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Sergio Blanes
Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN
Primer Encuentro Iberoamericano de Geometría, Mecánica y Control
Santiago de Compostela, 23-27 Junio 2008
Geometric Integrators: Composition, Splitting and Lie Group
Methods
To design numerical methods that share qualitative features of the differential equation
Introduction
Geometric Integration
Dynamical systems are frequently modelled by differential
equations
The models are usually simpler than the real problem
Good models must incorporate the important qualitative properties (conservation of mass, energy, momentum, unitariety, positivity, etc.)
Unfortunately, in general, no analytical solutions are known for models To use numerical methods for differential equations
Standard numerical methods (available at libraries, books, etc.) do not preserve such desired qualitative properties
The numerical solutions can seriously differ from the solutions for the model, and then from the real system
The Geometric Integration intends to incorporate some of the important qualitative properties on the methods
The numerical solutions are frequently
more accurate and reliable.
Hairer, Lubich and Wanner
Many different techniques have been developed in GI for numerically solving differential equations, depending on the structure of f(x,t). For simplicity, let us consider Hamiltonian systems with x=(q,p)
so f(x,t)=J H, where H is the (scalar) Hamiltonian fuction which describes the dynamical system. We have that
with H H and small.
real problem
(estándar method)
(model)
(symplectic method)
Example: Let us consider a Hamiltonian system x = (q , p)T
where is the Hamiltonian fuction. If the system is separable,H = T(p) + V(q) then, for a time step h
The harmonic oscillator:
Euler
sympl. Euler
h = 1/10
3 3.5 4 4.5
-2
-1.5
-1
-0.5
0
X
Y
h = 1/10
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
X
Y
The harmonic oscillator:
Euler
sympl. Euler
The harmonic oscillator:
The symplectic Euler has the invariant
Euler
sympl. Euler
The harmonic oscillator:
The symplectic Euler has the invariant
Euler
It is the exact solution of the perturbed Hamiltonian
sympl. Euler
The Kepler problem
i = 1, 2 where q = (y1, y2)T, p = (y’1, y’2)T
-2.5 -2 -1.5 -1 -0.5 0 0.5-1.5
-1
-0.5
0
0.5
1
1.5
X
Y
h = 1/10, e=0.6
3 periodos
-2.5 -2 -1.5 -1 -0.5 0 0.5-1.5
-1
-0.5
0
0.5
1
1.5
X
Y
h = 1/2, e=0.6
15 periodos
with H H and small.
real problem
(estándar method)
(model)
(symplectic method)
500 periods and 1500 evaluations per period: h 1/250
Euler
RK2
Example: Kepler Problem
Euler Simpléctico
Euler
RK2
Leapfrog Simpléctico
Example: Kepler Problem
500 periods and 1500 evaluations per period: h 1/250
The Lotka-Volterra equation:
Euler
Splitting
0 1 2 3 4 5 6 70
1
2
3
4
5
6
X
Y
0 1 2 3 4 5 6 70
1
2
3
4
5
6
X
Y
h = 1/10
The differential equation x’ = f (x) can be written as
and formal solution
A numerical emthod of order p for a time step h corresponds to a map where
Splitting methods
Autonomous system which can be split in two parts
Let us assume that x = fA(x), x = fB(x) are solvable, with et A and et B their associated flows. It is possible to build a composition method given by
where ai, bi have to satify a system of non-linear equations
. .
Composition Methods
Let xn+1 = h[p]
(xn) a numerical method of order p, where h[p] = h + O(h p+1), with
h the exact solution. It is possible to consider
Such that h[q] = h + O(hq+1 ) and q ≥ p. We have to look for the best set of
coefficients i.
. .
