Taylor Model-based Veri ﬁed Integrators Kyoko Makino and ... · Kyoko Makino and Martin Berz...

Taylor Model-based Verified Integrators

Kyoko Makino and Martin Berz

Michigan State University

The Lorenz Equations

The equations describe a simplified model of unpre-dictable turbulent flows in fluid dynamics.Exhibits sensitive dependence on initial conditions andchaoticity.

= ( − )

= (− )−

= −

The standard parameter values are

= 10 =8

3 = 28

and is often varied. The fixed points are

(0 0 0) (±p(− 1)±

p(− 1) − 1)

Study Trajectories of the Lorenz System

Using conventional Runge Kutta integrators, study tra-jectories of an initial point

( )|0 = (15 15 36)Integration from = 0 to = = 20

RK4: the 4th order RKRK4S: 4th order RK with automatic step size controlCOSY-RK8: 8th order RK implemented in COSY

-20 -15 -10 -5 0 5 10 15 20-25-20

-15-10

-5 0

5 10

15 20

25

5

10

15

20

25

30

35

40

45

z

Non-verified Runge-Kutta Integrations of Lorenz eqs. - Trajectory

cosy RKRK4S

RK4

x

y

z

-25

-20

-15

-10

-5

0

5

10

15

20

25

-20 -15 -10 -5 0 5 10 15 20

y

x

Non-verified Runge-Kutta Integrations of Lorenz eqs. - Trajectory

cosy RKRK4S

RK4

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20

x

Time

Non-verified Runge-Kutta Integrations of Lorenz eqs. - x Position

cosy RKRK4S

RK4


Using conventional Runge Kutta integrators, study tra-jectories of an initial point

( )|0 = (15 15 36)Integration from = 0 to = = 20

( ) ( ) ( ) single step error CPURK4 2.205 1.030 22.282 0.689RK4S -3.388 0.796 27.582 1.27e-2 0.904COSY-RK8 14.309 9.591 39.039 1.21e-7 1

RK4S: 4th order RK with automatic step size controlCOSY-RK8: 8th order RK implemented in COSY


Using COSY-RK8, study trajectories of initial points

( )|0 = (15 15 36)and

( )|0 = (15 15 36) + (±001±001±001)

��

� � ��

t: [ 0, 1.5589 ]center

---+---+-++---++-+-+++++

-20 -15 -10 -5 0 5 10 15 20x -25

-20-15

-10-5

05

1015

2025

y

51015202530354045

z

��

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Dis

tanc

e

Time

-00+000-0--0+-00+0-+0++000--0-+0-0--0+-00+-0++0+0-+0++

---+---+-++---++-+-+++++

��

� � ��

t: [ 0, 5.3260 ]center

---+---+-++---++-+-+++++

-20 -15 -10 -5 0 5 10 15 20x -25

-20-15

-10-5

05

1015

2025

y

51015202530354045

z

��

0

5

10

15

20

25

30

0 1 2 3 4 5 6

Dis

tanc

e

Time

-00+000-0--0+-00+0-+0++000--0-+0-0--0+-00+-0++0+0-+0++

---+---+-++---++-+-+++++

��

� � ��

t: [ 0, 10.000 ]center

---+---+-++---++-+-+++++

-20 -15 -10 -5 0 5 10 15 20x -25

-20-15

-10-5

05

1015

2025

y

51015202530354045

z

��

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10

Dis

tanc

e

Time

-00+000-0--0+-00+0-+0++000--0-+0-0--0+-00+-0++0+0-+0++

---+---+-++---++-+-+++++

Rigorous Integrations of the Lorenz System

As a test case, transport an area of initial condition

( )|0 = (15 15 36) + (±001±001±001)using rigorous ODE integrators.

AWA: An interval based ODE integrator by R. Lohner

COSY-VI: Taylormodel basedODE integrator inCOSY— Computed in 2001, using COSY-VI version 1

Taylor Model based Integrator COSY-VIversion 1 (1997—)

• High order expansion not only in time t but also in transversalvariables x.

