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On improvements in Exact Real Arithmetic for Initial Value Problems - - PowerPoint PPT Presentation

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On improvements in Exact Real Arithmetic for Initial Value Problems Franz Braue 1 Margarita Korovina 2 Norbert Mller 1 1 Universitt Trier 2 IIS Novosibirsk CCC, Kochel, 2015-09-17 1 Background and Setting Motivation iRRAM 2 Algorithm 3

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SLIDE 1

On improvements in Exact Real Arithmetic for Initial Value Problems

Franz Brauße1 Margarita Korovina2 Norbert Müller1

1 Universität Trier 2 IIS Novosibirsk

CCC, Kochel, 2015-09-17

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SLIDE 2

1 Background and Setting

Motivation iRRAM

2 Algorithm 3 Radius of Convergance

Picard-Lindelöf’s method Improved by Integrals Iterative Improvement

4 Countering wrapping effects

Lipschitz bounds reducing wrapping Taylor Models

5 Future Work and References

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 2 / 20

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SLIDE 3

Background and Setting Motivation

Why another approach to solving IVPs?

Goal

  • Provide reliable solutions up to arbitrary accuracy, efficiently!

Setting: Computable Analysis

  • Theoretical foundation for computations with continuous objects:

Real Numbers, Functions, Sets, . . . Result: IVP-solver in iRRAM

  • C++ implementation of concepts from Computable Analysis.
  • x ∈ R is represented as sequence (ci + eiI)i converging to x where

ci, ei dyadic and I = [−1, 1].

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 3 / 20

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SLIDE 4

Background and Setting iRRAM

Provide methods / tools suitable for use by engineers w/o in-depth knowledge about Real computation, but who assume to perform those. iRRAM works on names:

  • Cauchy: REAL
  • τTM: TM, linear multivariate poly w/ interval coeffs. T : Ik → Rd, k

can vary over course of computation due to polishing

  • τA for other wrappings A?

Algorithms dependant on actual rep- resentation of Reals (low level):

  • arithmetic
  • lim
  • evaluation of power series
  • Lipschitzify

Algorithms work independent of con- crete backend/representation (high level):

  • elementary funs
  • solving PIVP systems (including

bounds to R, M, L)

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 4 / 20

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SLIDE 5

Background and Setting iRRAM

Real Computation

computation

− − − − − − − →

accuracy

← − − − − − x ± 2−21 y ± 2−19 ↑ x ± 2−53 y ± 2−50 z ± 2−45 · · · x ± 2−140 y ± 2−136 z ± 2−128 · · · . . . . . . . . . ...

  • Computable Analysis: complete table
  • Numerical Computation: horizontal line
  • iRRAM: finite path

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

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SLIDE 6

Background and Setting iRRAM

Real Computation

computation

− − − − − − − →

accuracy

← − − − − − x ± 2−21 y ± 2−19 ↑ x ± 2−53 y ± 2−50 z ± 2−45 · · · x ± 2−140 y ± 2−136 z ± 2−128 · · · . . . . . . . . . ...

  • Computable Analysis: complete table
  • Numerical Computation: horizontal line
  • iRRAM: finite path

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

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SLIDE 7

Background and Setting iRRAM

Real Computation

computation

− − − − − − − →

accuracy

← − − − − − x ± 2−21 y ± 2−19 ↑ x ± 2−53 y ± 2−50 z ± 2−45 · · · x ± 2−140 y ± 2−136 z ± 2−128 · · · . . . . . . . . . ...

  • Computable Analysis: complete table
  • Numerical Computation: horizontal line
  • iRRAM: finite path

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

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Background and Setting Polynomial ODE systems

Polynomial ODE system d-dimensional polynomial f : R × Rd → Rd in d + 1 variables, then solution

  • y : R → Rd described by ODE system

d dt y(t) = f(t, y(t)) Usually we also have an initial value y0 = y(t0) at some t0.

  • virtually all real-world ODE systems are describable by poly right hand

side f

  • example systems: Van-der-Pol oscillator, n-Body problem, double

pendulum, Lorentz attractor, . . .

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 6 / 20

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Background and Setting Polynomial ODE systems

Example: Van-der-Pol equation, α = 3

˙ y1 = y2 y1(0) = 1 ˙ y2 = αy2 − y1 − αy2

1y2

y2(0) = 1

020406080 100

  • 2
  • 1

1 2

  • 4
  • 2

2 4 y2(t) vdp-3 t y1(t) y2(t)

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 7 / 20

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Algorithm

1 Background and Setting

Motivation iRRAM

2 Algorithm 3 Radius of Convergance

Picard-Lindelöf’s method Improved by Integrals Iterative Improvement

4 Countering wrapping effects

Lipschitz bounds reducing wrapping Taylor Models

5 Future Work and References

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 8 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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SLIDE 18

Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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SLIDE 19

Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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SLIDE 20

Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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SLIDE 21

Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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SLIDE 22

