Exponential Integrators - Numerical Analysis Innsbruck

Exponential Integrators
Marlis Hochbruck
Heinrich-Heine University Düsseldorf
Germany
Innsbruck, October 2004
– p.1

Motivation
Outline
Exponential integrators
general purpose integrators
2nd order equations
Schrödinger equations
Krylov approximations of the matrix exponential
– p.2

Definition
An exponential integrator is a numerical method which
involves an exponential function (or a related function) of
the Jacobian.
Possible applications: stiff or oscillatory problems
– p.3

Motivation
Exponential methods are an old idea
Certaine, 1960
Pope, 1963
Lawson, 1967
Nørsett, 1969
Ehle, Lawson, 1975
Lambert, Sigurdsson, 1972
Verwer, 1975
Friedli, 1978
Strehmel, Weiner, 1987
– p.4

Motivation
This old idea has not been regarded as practical
ϕ(hA) was only computable for small A
(via diagonalization)
in most approaches: ϕ(z) approximated by rational
function
(Padé approximation or uniform approximation)
– p.5

Motivation
This old idea became practical even for large problems
Krylov subspace approximations to ϕ(hA)v
Druskin, Knizhnerman, 1989–1995,
Gallopoulos and Saad, 1992,
H., Lubich, 1997
Moler and van Loan extended their 1979 paper
“19 dubious ways to compute the matrix exponential”
by method 20, Krylov subspace methods (2004)
– p.6

Motivation
Why might exponential integrators be a good idea?
semilinear problems
y ′ = −Ay + g(t, y),
variation of constants formula
y(t + h) = e −hA y(t) +
h
0
e −(h−τ)A g(tn + τ, y(tn + τ))dτ
approximating highly oscillatory functions by
polynomials or rational functions typically leads to step
size restrictions
hωmax ≤ c
– p.7

Stiff and nonstiff problems
Ode’s are characterized in nonstiff and stiff problems:
ode nonstiff stiff
method explicit implicit
stability h ∼ L −1 all h
implementation f(y) f ′ (y),
requires nonlinear solver
but it could be worse ...
– p.8

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
Oscillatory test problem
y ′ = −i(ω + g(t))y, ω = 100, g(t) = sin t/100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
expl. impl. exp.
steps 121 149 4
f(y) 745 723 29
LU decomp – 10 –
time (sec) 0.08 0.17 0.02
flops 82.904 120.417 9.370
– p.9

General purpose exponential integrators
Exponential Euler method
with
Properties
y ′ = f(y), y(0) = y0
yn+1 = yn + hϕ(hA)f(yn), A = f ′ (yn)
exact for f(y) = Ay + b
order two
ϕ(z) = ez − 1
z
– p.10

Higher order exponential integrators
(Motivated by Runge-Kutta- and Rosenbrock methods)
A ≈ f ′ (y0), ϕ(z) = ez −1
z
for i = 1, . . . , s : ki = ϕ(γhA) ⎝f(ui) + hA
ui = y0 + h
y1 = y0 + h
⎛
i−1
j=1
s
i=1
αijkj
biki .
explicit Runge-Kutta method: ϕ(z) ≡ 1 (A = 0)
Rosenbrock methods: ϕ(z) = 1/(1 − z)
(H., Lubich, Selhofer, 1998)
i−1
j=1
γijkj
⎞
⎠
– p.11

General purpose integrators
(Munthe-Kaas, 1999, in terms of Lie group methods)
Transformation
yields c(0) = 0 and
y(t) = e tA y0 + ϕ(tA)c(t)
c ′ = ψ(tA)(f(y) − Ay) − (ψ(tA) − 1) c(t)
t
Exponential integrator (RKMK methods):
solve ode for c with standard RK method
+ back transformation
– p.12

2nd order differential equations
y ′′ + Ω 2 (t, y)y = g(y), y(0) = y0 , y ′ (0) = y ′ 0
for constant g and Ω, exact solution satisfies
y(t+h)−2y(t)+y(t−h) = h 2 σ(hΩ) g−Ω 2 y(t) , σ(x) = sinc 2 x
2
(variation-of-constants formula)
Gautschi-type exponential integrator
yn+1 − 2yn + yn−1 = h 2 σ(hΩn) gn − Ω 2
n yn , Ωn = Ω(tn, yn)
choice of gn?
– p.13