Splitting methods
Autonomous system which can be split in two parts
Let us assume that x = fA(x), x = fB(x) are solvable, with et A and et B their associated flows. It is possible to build a composition method given by
where ai, bi have to satify a system of non-linear equations
Splitting Methods
Higher Orders
The well known fourth-order composition scheme
with
Higher Orders
The well known fourth-order composition scheme
with
Or in general
with
-Creutz-Gocksh(89) -Suzuki(90)-Yoshida(90)
Higher Orders
The well known fourth-order composition scheme
with
Or in general
with
-Creutz-Gocksh(89) Excellent trick!!-Suzuki(90)-Yoshida(90) costly methods with large errors!!
Order 4 6 8 10
3- For-Ru(89)
Yos(90),ect.
5-Suz(90)
McL(95)
Order 4 6 8 10
3- For-Ru(89) 7-Yoshida(90)
Yos(90),etc. 9-McL(95)
5-Suz(90) Kahan-Li(97)
McL(95) 11-13-Sof-Spa(05)
Order 4 6 8 10
3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90)
Yos(90),etc. 9-McL(95) Suz-Um(93)
5-Suz(90) Kahan-Li(97) 15-17-McL(95)
McL(95) 11-13-Sof-Spa(05) Kahan-Li(97)
19-21-Sof-Spa(05)
Order 4 6 8 10
3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90) 31-Suz-Um(93)
Yos(90),etc. 9-McL(95) Suz-Um(93) 31-33-Ka-Li(97)
5-Suz(90) Kahan-Li(97) 15-17-McL(95) 33-Tsitouras(00)
McL(95) 11-13-Sof-Spa(05) Kahan-Li(97) 31-35-Wanner(02)
19-21-Sof-Spa(05) 31-35-Sof-Spa(05)
Order 4 6 8 10
3- For-Ru(89) 7-Yoshida(90) 15-Yoshida(90) 31-Suz-Um(93)
Yos(90),etc. 9-McL(95) Suz-Um(93) 31-33-Ka-Li(97)
5-Suz(90) Kahan-Li(97) 15-17-McL(95) 33-Tsitouras(00)
McL(95) 11-13-Sof-Spa(05) Kahan-Li(97) 31-35-Wanner(02)
19-21-Sof-Spa(05) 31-35-Sof-Spa(05)
Processed
P3-17-McL(02) P5-15 P9-19 P15-25
B-Casas-Murua(06)
The Processing Technique
Let us consider N steps with the second order method Störmer/Verlet/leapfrog
A small modification at the beginning and at the end of the integration can transform a first order method into a second order one. The generalization corresponds to
where h,K is the kernel and h is the processor.
This technique can be used in tandem with the previous ones in order to improve the efficiency of the numerical methods.
Let us consider the following first order method for a system separable in N parts
Adjoint Method. Given h, its adjoint is defined as h* = (-h)-1. Then,
By symmetrization we obtain a second order method
This is a generalización del método Störmer/Verlet/leap-frog/Strang splitting.
Order 3 4 6 8
3- Ru(84) 3-Ru-Yos,etc 9-Forest(91) 27- ??
4,5-McL(95) 10-B-Moan(02)
6-B-Moan(02)
Processed
P2-7 P5-10 P14- ??
B-Casas-Murua(06)
Consider the linear time dependent SE
Which is also separable in its kinetic and potential parts. The solution of the discretised equation is given by
where c = (c1,…, cN)T N and H = T + V N×N Hermitian matrix.
It is easy to see that
Order 4 5 6 8
3S-Ru-Yos,etc 5NABA-Oku-Seel(94) 7SABA-For(91) 16SABA/BAB-?
4NAB-McL-At(92) 6NAB-McL-At(92) Oku-Skeel(94) 17SABA-Ok-Lu(94)
4NBAB-Cal-SS(93) 6NBAB-Chou-Sha(99) 7SBAB-For(91) 24SS-Ca-SS(93)
4,5SABA-McL(95) 8-15SABA/BAB
5SBAB-B-Moan(01) 11,14SBAB-B-M(02)
6SABA/BAB-B-Moan(02)
Processed
P3,4 P4-7 P9-11 B-Casas-Ros(01)
Consider again the linear time dependent SE
Which is also separable in its kinetic and potential parts. The solution of the discretised equation is given by
where c = (c1,…, cN)T N and H = T + V N×N Hermitian matrix.
