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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples On the Taylor models approach for solving ODEs Dr. Florian Bnger florian.buenger@tuhh.de Institute for Reliable

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

On the Taylor models approach for solving ODEs

  • Dr. Florian Bünger

florian.buenger@tuhh.de

Institute for Reliable Computing Hamburg University of Technology Schwarzenberg-Campus 3 D-21073 Hamburg Germany November 30th, 2017

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Outline

1

Introduction

2

Definition and arithmetic of Taylor models

3

Algorithm for solving ODEs

4

Accompanying methods

5

Numerical examples

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Taylor models have been used successfully to calculate verified inclusions of the solutions of initial value problems for ODEs. This was done by

  • Makino (PhD 1998) and Berz and their group

implementation: COSY INFINITY [Fortran]

  • Eble (PhD 2007, Univ. Karlsruhe, Alefeld, Lohner, Neher)

implementation: Riot [C++]

  • Dzetkuliˇ

c (2012) implementation: ODEintegrator [C++]

  • Xin Chen et al. (PhD 2015, RWTH Aachen)

implementation: FLOW [C++]

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Especially Berz, Makino, and their group focused on Taylor models and invented several accompanying methods like shrink wrapping blunting preconditioning error parametrization dynamic domain decomposition We give a short description of definition and arithmetic of Taylor models main algorithm steps for solving ODEs shrink wrapping and preconditioning numerical results

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Definition of a Taylor model

dimension: k ∈ N := {1, 2, . . . }

  • rder:

m ∈ N0 := {0, 1, 2, . . . } domain: D ∈ IRk := {[a, b] :=

k

×

i=1

[ai, bi] | a, b ∈ Rk, a ≤ b}

error interval: I ∈ IR polynomial part: p =

α∈Nk |α|≤m

pαxα ∈ R[x1, . . . , xk]

where |α| := k

i=1 αi

and xα := k

i=1 xαi i

evaluation point: x0 ∈ D Taylor model (of order m): p + I := {f ∈ C0(D, R) | ∀x ∈ D : f(x) ∈ p(x − x0) + I}

where p(x − x0) + I := {p(x − x0) + y | y ∈ I}

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underlying idea: p is the Taylor polynomial of order m with evaluation point x0

  • f some f ∈ Cm+1(D, R)

such that R(x) := f(x) − p(x − x0) ∈ I for all x ∈ D in short: p + I is an m-th order Taylor model for f ∈ Cm+1(D, R). This explains the name “Taylor model”. BUT: The formal definition does not (!) necessitate any Taylor expansion of some unknown function f in the background.

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Taylor model vectors are analogously denoted by p + I: p = (p1, . . . , pn) ∈ (R[x1, . . . , xk])n I = I1 × · · · × In ∈ IRn is an interval vector pi + Ii is a Taylor model as defined before, i = 1, . . . , n The range of p + I is p(D − x0) + I = {p(x − x0) + y | x ∈ D, y ∈ I} ⊆ Rn. Example:

k := 2 =: n, m := 3, x0 := (0, 0), D = [−1, 1]2, I := [−0.05, 0.05]2 p1(x1, x2) := −0.2x3

2 + 0.9x1,

p2(x1, x2) := 0.3x2

1 x2 + x2 + 0.1

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Now, arithmetic operations +, −, ×, ÷ standard functions exp, log, sin, cos, ... integration w.r.t. one of the n variables can be defined on m-th order Taylor models with fixed D and x0. Examples: a) Summation / Subtraction: (p + I) ± (q + J) := (p ± q) + (I ± J) b) Multiplication: (p + I) ∗ (q + J) := r + K c) Exponential function: exp(p + I) := r + K d) Antiderivative:

  • p + I dxi := r + K

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b) Multiplication: (p + I) ∗ (q + J) := r + K pq = r + s , deg(r) ≤ m , s: terms of order > m (p + I)(q + J) = pq + pJ + I(q + J) = r + (s + pJ + I(q + J))

  • to be contained in K

s(D − x0) + p(D − x0)J + I(q(D − x0) + J) ⊆ K1 ∈ IR By symmetry: s(D − x0) + q(D − x0)I + J(p(D − x0) + I) ⊆ K2 ∈ IR K := K1 ∩ K2

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c) Exponential function: exp(p + I) := r + K The image of p is thought to be centered around c := p(0). Thus, only a Taylor expansion of exp around c makes sense: exp(x) = exp(c)

m

  • i=0

(x − c)i i!

