Sergio Blanes Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia, SPAIN...

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.