Consider then
Hamiltonian system:
with formal solution
It is an orthogonal and symplectic operator.
Consider then
Hamiltonian system:
with formal solution
It is an orthogonal and symplectic operator.
Notice that
Consider the time dependent Maxwell Equations
electric & magnetic field vectors permeability and permitivity
Using dimensionless equations and a faithful spatial representation of E and B we have the discrete form
We can use splitting methods, i.e. Lie-Trotter/symplectic-Euler
Order 4 6 8 10 12
4,6 6 8 10 12
Gray-Manolopoulos(96)
Processed
P3,4 P3,4 P4-5 McL-Gray(97)
Order 4 6 8 10 12
4,6 6 8 10 12
Gray-Manolopoulos(96)
Processed
P3,4 P3,4 P4-5 McL-Gray(97)
P2-40 orders: 2-20 B-Casas-Murua(06)
Performance of a Method
versusError Computational Cost
Consider the linear time dependent SE
We have the discrete form
Consider then
Hamiltonian system:
with formal solution
Problem to Solve: To build splitting methods for the harmonic oscillator!!!
Exact solution (ortogonal and symplectic)
Let us consider the composition
Notice that
100 periods with initial conditions (q,p)=(1,1)
Example: Harmonic oscillator
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(EVAL)
LOG
(ER
RO
R)
100 periods with initial conditions (q,p)=(1,1)
Example: Harmonic oscillator
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(EVAL)
LOG
(ER
RO
R)
100 periods with initial conditions (q,p)=(1,1)
Example: Harmonic oscillator
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(EVAL)
LOG
(ER
RO
R)
100 periods with initial conditions (q,p)=(1,1)
Example: Harmonic oscillator
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(EVAL)
LOG
(ER
RO
R)
Schrödinger equation with a Morse potential
with
Initial conditions
Gaussian wave fuction in a Morse potential
Gaussian wave fuction in a Morse potential
4.4 4.6 4.8 5 5.2 5.4 5.6 5.8-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(N. FFTs)
LO
G(E
RR
. NO
RM
)S
2
S34
S17
8
GM44
GM88
GM12
12
Gaussian wave fuction in a Morse potential
4.4 4.6 4.8 5 5.2 5.4 5.6 5.8-9
-8
-7
-6
-5
-4
-3
-2
-1
0
LOG(N. FFTs)
LO
G(E
RR
. NO
RM
)S
2
S34
S17
8
GM44
GM88
GM12
12P
382
Consider , then write the solution as and look for
the DE for . To preserve the Lie algebra structure is easier than the Lie group.
We get an approximation in the Lie algebra and then
Lie Group Methods
X
In some cases it is convenient to write the differential equation in matrix form, e.g.
If and then the solution remains in the group.
Some families of methods
Runge-Kutta-Munthe-Kaas methods
Canonical coordinates of the second kind
Exponential integrators
etc.
Some other equations
Isospectral flows:
Equation on a sphere
Linear non-autonomous systems
with A(t) d×d
The Magnus expansion
where
Convergence condition:
Le us consider the Taylor expansio of A(t) about t1/2=t0+h/ 2 (to preserve time-symmetry)
with
If we take bi ai-1 hi, i = 1, 2, 3, then
b1, b2, b3 generators of a gradded Lie algebra with grades 1, 2, 3. All multidimensional integrals can be approximatted by using quadrature rules for unidimensional integrals (Iserles y Norsett).
These results can also be obtained by using the following unidimensional momentum integrals
These integrals are releted with the elements of the gradded Lie algebra bi and can be either evaluated analytically or numerically. A 4th-order method is
Higher order methods with a moderate number of commutators also exist.