• One time step integration via Picard iterations based on theShauder fixed point theorem.

• Shrink wrapping algorithm, a simplified version

The Wrapping Effect in Linear ODEs

Initial Condition Interval Box Solution Set

Solution Set in the Optimal Interval Box

Solution Set in Rotated Rectangles

( Here, the Right One is Optimal. )

Solution Set by Taylor Models

The Wrapping Effect in Nonlinear ODEs

Initial Condition Interval Box Solution Set

Solution Set in the Optimal Interval Box

Solution Set by Taylor Models

Solution Set in an Optimal Rotated Rectangle

Solution Set in an Optimal Eight-Corner Polygon

The Henon MapHenon Map: frequently used elementary example that exhibits many ofthe well-known effects of nonlinear dynamics, including chaos, periodic fixedpoints, islands and symplectic motion. The dynamics is two-dimensional,and given by

xn+1 = 1− αx2n + ynyn+1 = βxn.

It can easily be seen that the motion is area preserving for |β| = 1.Weconsider

α = 2.4 and β = −1,and concentrate on initial boxes of the from (x0, y0) ∈ (0.4, −0.4)+[−d, d]2.

1e-16

1e-14

1e-12

1e-10

1e-8

1e-6

1e-4

1e-2

1e+0

0 50 100 150 200 250 300

Err

or S

ize

Step Number

Henon (Area Preserving). Performance Comparison. TM order 13, IC width 4e-3, no domain split

naive TMInterval

��

��

�� ! "#" $�� !��%

-30

-20

-10

0

10

20

30

40

50

-20 -15 -10 -5 0 5 10 15 20 25

y

x

��& �'�

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 1 2 3 4 5 6

Ste

p S

ize

Time

fp RKAWA

COSY

+��, ��

14.985

14.99

14.995

15

15.005

15.01

15.015

14.985 14.99 14.995 15 15.005 15.01 15.015

y

x

initial condition box

8.99

8.995

9

9.005

9.01

9.015

9.02

9.025

9.03

14 14.002 14.004 14.006 14.008 14.01 14.012 14.014 14.016 14.018 14.02 14.022

y

x

step 2

3.73

3.735

3.74

3.745

3.75

3.755

3.76

3.765

11.604 11.606 11.608 11.61 11.612 11.614 11.616 11.618 11.62 11.622

y

x

step 4

-1.11

-1.105

-1.1

-1.095

-1.09

-1.085

-1.08

-1.075

5.818 5.82 5.822 5.824 5.826 5.828 5.83 5.832 5.834

y

x

step 8

-1.18

-1.175

-1.17

-1.165

-1.16

-1.155

-1.15

-1.145

1.175 1.18 1.185 1.19 1.195 1.2

y

x

step 12

-0.99

-0.98

-0.97

-0.96

-0.95

-0.94

-0.93

-0.92

-0.435 -0.43 -0.425 -0.42 -0.415 -0.41 -0.405 -0.4 -0.395

y

x

step 16

Taylor Model based Integrator COSY-VIversion 2 (2002-)

• Shrink wrapping algorithm including blunting to control ill-conditioned cases.

• Pre-conditioning algorithms based on the Curvilinear, QR de-composition, and blunting pre-conditioners.

• Capability of weighted order computation, allowing to suppressthe expansion order in transversal variables x.

1e-16

1e-14

1e-12

1e-10

1e-8

1e-6

1e-4

1e-2

1e+0

0 50 100 150 200 250 300

Err

or S

ize

Step Number


PreConditioningnaive TMInterval

Taylor Model based Integrator COSY-VIversion 3 (2007-)

•More economical one time step integration using the referencetrajectory and the Lie derivative based flow operator on thedeviation equations.

• Non aborting mechanism when prohibited arithmetic happenssuch as 1/f for 0 ∈ f.

