eigenvalues
Nonoverdamped problems
For nonoverdamped eigenproblems the natural ordering to call the smallesteigenvalue the first one, the second smallest the second one, etc., is notappropriate.
This is obvious if we make a linear eigenvalue
T (λ)x := (λI − A)x = 0
nonlinear by restricting it to an interval J which does not contain the smallesteigenvalue of A.
Then all conditions are satisfied, p is the restriction of the Rayleigh quotientRA to
D := x 6= 0 : RA(x) ∈ J,
and infx∈D p(x) will in general not be an eigenvalue.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 50 / 61
Variational characterization of eigenvalues
Nonoverdamped problems
For nonoverdamped eigenproblems the natural ordering to call the smallesteigenvalue the first one, the second smallest the second one, etc., is notappropriate.
This is obvious if we make a linear eigenvalue
T (λ)x := (λI − A)x = 0
nonlinear by restricting it to an interval J which does not contain the smallesteigenvalue of A.
Then all conditions are satisfied, p is the restriction of the Rayleigh quotientRA to
D := x 6= 0 : RA(x) ∈ J,
and infx∈D p(x) will in general not be an eigenvalue.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 50 / 61
Variational characterization of eigenvalues
Enumeration of eigenvalues
If λ ∈ J is an eigenvalue of T (·) then µ = 0 is an eigenvalue of the linearproblem T (λ)y = µy , and therefore there exists ` ∈ N such that
0 = maxV∈H`
minv∈V\0
vHT (λ)v‖v‖2
where H` denotes the set of all `–dimensional subspaces of Cn.
In this case λ is called an `-th eigenvalue of T (·).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 51 / 61
Variational characterization of eigenvalues
Enumeration of eigenvalues
If λ ∈ J is an eigenvalue of T (·) then µ = 0 is an eigenvalue of the linearproblem T (λ)y = µy , and therefore there exists ` ∈ N such that
0 = maxV∈H`
minv∈V\0
vHT (λ)v‖v‖2
where H` denotes the set of all `–dimensional subspaces of Cn.
In this case λ is called an `-th eigenvalue of T (·).
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 51 / 61
Variational characterization of eigenvalues
Minmax characterization (V., Werner 1982, V. 2009)Under the conditions given above it holds:
(i) For every ` ∈ N there is at most one `-th eigenvalue of T (·) which can becharacterized by
λ` = minV∈H`, V∩D 6=∅
supv∈V∩D
p(v). (∗)
The set of eigenvalues of T (·) in J is at most countable.
(ii) Ifλ` := inf
V∈H`, V∩D 6=∅sup
v∈V∩Dp(v) ∈ J
for some ` ∈ N then λ` is the `-th eigenvalue of T (·) in J, and (*) holds.
(iii) λ is an `-th eigenvalue if and only if µ = 0 is the ` largest eigenvalue ofthe linear eigenproblem T (λ)x = µx .
(iv) The minimum in (*) is attained for the invariant subspace of T (λ`)corresponding to its ` largest eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 52 / 61
Variational characterization of eigenvalues
Minmax characterization (V., Werner 1982, V. 2009)Under the conditions given above it holds:
(i) For every ` ∈ N there is at most one `-th eigenvalue of T (·) which can becharacterized by
λ` = minV∈H`, V∩D 6=∅
supv∈V∩D
p(v). (∗)
The set of eigenvalues of T (·) in J is at most countable.
(ii) Ifλ` := inf
V∈H`, V∩D 6=∅sup
v∈V∩Dp(v) ∈ J
for some ` ∈ N then λ` is the `-th eigenvalue of T (·) in J, and (*) holds.
(iii) λ is an `-th eigenvalue if and only if µ = 0 is the ` largest eigenvalue ofthe linear eigenproblem T (λ)x = µx .
(iv) The minimum in (*) is attained for the invariant subspace of T (λ`)corresponding to its ` largest eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 52 / 61
Variational characterization of eigenvalues
Minmax characterization (V., Werner 1982, V. 2009)Under the conditions given above it holds:
(i) For every ` ∈ N there is at most one `-th eigenvalue of T (·) which can becharacterized by
λ` = minV∈H`, V∩D 6=∅
supv∈V∩D
p(v). (∗)
The set of eigenvalues of T (·) in J is at most countable.
(ii) Ifλ` := inf
V∈H`, V∩D 6=∅sup
v∈V∩Dp(v) ∈ J
for some ` ∈ N then λ` is the `-th eigenvalue of T (·) in J, and (*) holds.
(iii) λ is an `-th eigenvalue if and only if µ = 0 is the ` largest eigenvalue ofthe linear eigenproblem T (λ)x = µx .