Commutator-free Magnus integrators
For many problem it is simpler to compute
exp(A(ti)) than exp([A(ti), A(tj)])
To look for variants of the Magnus expansion without commutatos.
Some 4th-order commutator-free Magnus integrators are
They preserve time-symmetry and have shown good performance on a number of examples.
Example: The skew-symmetric matrix (N=10)
Example: The Mathieu equation
in. conds.T
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, (2006).
A. Iserles, H.Z. Munthe-Kaas, S.P. Nörsett and A. Zanna, Lie group methods, Acta Numerica, 9 (2000), 215-365.
B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, (2004).
R.I. McLachlan and R. Quispel,Splitting methods, Acta Numerica, 11 (2002), 341-434.
Basic References
To build efficient geometric integrators require to analyse in some detail the algebraic structure of the problem:
Linear ― Non Linear Autonomous ― Non autonomous Separable ― Non Separable If Separable: near-Integrable, Quadratic Kinetic Energy Evolution on Different Time Scales
To build efficient geometric integrators require to analyse in some detail the algebraic structure of the problem:
Linear ― Non Linear Autonomous ― Non autonomous Separable ― Non Separable If Separable: near-Integrable, Quadratic Kinetic Energy Evolution on Different Time Scales
Integración Geométrica
Sistemas Autónomos Sistemas No Autónomos
PasoVariable
ExtrapolaciónSplitting y
ComposiciónSistemas
No LinealesSistemasLineales
ConProcesado
Sin Procesado
SistemasSeparables
Desarrollode Magnus
Desarrollode Fer
Sergio Blanes
Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN
Primer Encuentro Iberoamericano de Geometría, Mecánica y Control
Santiago de Compostela, 23-27 Junio 2008
Geometric Integrators: Composition, Splitting and Lie Group
Methods
Conclusions
Splitting methods are powerful methods for many problems
ConclusionsConclusions
Conclusions
Splitting methods are powerful methods for many problems
But if using appropriate methods for each problem
ConclusionsConclusions
Conclusions
Splitting methods are powerful methods for many problems
But if using appropriate methods for each problem
Alternatively, we can build methods tailored for particular problems
ConclusionsConclusions
Splitting MethodsSplitting Methods
Splitting MethodsSplitting Methods
Order 3 4 6 8
2NAB-Ruth(84) 2SBAB-Koseleff(93) 4,5S-O-M-F(02) 11S-O-M-F(02)
Chin(97)
Mik-Pu(99)
4SBAB-Su(95)
3,4S-Chin(02)/O-M-F(02)
Processed
P1-Rowlands(91) P3-B-Ca-Ros(01) P5-B-Ca-Ros(01)
Wis-Ho-To(96)
B-Casas-Ros(99)
P2-LM-SS-Skeel(97)
Splitting MethodsSplitting Methods
Splitting MethodsSplitting Methods
Order (n,2) (n,4) (n,5)
Ki-Yo-Na(91) 4(6,4)BAB
1(2,2)- Wis-Hol(91) 5(8,4) ABA/BAB
n(2n,2), n=2,...,5 ABA/BAB McLachlan (95)
Processed
P1(32,2)-W-H-T(96) P3(7,6,4)ABA-BCR(00)
RKN-Methods P2(6,4)AB P3(7,6,5)AB
Mod. Pot P1(6,4)ABA P2(7,6,5)AB B-Casas-Ros(00)
Pn(2n,4)–Lask-Ro(01)
Introducción y PreliminaresMétodos de Splitting y Composición para sistemas separablesMétodo adjuntoSistemas separables en dos partesMétodos de órdenes superiores
Familias de métodos
Técnica de Composición con ProcesadoÁlgebras de Lie GraduadasCondiciones de ordenAhorro computacional de la Técnica del ProcesadoConstrucción de NúcleosConstrucción de Procesadores
Técnica de Extrapolación
Ejemplos NuméricosEjemplo 1: El problema de Lotka-VolterraEjemplo 2: El problema del flujo-ABCEjemplo 3: El problema de Kepler
Conclusiones y Líneas de Trabajo
Incremento del Orden de Integradores para EDOs utilizando Extrapolación y Composición con
Procesado
To design numerical methods that share qualitative features of the differential equation
Introduction
La mayoría de los modelos que describen los procesos dinámicos de la naturaleza vienen dados por ecuaciones diferenciales. La solución analítica suele ser desconocida y los métodos numéricos son indispensables.