2nd order differential equations
yn+1 − 2yn + yn−1 = h 2 σ(hΩn) gn − Ω 2
n yn
obvious choice: gn = g(yn) −→ Gautschi ’61
resonance problems for hωk ≈ jπ, ωk eigenvalue of Ωn
– p.14

2nd order differential equations
yn+1 − 2yn + yn−1 = h 2 σ(hΩn) gn − Ω 2
n yn
obvious choice: gn = g(yn) −→ Gautschi ’61
resonance problems for hωk ≈ jπ, ωk eigenvalue of Ωn
better choice: gn = g(φ(hΩn)yn), φ filter function
φ(0) = 1 , φ(kπ) = 0 , k = 1, 2, 3, . . .
– p.14

2nd order differential equations
yn+1 − 2yn + yn−1 = h 2 σ(hΩn) gn − Ω 2
n yn
obvious choice: gn = g(yn) −→ Gautschi ’61
resonance problems for hωk ≈ jπ, ωk eigenvalue of Ωn
better choice: gn = g(φ(hΩn)yn), φ filter function
φ(0) = 1 , φ(kπ) = 0 , k = 1, 2, 3, . . .
convergence result: (H., Lubich, 1999, Grimm 2002)
Assumptions: g smooth, bounded energy:
yn − y(tn) ≤ h 2 C(tn)ℓ(n, N), C(tn) ∼ e tnL
– p.14

Schrödinger equations
Quantum dynamics simulations:
ψ ′ (t) = −iH(t)ψ(t), H(t) = U + V (t), U ∼ −∆∆x
(hence H(t) large, ∼ 1/∆x d )
exponential midpoint rule
(exact for constant H)
ψn+1 = exp
−ihH(tn+1/2 ψn
– p.15

Magnus integrators
y ′ = A(t)y(t), y(0) = y0
Magnus, 1954: determine Ω(t) such that
y(t) = exp(Ω(t))y0
Magnus expansion (valid for Ω(t) < π)
Ω(t) =
t
0
A(τ)dτ − 1
2
+ 1
4
+ 1
12
t
0
t
0
t
0
τ
[
[
0
τ
[
0
τ
0
A(σ)dσ, A(τ)] dτ
σ
[
0
A(µ)dµ, A(σ)] dσ, A(τ)] dτ
A(σ)dσ, [
τ
0
A(µ)dµ, A(τ)]] dτ + . . .
– p.16

Numerical methods
Magnus integrators
yn+1 = exp(Ωn)yn, Ωn ≈ Ω(h)
(review: Iserles, Munthe-Kaas, Nørsett, Zanna, ’00)
Approximation involves
truncating the Magnus expansion (after k terms)
k = 1 : Ω(t) ≈
k = 2 : Ω(t) ≈
t
0
t
0
A(tn + τ)dτ
A(tn + τ)dτ − 1
2
t
0
τ
[
0
A(tn + σ)dσ, A(tn + τ)] dτ
approximating integrals by replacing A(t) by
interpolation polynomial A(t) for quadrature nodes
tn + cjh
– p.17

Examples of Magnus integrators
k = 1, exponential midpoint rule
Ωn = hA(tn + h/2).
k = 2, two-point Gauß quadrature rule:
Ωn = h
2 (A1
√
3h2 + A2) +
12 [A2, A1], Aj = A(tn + cjh),
cj nodes of Gauß quadrature rule
k = 2, method by Blanes, Casas, Ros ’00
Ωn = h
A(tn)+4A(t
6
n+1/2)+A(tn+1) − h2
12 [A(tn), A(tn+1)].
– p.18

Convergence for Schrödinger equations
Theorem (H., Lubich, 2003)
Exponential midpoint rule
yn − y(tn) ≤ Ch 2 tn max
0≤t≤tn
4th-order Gauß method: for hD ≤ c
yn − y(tn) ≤ Ch 4 tn max
0≤t≤tn
Dy(t)
D 3 y(t)
– p.19

Convergence for Schrödinger equations
Theorem (H., Lubich, 2003)
Exponential midpoint rule
yn − y(tn) ≤ Ch 2 tn max
0≤t≤tn
4th-order Gauß method: for hD ≤ c
yn − y(tn) ≤ Ch 4 tn max
0≤t≤tn
Dy(t)
D 3 y(t)
error bound for classical implicit midpoint rule:
yn − y(tn) ≤ Ch 2 tn max
0≤t≤tn
d3
y(t)
dt3 – p.19