(iv) The minimum in (*) is attained for the invariant subspace of T (λ`)corresponding to its ` largest eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 52 / 61
Variational characterization of eigenvalues
Minmax characterization (V., Werner 1982, V. 2009)Under the conditions given above it holds:
(i) For every ` ∈ N there is at most one `-th eigenvalue of T (·) which can becharacterized by
λ` = minV∈H`, V∩D 6=∅
supv∈V∩D
p(v). (∗)
The set of eigenvalues of T (·) in J is at most countable.
(ii) Ifλ` := inf
V∈H`, V∩D 6=∅sup
v∈V∩Dp(v) ∈ J
for some ` ∈ N then λ` is the `-th eigenvalue of T (·) in J, and (*) holds.
(iii) λ is an `-th eigenvalue if and only if µ = 0 is the ` largest eigenvalue ofthe linear eigenproblem T (λ)x = µx .
(iv) The minimum in (*) is attained for the invariant subspace of T (λ`)corresponding to its ` largest eigenvalues.
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Variational characterization of eigenvalues
Safeguarded iteration
The minimum inλ` = min
V∈H`V∩D 6=∅
supv∈V∩D
p(v)
is attained by the invariant subspace of T (λ`) corresponding to the ` largesteigenvalues, and the maximum by every eigenvector corresponding to theeigenvalue 0. This suggests
Safeguarded iteration1: Start with an approximation µ1 to the `-th eigenvalue of T (λ)x = 02: for k = 1,2, . . . until convergence do3: determine eigenvector u corresponding to the `-largest eigenvalue of
T (µk )4: evaluate µk+1 = p(u)5: end for
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Variational characterization of eigenvalues
Safeguarded iteration
The minimum inλ` = min
V∈H`V∩D 6=∅
supv∈V∩D
p(v)
is attained by the invariant subspace of T (λ`) corresponding to the ` largesteigenvalues, and the maximum by every eigenvector corresponding to theeigenvalue 0. This suggests
Safeguarded iteration1: Start with an approximation µ1 to the `-th eigenvalue of T (λ)x = 02: for k = 1,2, . . . until convergence do3: determine eigenvector u corresponding to the `-largest eigenvalue of
T (µk )4: evaluate µk+1 = p(u)5: end for
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 53 / 61
Variational characterization of eigenvalues
Convergence of safeguarded iteration
For ` = 1 the safeguarded iteration converges globally to λ1
If λ` is a simple eigenvalue then the (local) convergence is quadraticIf T ′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof
T (σk )x = µT ′(σk )x
corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.A variant exists which is globally convergent also for higher eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 54 / 61
Variational characterization of eigenvalues
Convergence of safeguarded iteration
For ` = 1 the safeguarded iteration converges globally to λ1
If λ` is a simple eigenvalue then the (local) convergence is quadraticIf T ′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof
T (σk )x = µT ′(σk )x
corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.A variant exists which is globally convergent also for higher eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 54 / 61
Variational characterization of eigenvalues
Convergence of safeguarded iteration
For ` = 1 the safeguarded iteration converges globally to λ1
If λ` is a simple eigenvalue then the (local) convergence is quadraticIf T ′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof
T (σk )x = µT ′(σk )x
corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.A variant exists which is globally convergent also for higher eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 54 / 61
Variational characterization of eigenvalues
Convergence of safeguarded iteration
For ` = 1 the safeguarded iteration converges globally to λ1