  • q(x):=

+ (x − c)(m+1) (m + 1)! exp(ξ)

  • R(x):=

, ξ ∈ c+[0, 1](x −c). q(p + I) = r + J , deg(r) ≤ m , J ∈ IR exp(p + I) = q(p + I) + R(p + I) = r + (J + R(p + I))

  • to be contained in K

p(D − x0) + I ⊆ K1 ∈ IR R(K1) = (K1−c)(m+1)

(m+1)!

exp(c) exp([0, 1](K1 − c)) ⊆ K2 ∈ IR K := J + K2

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d) Antiderivative:

xi

  • x0i

p + I dxi := r + K, xi ∈ [xi, xi] Integrate p with respect to xi. The resulting polynomial ˆ p has degree at most m + 1. Split ˆ p = r + s such that s contains all terms of degree m + 1.

xi

  • x0i

p + I dxi ⊆ r + s(D − x0) + I([xi, xi] − x0i)

  • to be contained in K ∈ IR

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IIVP: y′ = f(y, t) ∈ Rn, t ∈ (t0, te) (1) y(t0) ∈ Y0 ∈ IRn Solve ODE stepwise on appropriate subintervals [ti, ti+1],

t0 < t1 < · · · < tN = te, N ∈ N.

A Taylor model n-vector p(i)(x, t) + I(i) is supplied such that for all y0 ∈ Y0 the solution ˜ y of (1) with ˜ y(t0) = y0 fulfills ˜ y(t) ∈ p(i)([−1, 1]n, t − ti) + I(i) ∀t ∈ [ti, ti+1], i = 0, . . . , N − 1.

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Example: Lotka-Volterra equations: y′

1 = 2y1(1 − y2)

y′

2 = y2(y1 − 1)

y1(0) ∈ [0.95, 1.05] y2(0) ∈ [2.95, 3.05]

Remark: The Taylor model method supplies enclosures at all points t ∈ [ti, ti+1]

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Standard form: dimension: k := n + 1 dim of ODE + 1 domain: D := [−1, 1]n × [ti, ti+1], B := [−1, 1]n evaluation point: x0 := (0, . . . , 0, ti) The range p(i)(B, ti+1 − ti) + I(i) = p(i+1)(B, 0) + I(i+1) (only implicitly given) is taken as set of initial conditions for the next integration step from ti+1 to ti+2. The error interval vectors I(0), I(1), . . . , I(N), which cover all numerical and rounding errors, may strongly grow due to the wrapping effect of interval arithmetic.

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Main steps of integration algorithm

IIVP: y′ = f(y, t) ∈ Rn, t ∈ (t0, te) y(t0) ∈ Y0 = y0 + [−r, r] ∈ IRn, r ∈ Rn

>0

1

q(x, t) = q(x) := y0 + diag(r)x = (y0i + rixi)1≤i≤n

2

J := [0, 0] ∈ IRn

3

q(B) = Y0 Range of q + J

4

K(p + I) := q + J +

t

  • t0

f(p + I, τ) dτ Integral operator

5

K(p + I) ⊆ p + I ⇒

Schauder

y(t) ∈ p(B, t − t0) + I ∀t ∈ [t0, t1]

6

Set q(x) := p(x, t1 − t0), J := I for next integration on [t1, t2] .

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Construction of p + I satisfying K(p + I) ⊆ p + I Step 1: Picard-Iteration p(0) + I(0) := q + J p(i+1) + I(i+1) := K(p(i) + I(i)), i = 1, . . . m − 1 p := p(m) polynomial part K(p + I′) = p + I′′ Step 2: Error inclusion by stepwise ε-inflation I(0) := [0, 0] ∈ IRn p + I′

(i) := K(p + I(i))

ε-inflation with εrel, εabs > 0: I(i+1) := [1 − εrel, 1 + εrel] · I′

(i) + [−εabs, εabs]

If I′

(i) ⊆ I(i)

⇒ I := I′

(i)

If inclusion cannot be obtained, then the step size is decreased.