Algorithm

Approach to solving such IVP systems

Classical power series method, e.g. derivable through Picard iteration

3m + 1 initial value ym → sequence of Taylor series coeffs ( an)n 3m + 2 bounds R on radius of convergence and M : an ≤ MR−n ❀ bounded truncation error ❀ power series for y(t) while t ∈ (tm ± R) 3m + 3 choose tm+1 and evaluate ym+1 =

n

antn

m+1

recursion

  • n

coefficients summation power series condition initial evaluation inside of circle of convergence sum function

solution y(t) time t

Good bounds on radius are crucial:

  • step size
  • fewer Taylor series coefficients need to be computed

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

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Radius of Convergance

1 Background and Setting

Motivation iRRAM

2 Algorithm 3 Radius of Convergance

Picard-Lindelöf’s method Improved by Integrals Iterative Improvement

4 Countering wrapping effects

Lipschitz bounds reducing wrapping Taylor Models

5 Future Work and References

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 10 / 20

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Radius of Convergance Picard-Lindelöf’s method

Original approach by P-L

Fix some δ and select compact region around initial value: Cε = {(t0 + t, w0 + w) : |t| ≤ δ ∧ w ≤ ε} p(ε) = max f(Cε) ❀ RPL(ε) = min{δ, ε/p(ε)} Leaves option to choose ε. . .

δ R ε

  • w0

t0 p(ε)

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 11 / 20

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SLIDE 25

Radius of Convergance Picard-Lindelöf’s method

Original approach by P-L

Fix some δ and select compact region around initial value: Cε = {(t0 + t, w0 + w) : |t| ≤ δ ∧ w ≤ ε} p(ε) = max f(Cε) ❀ RPL(ε) = min{δ, ε/p(ε)} Leaves option to choose ε. . .

δ R ε

  • w0

t0 p(ε)

020406080 100

  • 2
  • 1

1 2

  • 4
  • 2

2 4 y2(t) vdp-3 t y1(t) y2(t)

0.2 0.4 11 13 15 17

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 11 / 20

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SLIDE 26

Radius of Convergance Improved by Integrals

p(ε) monotonically increasing ❀ ε

0 1/p(s) ds ≥ ε/p(ε)

Worst-case is on the boundary. So Rint(0, ε) = min{δ, ε

0 1/p(s) ds}.

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 12 / 20

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Radius of Convergance Improved by Integrals

p(ε) monotonically increasing ❀ ε

0 1/p(s) ds ≥ ε/p(ε)

Worst-case is on the boundary. So Rint(0, ε) = min{δ, ε

0 1/p(s) ds}.

RPL Rint t0

  • w0

ε δ

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 12 / 20

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SLIDE 28

Radius of Convergance Improved by Integrals

p(ε) monotonically increasing ❀ ε

0 1/p(s) ds ≥ ε/p(ε)

Worst-case is on the boundary. So Rint(0, ε) = min{δ, ε

0 1/p(s) ds}.

RPL Rint t0

  • w0

ε δ

0.2 0.4 11 13 15 17

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 12 / 20

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Radius of Convergance Iterative Improvement

Have some R

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 30

Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

  • y usually fluctuates somewhat.

p(s) nor Rint catch that. Taylor series for y(t0+t) defined for |t| < R, t ∈ C.

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 33

Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

  • y usually fluctuates somewhat.

p(s) nor Rint catch that. Taylor series for y(t0+t) defined for |t| < R, t ∈ C. RPL Rint y(t0 + t) t0

  • w0

ε δ

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 34

Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

  • y usually fluctuates somewhat.

p(s) nor Rint catch that. Taylor series for y(t0+t) defined for |t| < R, t ∈ C. Compute truncated series at

c_int(INTERVAL(t0-t,t0+t), INTERVAL(-t,+t))

and get interval enclosing the solution. RPL Rint y(t0 + t) t0

  • w0

ε δ

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 35

Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

  • y usually fluctuates somewhat.

p(s) nor Rint catch that. Taylor series for y(t0+t) defined for |t| < R, t ∈ C. Compute truncated series at

c_int(INTERVAL(t0-t,t0+t), INTERVAL(-t,+t))

and get interval enclosing the solution. Use that to restart Rint compu- tation ❀ Ritr. RPL Rint y(t0 + t) Ritr t0

  • w0

ε δ

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 36

Radius of Convergance Iterative Improvement

Have some R ❀ y([t0 ± R]) evaluable ❀ Iterative increase

  • y usually fluctuates somewhat.

p(s) nor Rint catch that. Taylor series for y(t0+t) defined for |t| < R, t ∈ C. Compute truncated series at

c_int(INTERVAL(t0-t,t0+t), INTERVAL(-t,+t))

and get interval enclosing the solution. Use that to restart Rint compu- tation ❀ Ritr. RPL Rint y(t0 + t) Ritr t0

  • w0

ε δ

0.2 0.4 11 13 15 17

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 13 / 20

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SLIDE 37

Countering wrapping effects

1 Background and Setting

Motivation iRRAM

2 Algorithm 3 Radius of Convergance

Picard-Lindelöf’s method Improved by Integrals Iterative Improvement

4 Countering wrapping effects

Lipschitz bounds reducing wrapping Taylor Models

5 Future Work and References

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 14 / 20

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Countering wrapping effects Lipschitz bounds reducing wrapping

Retrieving and using Lipschitz bounds

Perturbation µ in initial value: w0 ∈ [ w′

0 ± µ]. Affects

y(·, w0) ❀ Lipschitz bound wrt. initial value:

  • y(t,

w0) − y(t, w′

0) ≤ L ·

w0 − w′ Using: operator L : [Rd → [R → Rd]] × Rd × R → [R → Rd], used as

  • y(t) = L(TC( · , R, M), w0, L)(t); essentially L() is apply().