• Improvement of step size control.• Error parametrization of Taylor models.• Dynamic domain decomposition.

Error Parametrization of Taylor modelsMotivation: Is it possible to absorb the remainder error boundintervals of Taylor models into the polynomial parts using addi-tional parameters?

Phrase the question as the following problem:

1. Have Taylor models with 0 remainder error interval, which de-pend on the independent variables x and the parameters α.

T0 = P0(x, α) +−−→[0, 0].

2. Perform Taylor model arithmetic on T0, namely F (T0)

F (T0) = P (x, α) + IF , where IF 6=−−→[0, 0].

3. Try to absorb IF into the polynomial part that depends on α

P (x, α) + IF ⊆ P 0(x, α) +−−→[0, 0]. (A)

Error AbsorptionWe limit the explicitly α-dependent part Pα(x, α) to be onlylinearly dependent on α, and express IF by the matrix form.

Pα(x, α) + IF ⊆³cM + cM(x)´ · α + ³bIF + bIF (x)´ · β.

where (bIF )ii = |IFi| , bIF (x) = 0. The problem is now to find a setsum of two parallelepipeds. Choose a favoured collection ofv column vectors bL + bL(x) using the Psum algorithm.

Pα(x, α) + IF ⊆³bL + bL(x)´ · α + ³ bE + bE(x)´ · β

⊆ bL ◦ h³bI + bL−1 ◦ bL(x)´ · α + bB · βiwhere bB is diagonal, ( bB)ii = |bound((bL−1 ◦ ( bE + bE(x)) · β)i)|.If the diagonal terms of

³bI + bL−1 ◦ bL(x)´ are positive,Pα(x, α) + IF ⊆

³bL+ bL(x) + bL ◦ bB´ · α.

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Psum of Org Parallelpiped (0.4,0.15)-(0.2,0.13) and I-box 0.05-0.05

OrgI-boxbasisPsum

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Psum of Org Parallelpiped (0.4,0.15)-(0.2,0.13) and I-box 0.07-0.07

OrgI-boxbasisPsum

Cost of Additional Parameters

For a v dimensional system, we need v parameters α to absorbTaylor model remainder error bound intervals. The dependenceon α is limited to linear. So, we use weighted DA. Choose anappropriate weight order w for α.

• The dependence on α has to be kept linear. Namely 2 ·w > n,where n is the computational order of Taylor models. Choose

w = Int³n2

´+ 1.

Maximum size necessary for DA and TM for v = 2.

n v DA TM |13 2 105 140 |21 2 253 304 |33 2 595 670 |

v DA TM2 + 2 2380 24192 + 2 12650 127052 + 2 66045 66124

⇒w vw DA TM7 2 + 2w 161 20011 2 + 2w 385 44017 2 + 2w 901 980

1e-16

1e-14

1e-12

1e-10

1e-8

1e-6

1e-4

1e-2

1e+0

0 50 100 150 200 250 300

Err

or S

ize

Step Number


Error parametrizationPreConditioningnaive TMInterval

Dynamic Domain DecompositionFor extended domains, this is natural equivalent to step sizecontrol. Similarity to what’s done in global optimization.1. Evaluate ODE for ∆t = 0 for current flow.

2. If resulting remainder boundR greater than ε, split the domainalong variable leading to longest axis.

3. Absorb R in the TM polynomial part using the error parame-trization method. If it fails, split the domain along variableleading to largest x dependence of the error.

4. Put one half of the box on stack for future work.Things to consider:• Utilize "First-in-last-out" stack; minimizes stack length. Spe-cial adjustments for stack management in a parallel environ-ment, including load balancing.

• Outlook: also dynamic order control for dependence on initialconditions

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

4.08 4.1 4.12 4.14 4.16 4.18 4.2

The Duffing EquationThe equation describes a damped and driven oscillator.Exhibits sensitive dependence on initial conditions and chaoticity.