estructuras y simetrías propiedades cualitativas
Los métodos numéricos tradicionales no suelen respetar estas propiedades, haciéndolos poco eficientes en muchos casos.
La Integración Geométrica (IG) es un área reciente de la Matemática Aplicada donde estas propiedades son tenidas en cuenta.
Historial Investigador Historial Investigador
Geometric Integration
Desarrollo de Fer
with Fn(t) and An(t) given by
Líneas de Investigación Líneas de Investigación
Non-autonomous systemsLinear System Let us consider the linear equation
with A(t) d×d, and x (t)=t x0 denotes its solution. If A g (Lie algebra)
G (associated Lie group).
Líneas de Investigación Líneas de Investigación
Non-autonomous systemsLinear System Let us consider the linear equation
with A(t) d×d, and x (t)=t x0 denotes its solution. If A g (Lie algebra)
G (associated Lie group).
Aproximations:Magnus expansion
with
Líneas de Investigación Líneas de Investigación
The Magnus series presents a better structure to build numerical methods. Le us consider the Taylor expansio of A(t) about t1/2 = t0 + h/ 2 (to preserve time-symmetry)
with
If we take bi ai-1 hi, i = 1, 2, 3, then
b1, b2, b3 can be considered as generators of a gradded Lie algebra with grades 1, 2, 3. All the multidimensional integrals can be approximatted by using quadrature rules for unidimensional integrals (Iserles y Norsett).
Líneas de Investigación Líneas de Investigación
These results can also be obtained by using the following unidimensional momentum integrals (for a sixth-order method)
These integrals are releted with the elements of the gradded Lie algebra bi and can be either evaluated analytically or numerically. A 4th-order method is
Higher order methods with a moderate number of commutators also exist.
Líneas de Investigación Líneas de Investigación
Commutator-free Magnus integrators
In many problem, the structure of the matrix A(t) is relatively simple while the matrices obtained from the commutators can be rather involved it is simpler to evaluate the exponential of A(ti) but the exponential of [ A(ti), A(tj) ] can be very costly. It seems convenients to look for variants of the Magnus expansion which do not involve commutatos.
Ejemplo: La ecuación de Schrödinger (con FFTs).
Some 4th-order commutator-free Magnus integrators are
They preserve time-symmetry and have shown good performance on a number of examples.
Líneas de Investigación Líneas de Investigación
Consider the Strang-splitting or leap-frog second order method
Notice that
the exponentials are computed only once and are stored at the beginning. Similarly, for the kinetic part we have
Consider the Strang-splitting or leap-frog second order method
Notice that
the exponentials are computed only once and are stored at the beginning. Similarly, for the kinetic part we have
Considering that we have
Then it is easy to see that with
and we can use Nyström methods
Consider now the one-dimensional cubic NLES
We solve: in the coordinate space and
in the momentum space
Periodic boundary conditions are assumed and FFTs are used to carry out the coordinate transforms. Observe that
where
Consider the linear system of ODEs (or semidiscretized PDEs)
Let us define z = (q,p)T , then
The first order splitting method is given by
Higher order splitting methods can be obtained as follows
with chosen such that
However, our problem has a very particular structure
Linear non-autonomous systemsLinear System Let us consider the linear equation
with A(t) d×d, and x (t)=t x0 denotes its solution. If A g (Lie algebra)
G (associated Lie group).
The Magnus series presents a better structure to build numerical methods.