Retrieving: Choose t, |t| ≤ R(µ, ε) and use remaining state space up to η s.t. (t, y(t, [ w0 ± η])) ⊂ Cε to bound fluctuation. Idea: Cauchy on polydisc D := { w0 + w : w∞ ≤ η} ⊂ Cd Fix t, vary just w ∈ [ w0 ± µ], then (ν = 1, . . . , d): D

eiyν(t,

w) =

  • 1

(2πi)

1

  • ∂D

yν(t, ξ) ( ξ − w)

ei+ 1 d

ξ

  • ≤

M · ηd (η − √ 2µ)d+1 =: Lµ(t)

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 15 / 20

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Countering wrapping effects Taylor Models

Taylor Models

Classic version by Makino/Berz, named wrt. Taylor expansion of functions. T( λ) =

  • n c

n

λ

n polynomial in vector of error variables

λ ∈ Ik plus a real remainder interval I; c

n ∈ R

  • Representation of real intervals: ∃λ : T(λ) ∋ x
  • Allows for cancellation: T, T ′ both represent x

= ⇒ T( λ) − T ′( λ) = I − I′ ≈ 0 Usual interval arithmetic for I − I′, though

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 16 / 20

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Countering wrapping effects Taylor Models

Taylor Models

Classic version by Makino/Berz, named wrt. Taylor expansion of functions. T( λ) =

  • n c

n

λ

n polynomial in vector of error variables

λ ∈ Ik plus a real remainder interval I; c

n ∈ R

  • Representation of real intervals: ∃λ : T(λ) ∋ x
  • Allows for cancellation: T, T ′ both represent x

= ⇒ T( λ) − T ′( λ) = I − I′ ≈ 0 Usual interval arithmetic for I − I′, though Generalization: c

n ⊆ R intervals, no remainder interval necessary

  • Allows transparent change of representation T →

T (“polish”) Example: order reduction: c2,1λ2

1λ2 ❀ c2,1I2

  • c0,1

λ2

  • Naturally implementable by types REAL or INTERVAL in iRRAM
  • Integrate seamlessly to Taylor series evaluation: truncation error → ˜

c0

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 16 / 20

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SLIDE 41

Countering wrapping effects Back to Van-der-Pol example, α = 3

TMs allow to catch rapid convergance near attractor

Van-der-Pol, α = 3, different “polishing” strategies

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 17 / 20

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SLIDE 42

Countering wrapping effects Back to Van-der-Pol example, α = 3

TMs allow to catch rapid convergance near attractor

Van-der-Pol, α = 3, different “polishing” strategies

  • 200
  • 180
  • 160
  • 140
  • 120
  • 100
  • 80
  • 60

2000 4000 6000 8000 10000 12000 14000 Precision Steps

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 17 / 20

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SLIDE 43

Countering wrapping effects Back to Van-der-Pol example, α = 3

Timings

method tend steps time initial final bits / [s] precision tend Lipschitz 10 139 131 2−601 2−100 51.1 20 285 655 2−1151 2−94 52.8 50 – – – – – 100 – – – – – Taylor 10 139 37 2−207 2−190 1.7 model, 20 285 71 2−207 2−179 1.4

  • lder sweep

50 711 180 2−207 2−174 0.62 100 1440 344 2−207 2−156 0.51

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 18 / 20

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SLIDE 44

Future Work and References

1 Background and Setting

Motivation iRRAM

2 Algorithm 3 Radius of Convergance

Picard-Lindelöf’s method Improved by Integrals Iterative Improvement

4 Countering wrapping effects

Lipschitz bounds reducing wrapping Taylor Models

5 Future Work and References

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 19 / 20

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SLIDE 45

Future Work and References

Improve speed of IVP-Solver to match double precision implementations. Generalize PIVP-Solver to ˜ HIVP-Solver w/ ˜ H = holom. fns w/ additionally provided comp. modulus of continuity Radius of Convergance:

  • Use higher-order TMs to get better approx. for derivatives.
  • Refine computation towards tighter bounds than Ritr.

Taylor Models:

  • Allow to catch attractors ❀ need better polishing heuristics.
  • Formalize TMs to reason about behaviour in ERA setting.
  • Generalize iRRAM’s methods to TMs (and other representations

besides τTM?)

❀ What properties are necessary to enable efficient handling?

iRRAM Documentation: http://irram.uni-trier.de/ iRRAM Code: https://github.com/norbert-mueller/iRRAM

Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 20 / 20