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Shrink wrapping aims to eliminate the error terms I(i) at t = ti. The coefficients of the polynomials p(i)

j (x) := p(i) j (x, 0), j = 1, . . . , n,

are slightly perturbed so that the perturbation ˜ p(i) fulfills p(i)(B) + I(i) ⊆ ˜ p(i)(B) + J, J ≈ [0, 0]n ∈ IRn. Example:

p1(x1, x2) := −0.2x3

2 + 0.9x1,

p2(x1, x2) := 0.3x2

1 x2 + x2 + 0.1,

I := 0.05B ˜ p1(x1, x2) := −0.23x3

2 + 1.035x1,

˜ p2(x1, x2) := 0.345x2

1 x2 + 1.15x2 + 0.115

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Definition (Makino and Berz [9], Definition 2)

Let M = I + S + I, where S is a polynomial and I is a small interval. We include I into the interval box d · [−1, 1]v. We pick numbers s and t satisfying s ≥ |Si(x)| ∀x ∈ B, 1 ≤ i ≤ v, (2) t ≥

  • ∂Si(x)

∂xj

  • ∀x ∈ B, 1 ≤ i, j ≤ v.

(3) We call a map M shrinkable if (1 − vt) > 0 and (1 − s) > 0; (4) both of which can be achieved if S (and since it is a polynomial, also its derivative) is sufficiently small in magnitude. Then we define q, the so-called shrink wrap factor, as q = 1 + d · 1 (1 − (v − 1)t) · (1 − s) . (5)

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Explanation of notation: dimension: v ∈ N (corresponds to n) domain: B := [−1, 1]v radius: d ∈ R>0 error “interval”: I ⊆ d · B = d · [−1, 1]v = [−d, d]v ⊆ IRv polynomials: I, S ∈ (R[x1, . . . , xv])v I(x1, . . . , xv) := (x1, . . . , xv) “identity” Taylor model: M = I + S + I, P := I + S polynomial part, I error interval vector

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Theorem (Makino, Berz [9], Theorem 3, Shrink Wrapping)

Let M = I + S(x), where I is the identity. Let I = d · [−1, 1]v, and R = I +

  • x∈B

M(x) be the set sum of the interval I = [−d, d]v and the range of M over the original domain box B. Let q be the shrink wrap factor of M; then we have R ⊆

  • x∈B

(qM)(x) (6) and hence multiplying M with the number q allows to set the remainder bound to zero.

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Explanation of notation: polynomial part: M = I + S (in contrast to Def. 2) error “interval”: I = [−d, d]v (enclosure of I in Def. 2) Taylor model: M + I = I + S + I Conclusion of Theorem 3: M(B) + I ⊆ (qM)(B) The range of M + I is contained in that of the shrink wrapped Taylor model (with zero error interval vector) qM := (qM1, . . . , qMv) where q is the shrink wrap factor as in (5) of Def. 2.

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Shrink wrapping as described by Makino and Berz starts with transforming an arbitrary Taylor model to a normalized form M = I + S such that the following additional condition holds true: S(0) = 0 and S has “small” linear coefficients in magnitude. (7) This is not included in Def. 2 and also not used in the proof of Th. 3. We state two counterexamples to Theorem 3. For the first, condition (7) is not satisfied. The second fulfills (7) in the strong way that the constant and all linear coefficients of S are zero.

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Example 1 v := 2, S1 :≡ 0, S2(x1, x2) := 0.24x2

1 − 0.99,

d := 0.0052 s := 0.99 and t := 0.48 fulfill conditions (2), (3), (4) of Def. 1. The corresponding shrink wrap factor is: q := 1+ d (1 − (v − 1)t)(1 − s) = 1+ d (1 − t)(1 − s) = 1+ 0.0052 0.52 · 0.01 = 2 For M := I + S we have M(B) (qM)(B). Thus, M(B) + I (qM)(B) where I := [−d, d]2. This contradicts Theorem 3.

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M(B) (qM)(B) y = M(1, 1)

0.5 1 1.5 x 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x 2

M(B) (qM)(B) y = M(1, 1)

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Example 2 v := 2, α ∈ (0, 1/4), S1 :≡ 0, S2(x1, x2) := α(x2

1 − x2 2), d ∈ (0, 1/20]

Note that the constant and linear terms of S1 and S2 are zero! s := α and t := 2α fulfill conditions (2), (3), (4) of Def. 1. The corresponding shrink wrap factor is: q := 1 + d (1 − (v − 1)t)(1 − s) = 1 + d (1 − 2α)(1 − α) For M := I + S and I := [−d, d]2 we have M(B) + I (qM)(B). This contradicts Theorem 3.