If λ` is a simple eigenvalue then the (local) convergence is quadraticIf T ′(λ) is positive definite and xk in Step 3 is replaced by an eigenvectorof
T (σk )x = µT ′(σk )x
corresponding to the `-th largest eigenvalue, then the convergence iseven cubic.A variant exists which is globally convergent also for higher eigenvalues.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 54 / 61
Numerical example
Outline
1 Introduction
2 Numerical methods for dense problems
3 Iterative projection methods
4 Variational characterization of eigenvalues
5 Numerical example
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Numerical example
Numerical example
pyramidal quantum dot: baselength: 12.4 nm, height: 6.2 nmcubic matrix: 24.8× 24.8× 18.6 nm3
Parameters (Hwang, Lin, Wang, Wang 2004)P1 = 0.8503, g1 = 0.42, δ1 = 0.48, V1 = 0.0P2 = 0.8878, g2 = 1.52, δ2 = 0.34, V1 = 0.7
Discretization by FEM or FVM yields rational eigenproblem
T (λ)x = λMx − 1m1(λ)
A1x − 1m2(λ)
A2x − Bx = 0
where T (λ) is symmetric and satisfies conditions of minmax characterizationfor λ ≥ 0
All timings for MATLAB 7.0.4 on AMD Opteron Processor 248× 860 64 with2.2 GHz and 4 GB RAM
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Numerical example
Numerical example
pyramidal quantum dot: baselength: 12.4 nm, height: 6.2 nmcubic matrix: 24.8× 24.8× 18.6 nm3
Parameters (Hwang, Lin, Wang, Wang 2004)P1 = 0.8503, g1 = 0.42, δ1 = 0.48, V1 = 0.0P2 = 0.8878, g2 = 1.52, δ2 = 0.34, V1 = 0.7
Discretization by FEM or FVM yields rational eigenproblem
T (λ)x = λMx − 1m1(λ)
A1x − 1m2(λ)
A2x − Bx = 0
where T (λ) is symmetric and satisfies conditions of minmax characterizationfor λ ≥ 0
All timings for MATLAB 7.0.4 on AMD Opteron Processor 248× 860 64 with2.2 GHz and 4 GB RAM
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 56 / 61
Numerical example
Numerical example
pyramidal quantum dot: baselength: 12.4 nm, height: 6.2 nmcubic matrix: 24.8× 24.8× 18.6 nm3
Parameters (Hwang, Lin, Wang, Wang 2004)P1 = 0.8503, g1 = 0.42, δ1 = 0.48, V1 = 0.0P2 = 0.8878, g2 = 1.52, δ2 = 0.34, V1 = 0.7
Discretization by FEM or FVM yields rational eigenproblem
T (λ)x = λMx − 1m1(λ)
A1x − 1m2(λ)
A2x − Bx = 0
where T (λ) is symmetric and satisfies conditions of minmax characterizationfor λ ≥ 0
All timings for MATLAB 7.0.4 on AMD Opteron Processor 248× 860 64 with2.2 GHz and 4 GB RAM
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 56 / 61
Numerical example
Numerical example
pyramidal quantum dot: baselength: 12.4 nm, height: 6.2 nmcubic matrix: 24.8× 24.8× 18.6 nm3
Parameters (Hwang, Lin, Wang, Wang 2004)P1 = 0.8503, g1 = 0.42, δ1 = 0.48, V1 = 0.0P2 = 0.8878, g2 = 1.52, δ2 = 0.34, V1 = 0.7
Discretization by FEM or FVM yields rational eigenproblem
T (λ)x = λMx − 1m1(λ)
A1x − 1m2(λ)
A2x − Bx = 0
where T (λ) is symmetric and satisfies conditions of minmax characterizationfor λ ≥ 0
All timings for MATLAB 7.0.4 on AMD Opteron Processor 248× 860 64 with2.2 GHz and 4 GB RAM
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 56 / 61
Numerical example
Numerical example ct.
FVM: Hwang, Lin, Wang, Wang (2004)
dim λ1 λ2/3 λ4 λ5 CPU time2′475 0.41195 0.58350 0.67945 0.70478 0.68 s
22′103 0.40166 0.57668 0.68418 0.69922 8.06 s186′543 0.39878 0.57477 0.68516 0.69767 150.92 s
1′532′255 0.39804 0.57427 0.68539 0.69727 4017.67 s12′419′775 0.39785 0.57415 overnight
FEM: Cubic Lagrangian elements on tetrahedal grid
dimension: 96’640 ((DofQD,Dofmat,Dofinterf) = (43′615,43′897,9′128)
dim λ1 λ2 λ3 λ4 λ5 CPU time96′640 0.39779 0.57411 0.57411 0.68547 0.69714Arnoldi 44 it. 29 it. 29 it. 24 it. 21 it. 188.8 s
JD 15 it. 9 it. 1 it. 7 it. 7 it. 204.4 sHLWW 45 it. 49 it. 5 it. 24 it. 21 it. 226.7 s
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 57 / 61
Numerical example
Numerical example ct.