+ + + 3 = cos()

Example: Study

=

= − − 3 + cos()

with = 025 = 03

for ∈ [0 ] ( ) ∈ [−2 2]× [−2 2]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing. IC split map. 12x12 ICs. VIRDA=0.50. 343 Objs. min_length=2.083e-2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing. Time 0 to pi. 12x12 ICs. VIRDA=0.50. 343 Objs

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing eq. x’=y, y’=x-delta*y-x^3+gamma*cos(t), delta=0.25, gamma=0.3, 12x12 boxes in [-2,2]^2, T=0 (IC)

box 1-3box 27

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing eq. x’=y, y’=x-delta*y-x^3+gamma*cos(t), delta=0.25, gamma=0.3, 12x12 boxes in [-2,2]^2, T=pi/4

box 1-3box 27

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing eq. x’=y, y’=x-delta*y-x^3+gamma*cos(t), delta=0.25, gamma=0.3, 12x12 boxes in [-2,2]^2, T=pi/2

box 1-3box 27

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing eq. x’=y, y’=x-delta*y-x^3+gamma*cos(t), delta=0.25, gamma=0.3, 12x12 boxes in [-2,2]^2, T=3*pi/4

box 1-3box 27

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Duffing eq. x’=y, y’=x-delta*y-x^3+gamma*cos(t), delta=0.25, gamma=0.3, 12x12 boxes in [-2,2]^2, T=pi

box 1-3box 27


Rigorous flow integrations of large ranges of initial con-ditions have been computed using Taylor model basedODE integrators, particularly by COSY-VI version 3.

Example: Flow computations of the standard Lorenzequations for an area of initial condition

( )|0 = ([−40 40] [−50 50] [−25 75])

-40 -30 -20 -10 0 10 20 30 40

-60-40-20 0 20 40 60-40

-20

0

20

40

60

80

z

Lorenz IC:[-40,40]x[-50,50]x[-25,75] T=0.1

xy

z

-40 -30 -20 -10 0 10 20 30 40-60

-40-20

0 20

40 60-40

-20

0

20

40

60

80

z

Lorenz IC:[-40,40]x[-50,50]x[-25,75] T=0.1

x

y

z


Rigorous flow integrations of large ranges of initial con-ditions have been computed using Taylor model basedODE integrators, particularly by COSY-VI version 3.

Example: Flow computations of the standard Lorenzequations for an area of initial condition

( )|0 = ([−40 40] [−50 50] [−25 75])

Example: Rigorous mapping of the Lorenz system tostudy dynamics onPoincare surfaces. Particularly com-pute the second full return Poincaremaps to the surface = − 1 with a wide range of dependence.

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-15-10

-5 0

5 10

15 20

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z

Lorenz, point integrations for the 2nd return to z=27 plane from the top

x

y

z

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

20 30 40 50 60 70 80 90 100

retu

rn ti

me

rho

Return time of BOXY boxes

2 Return min2 Return max1 Return min1 Return max

Rigorous Flows and Poincare Projections

Rigorous flow computations using COSY-VI ver 3• Integrate until an approximate return time •The last time step is arranged such that the initialTaylor models () of the time step is entirely beforecrossing the surface and the solution Taylor models() is entirely after crossing the surface.

•Using the explicit time dependent TMsolution ( )of the last time step, generate the position dependentreturn time () as a Taylor model.

• ( ()) provides a projected solution up to thecomputation order; finally the small TM remainderis projected.

-20 -15 -10 -5 0 5 10 15 20-25-20

-15-10

-5 0

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Lorenz, VI integration of IC1 piece and corner point integrations

p1p2p3p4

IC1

x

y

z

Outlook

•Conduct Poincare projections more frequently,possibly at every time step.

• Improvement of the Taylor model arithmetic packagein COSY to allow arbitrarily high precision Taylormodel computations.

• Improvement of COSY-VI,associated to above and else.

Taylor Model-based Veri ﬁed Integrators Kyoko Makino and ... · Kyoko Makino and Martin Berz...

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