Let 0 < ɛ ≪ 1
Generalizations
2nd order equations: Lorentz, 2004
y ′′ + 1
ɛ2 Ω2 (t)x(t) = 1
f(t)
ɛ2 Schrödinger equations: Jahnke, Lubich, 2000-2004
iψ ′ = 1
ɛ H(t)ψ(t)
– p.20

Schrödinger equation
iψ ′ = 1
ɛ H(t)ψ(t)
Main idea for construction and error analysis:
transformation to adiabatic variables
Q(t) T
ψ(t) = exp − i
ɛ Φ(t)
η(t)
where
H(t) = Q(t)Λ(t)Q(t) T , Λ diagonal
Φ(t) =
t
t0
Λ(s)ds
−→ ode for η(t) (advantage: η ′ (t) bounded uniformly in ɛ)
– p.21

for all these problems:
symmetric methods
Convergence
require only one evaluation of Ω or H, resp. per time
step
error bounds valid for ɛ ≤ h < √ ɛ
error bounded by Ch 2 , where C is independent of ɛ
implementation requires (partial) diagonalization of Ω, H,
resp.
– p.22

Other approaches / other applications
(Commutator-free) Lie group methods
Crouch, Grossman, 1993
Munthe-Kaas, 1995–2004
Celledoni, Marthinsen, Owren, 2003
. . .
Methods for nonlinear Schrödinger equations
Strang splitting (analysis: Jahnke, Lubich 2000)
Berland, Owren, Skaflestad, 2004
– p.23

Stiff and oscillatory problems
ode nonstiff stiff oscillatory
method explicit implicit exponential
stability h ∼ L −1 all h all h
implementation f(y) f ′ (y), f ′ (y)
requires nonlinear solver ϕ(A)b
– p.24

Implementation of exponential integrators
exponential integrators require approximation of
assume:
ϕ(A)b
A complex N × N matrix with large N
matrix-vector multiplication Av cheap
b = 1
– p.25

Krylov subspace methods
Idea: approximate ϕ(A)b in subspace of low dimension
suitable subspace:
Kn(A, b) := span {b, Ab, . . . , A n−1 b}
nth Krylov subspace w.r.t. A and b, (n ≪ N)
two essential steps to compute nth iterate:
xn = Vnyn ≈ ϕ(A)b, xn ∈ Kn(A, b)
construction of a basis Vn of Kn(A, b)
(Arnoldi, Lanczos, Tschebyscheff)
characterization of yn
– p.26

Approximation of matrix functions
Krylov approximation to ϕ(A)b:
ϕ(A)b = 1
ϕ(λ) (λI − A)
2πi Γ
−1 ≈
b dλ
1
2πi
Γ
= Vnϕ(Hn)e1
ϕ(λ) Vn(λI − Hn) −1 e1 dλ
other derivations:
Park, Light 1985, Druskin, Knizhnerman 1991,
Gallopoulos, Saad 1992
– p.27

Convergence
derive error bounds for Ax = b and thus for (λI − A)x = b
(complex approximation theory using conformal
mappings and Faber polynomials)
multiply by ϕ(λ)
integrate over suitable contour Γ
always obtain superlinear convergence for n ≥ n0
– p.28

Example
A : 1001 × 1001 (symmetric), eigenvalues in [−40, 0]
(uniformly distributed)
10 0
10 −5
10
0 20 40 60 80
−10
error and error bounds
for approximations to
(I − A) −1 b and exp(A)b
– p.29

Example
A : 1001 × 1001 (skew symmetric), eigenvalues i[−20, 20]
(uniformly distributed)
10 0
10 −5
10
0 20 40 60 80
−10
error and error bounds
for approximations to
(I − A) −1 b and exp(A)b
– p.30

Start of superlinear convergence behavior
n0 ∼ hA
n0 ∼ hA
θπ n0 ∼ hA α , 1 2
≤ α = 1
2−θ
≤ 1
– p.31

exponential integrators
go back to old ideas
can be practical
Summary
can be used with step sizes larger than 1/ωmax for
highly oscillatory problems
can be implemented using Krylov subspace methods
http://www.am.uni-duesseldorf.de/∼marlis
– p.32