FVM: Hwang, Lin, Wang, Wang (2004)
dim λ1 λ2/3 λ4 λ5 CPU time2′475 0.41195 0.58350 0.67945 0.70478 0.68 s
22′103 0.40166 0.57668 0.68418 0.69922 8.06 s186′543 0.39878 0.57477 0.68516 0.69767 150.92 s
1′532′255 0.39804 0.57427 0.68539 0.69727 4017.67 s12′419′775 0.39785 0.57415 overnight
FEM: Cubic Lagrangian elements on tetrahedal grid
dimension: 96’640 ((DofQD,Dofmat,Dofinterf) = (43′615,43′897,9′128)
dim λ1 λ2 λ3 λ4 λ5 CPU time96′640 0.39779 0.57411 0.57411 0.68547 0.69714Arnoldi 44 it. 29 it. 29 it. 24 it. 21 it. 188.8 s
JD 15 it. 9 it. 1 it. 7 it. 7 it. 204.4 sHLWW 45 it. 49 it. 5 it. 24 it. 21 it. 226.7 s
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 57 / 61
Numerical example
Convergence history
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 58 / 61
Numerical example
Preconditioning
incomplete LU with cut-off threshold τ
τ JD Arnoldi HLWW precond.0.1 261.4 1084.1 1212.4 3.4
0.01 132.7 117.1 155.7 71.70.001 118.9 61.2 96.0 246.6
0.0001 155.6 46.6 71.1 665.6
Sparse approximate inverse
τ JD Arnoldi HLWW precond.0.4 968.6 3105.7 4073.4 314.00.3 819.9 2027.6 2744.7 641.20.2 718.1 1517.5 2157.2 1557.00.1 694.9 1461.5 2124.4 1560.9
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 59 / 61
Numerical example
Preconditioning
incomplete LU with cut-off threshold τ
τ JD Arnoldi HLWW precond.0.1 261.4 1084.1 1212.4 3.4
0.01 132.7 117.1 155.7 71.70.001 118.9 61.2 96.0 246.6
0.0001 155.6 46.6 71.1 665.6
Sparse approximate inverse
τ JD Arnoldi HLWW precond.0.4 968.6 3105.7 4073.4 314.00.3 819.9 2027.6 2744.7 641.20.2 718.1 1517.5 2157.2 1557.00.1 694.9 1461.5 2124.4 1560.9
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 59 / 61
Numerical example
Conclusions
Iterative projection methods of Jacobi-Davidson and Arnoldi type areefficient methods for solving sparse nonlinear eigenvalue problems.If accurate preconditioners are available the nonlinear Arnoldi methodseems to be superior to Jacobi-Davidson methodA cruical task when computing more than one eigenpair is to inhibit themethod from converging to an eigenpair which was detected alreadypreviously.If T (λ) allows for a minmax characterization of its eigenvalues they canbe determined safely one after the other by solving the projectedproblems with safeguarded iteration.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 60 / 61
Numerical example
Conclusions
Iterative projection methods of Jacobi-Davidson and Arnoldi type areefficient methods for solving sparse nonlinear eigenvalue problems.If accurate preconditioners are available the nonlinear Arnoldi methodseems to be superior to Jacobi-Davidson methodA cruical task when computing more than one eigenpair is to inhibit themethod from converging to an eigenpair which was detected alreadypreviously.If T (λ) allows for a minmax characterization of its eigenvalues they canbe determined safely one after the other by solving the projectedproblems with safeguarded iteration.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 60 / 61
Numerical example
Conclusions
Iterative projection methods of Jacobi-Davidson and Arnoldi type areefficient methods for solving sparse nonlinear eigenvalue problems.If accurate preconditioners are available the nonlinear Arnoldi methodseems to be superior to Jacobi-Davidson methodA cruical task when computing more than one eigenpair is to inhibit themethod from converging to an eigenpair which was detected alreadypreviously.If T (λ) allows for a minmax characterization of its eigenvalues they canbe determined safely one after the other by solving the projectedproblems with safeguarded iteration.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 60 / 61
Numerical example
Conclusions
Iterative projection methods of Jacobi-Davidson and Arnoldi type areefficient methods for solving sparse nonlinear eigenvalue problems.If accurate preconditioners are available the nonlinear Arnoldi methodseems to be superior to Jacobi-Davidson methodA cruical task when computing more than one eigenpair is to inhibit themethod from converging to an eigenpair which was detected alreadypreviously.If T (λ) allows for a minmax characterization of its eigenvalues they canbe determined safely one after the other by solving the projectedproblems with safeguarded iteration.
TUHH Heinrich Voss Iterative projection methods Shanghai, May 27, 2012 60 / 61
Numerical example
References
H. Voss. An Arnoldi method for nonlinear symmetric eigenvalue problems. OnlineProc. SIAM Conf. Applied Linear Algebra, Williamsburg, 2003.T. Betcke, H. Voss. A Jacobi–Davidson–type projection method for nonlineareigenvalue problems. Future Generation Comput. Syst. 20, 363 – 372 (2004)H. Voss. An Arnoldi method for nonlinear eigenvalue problems. BIT NumericalMathematics 44, 387 – 401 (2004)H. Voss. A new justification of the Jacobi–Davidson method for largeeigenproblems. Linear Algebra Appl. 424, 448 – 455 (2007)H. Voss. A Jacobi–Davidson method for nonlinear and nonsymmetriceigenproblems. Computers & Structures 85, 1284 – 1292 (2007)M.M. Betcke, H. Voss. Restarting projection methods for rational eigenproblemsarising in fluid-solid vibrations. Math. Modelling Anal. 13, 171 – 182 (2008)H. Voss. Iterative projection methods for large-scale nonlinear eigenvalueproblems. pp. 187 – 214 in B.H.V. Topping et al. (eds.), ComputationalTechnology Reviews, vol. 1, 2010M.M. Betcke, H. Voss. Analysis and efficient solution of stationary Schrodingerequation governing electronic states of quantum dots and rings in magnetic fieldCommun.Comput.Phys. 12, 1591 – 1617 (2012)
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