on improvements in exact real arithmetic for initial … improvements in exact real arithmetic for...

On improvements in Exact Real Arithmeticfor Initial Value Problems

Franz Brauße1 Margarita Korovina2 Norbert Müller1

1 Universität Trier

2 IIS Novosibirsk

CCC, Kochel, 2015-09-17

1 Background and Setting

Motivation

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

Background and Setting Motivation

Why another approach to solving IVPs?

• 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 whereci, ei dyadic and I = [−1, 1].

Background and Setting iRRAM

Provide methods / tools suitable for use by engineers w/o in-depthknowledge 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, kcan 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 (highlevel):

• elementary funs

• solving PIVP systems (includingbounds to R,M, L)

Real Computation

computation−−−−−−−→ac

cy←−−−

−− 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

Real Computation

cy←−−−

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

......

.... . .

Real Computation

cy←−−−

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

......

.... . .

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

~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 handside ~f

• example systems: Van-der-Pol oscillator, n-Body problem, doublependulum, Lorentz attractor, . . .

Background and Setting Polynomial ODE systems

Example: Van-der-Pol equation, α = 3

y1 = y2 y1(0) = 1

y2 = αy2 − y1 − αy21y2 y2(0) = 1

020406080100 -2 -1 0 1 2

Algorithm

Motivation

2 Algorithm

Taylor Models

Algorithm

Approach to solving such IVP systemsClassical power series method, e.g. derivable through Picard iteration

3m+ 1 initial value ~ym 7→ sequence of Taylor series coeffs (~an)n3m+ 2 bounds R on radius of convergence and M : ‖~an‖ ≤MR−n ;

bounded truncation error ; power series for ~y(t) whilet ∈ (tm ± R)

3m+ 3 choose tm+1 and evaluate ~ym+1 =∑n ~ant

recursionon

coefficients

summation

powerseries

conditioninitial

evaluationinside ofcircle of

convergence

sumfunction

solution y(t)

time t

Good bounds on radius are crucial:

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

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Algorithm

recursionon

coefficients

summation

powerseries

conditioninitial

convergence

sumfunction

solution y(t)

time t

Radius of Convergance

Motivation

2 Algorithm

Taylor Models

Radius of Convergance Picard-Lindelöf’s method

Original approach by P-L

Fix some δ and select compact region aroundinitial 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. . .

020406080100 -2 -1 0 1 2

11 13 15 17

Radius of Convergance Picard-Lindelöf’s method

Original approach by P-L

Fix some δ and select compact region aroundinitial 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. . .

020406080100 -2 -1 0 1 2

11 13 15 17

Radius of Convergance Improved by Integrals

p(ε) monotonically increasing ;∫ε01/p(s)ds ≥ ε/p(ε)

Worst-case is on the boundary. So Rint(0, ε) = min{δ,∫ε0 1/p(s)ds}.

RPL Rint

11 13 15 17

RPL Rint

11 13 15 17

RPL Rint

11 13 15 17Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 12 / 20

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) definedfor |t| < R, t ∈ C.

Compute truncated series atc_int(INTERVAL(t0-t,t0+t),

INTERVAL(-t,+t)) and getinterval enclosing the solution.

Use that to restart Rint compu-tation ; Ritr.

RPL Rint

y(t0 + t)

11 13 15 17

Have some R ; y([t0 ± R]) evaluable

; Iterative increase

RPL Rint

y(t0 + t)

11 13 15 17

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

RPL Rint

y(t0 + t)

11 13 15 17

RPL Rint

y(t0 + t)

11 13 15 17

RPL Rint

y(t0 + t)

11 13 15 17

~y usually fluctuates somewhat.p(s) nor Rint catch that.Taylor series for y(t0+t) definedfor |t| < R, t ∈ C.Compute truncated series atc_int(INTERVAL(t0-t,t0+t),

RPL Rint

y(t0 + t)

11 13 15 17

INTERVAL(-t,+t)) and getinterval enclosing the solution.Use that to restart Rint compu-tation ; Ritr.

RPL Rint

y(t0 + t)

11 13 15 17

INTERVAL(-t,+t)) and getinterval enclosing the solution.Use that to restart Rint compu-tation ; Ritr.

RPL Rint

y(t0 + t)

11 13 15 17

Countering wrapping effects

Motivation

2 Algorithm

Taylor Models

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 ηs.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+~1d~ξ

∥∥∥∥∥ ≤ M · ηd

(η−√2µ)d+1

=: Lµ(t)

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) 3 x• Allows for cancellation: T, T ′ both represent x=⇒ T(~λ) − T ′(~λ) = I− I ′ ≈ 0Usual interval arithmetic for I− I ′, though

Generalization: c~n ⊆ R intervals, no remainder interval necessary

• Allows transparent change of representation T 7→ T (“polish”)Example: order reduction: c2,1λ21λ2 ; c2,1I2︸︷︷︸

• Naturally implementable by types REAL or INTERVAL in iRRAM• Integrate seamlessly to Taylor series evaluation: truncation error 7→ c0

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) 3 x• Allows for cancellation: T, T ′ both represent x=⇒ T(~λ) − T ′(~λ) = I− I ′ ≈ 0Usual interval arithmetic for I− I ′, though

Generalization: c~n ⊆ R intervals, no remainder interval necessary

• Allows transparent change of representation T 7→ T (“polish”)Example: order reduction: c2,1λ21λ2 ; c2,1I2︸︷︷︸

• Naturally implementable by types REAL or INTERVAL in iRRAM• Integrate seamlessly to Taylor series evaluation: truncation error 7→ c0

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

TMs allow to catch rapid convergance near attractorVan-der-Pol, α = 3, different “polishing” strategies

0 2000 4000 6000 8000 10000 12000 14000

Precision

TMs allow to catch rapid convergance near attractorVan-der-Pol, α = 3, different “polishing” strategies

0 2000 4000 6000 8000 10000 12000 14000

Precision

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

older sweep 50 711 180 2−207 2−174 0.62

100 1440 344 2−207 2−156 0.51

Future Work and References

Motivation

2 Algorithm

Taylor Models

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/ additionallyprovided 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 representationsbesides τTM?); What properties are necessary to enable efficient handling?

iRRAM Documentation: http://irram.uni-trier.de/

iRRAM Code: https://github.com/norbert-mueller/iRRAM

on improvements in exact real arithmetic for initial … improvements in exact real arithmetic for...

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