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Ansatz: y := M(1, 1) = (1, 1)T ∈ M(B) z := y + (−d, d)T = (1 − d, 1 + d)T ∈ M(B) + I BUT: z ∈ (qM)(B) Illustration: α := 0.1, d := 0.005

0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 x 1 0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 x 2

y z

M(B) (qM)(B) z=(1-d,1+d)

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New shrink wrapping version Difference (to shrink wrapping by Makino and Berz): B is stretched by a new shrink wrap factor q before M is applied. ⇒ shrink wrapped range is M(qB) and not qM(B). The new shrink wrapping method is firstly presented in a simple form: partial derivatives are uniformly bounded by some t > 0 (This is similar to Def. 2) simple, “uniform” error interval vector I := rB, r > 0

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Lemma

Let n ∈ N, B := [−1, 1]n, r ∈ R≥0, t ∈ [0, 1/n), q := 1 +

r 1−nt .

Suppose f : qB → Rn is a continuously differentiable function such that g(x) := f(x) − x satisfies

  • ∂gi(x)

∂xj

  • ≤ t

for all x ∈ qB and all i, j ∈ {1, . . . , n}. Then f(B) + rB ⊆ f(qB) . Proof: Take ˆ x ∈ B and v ∈ rB and note that ˆ x + (q − 1)B ⊆ qB. Define h : ˆ x + (q − 1)B → Rn, x → f(ˆ x) − g(x) + v. For x ∈ ˆ x + (q − 1)B the mean value theorem yields: h(x)−ˆ x = g(ˆ x)−g(x)+v ≤ ntˆ x−x+v ≤ nt(q−1)+r = q−1 Hence, h(ˆ x + (q − 1)B) ⊆ ˆ x + (q − 1)B. Brouwer’s fixed point theorem supplies an ˜ x ∈ ˆ x + (q − 1)B ⊆ qB s.t. ˜ x = h(˜ x) = f(ˆ x) − g(˜ x) + v. This means f(ˆ x) + v = ˜ x + g(˜ x) = f(˜ x) ∈ f(qB).

  • Florian Bünger

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General form of new shrink wrapping method: more separate restrictions on partial derivatives zero-symmetric error interval vector with distinct component radii Notation is less obvious but much better results are obtained.

Lemma

Let n ∈ N, B := [−1, 1]n, r, s ∈ Rn

≥0,

q := ✶ + r + s. Suppose f : qB → Rn is continuously differentiable such that g(x) := f(x) − x satisfies |g′(x)|(q − ✶) ≤ s for all x ∈ qB.

  • ⇔ sup

x∈qB

|g′

i (x)|(q − ✶) = sup x∈qB n

  • j=1
  • ∂gi

∂xj (x)

  • (qj − 1) ≤ si

for all i ∈ {1, . . . , n}

  • Then

f(B) + rB ⊆ f(qB) . published article:

Shrink wrapping for Taylor models revisited, Numerical Algorithms,

  • pp. 1-17, First Online: 30 Sept. 2017

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Preconditioning of Taylor models p(x) + I := p(i)(x, 0) + I(i), x ∈ B = [−1, 1]n p(B) + I initial values for integration on [ti, ti+1] Factorization: p(x) + I ∈ (pℓ(x) + Iℓ)

  • left Taylor Model
  • (pr(x) + Ir)
  • right Taylor Model

∀x ∈ B important: pr(B) + Ir ⊆ B = domain of pℓ(x) p(B) + I = pℓ(pr(B) + Ir) + Iℓ ⊆ pℓ(B) + Iℓ Integrating pℓ + Iℓ on [ti, ti+1] yields Taylor model p∗

ℓ (x, t) + I∗ ℓ , s.t.

y(t) ∈ p∗

ℓ (pr(B) + Ir, t − ti) + I∗ ℓ

y′ = f(t, y), y0 ∈ Y0, t ∈ [ti, ti+1]. in short: (p∗

ℓ + I∗ ℓ ) ◦ (pr + Ir) factorizes the ODE-flow on [ti, ti+1].

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p∗

ℓ (x) := p∗ ℓ (x, ti+1 − ti),

x ∈ B = [−1, 1]n Write p∗

ℓ (x) =

Ax

  • linear part, A ∈ Rn,n

+ q(x)

  • nonlinear part

+ c

  • constant part = p∗

ℓ (0)

. Factorize A = QR. Possible choices: a) parallelepiped method: Q := A, R := eye(n) b) QR-factorization c) curvilinear coordinates chosen w.r.t f. r+J := Q−1((p∗

ℓ −c+I∗ ℓ )◦(pr+Ir)) = (Rx+Q−1q(x))◦(pr+Ir)+Q−1I∗ ℓ

Choose diagonal matrix S s.t. S(r(B) + J) ⊆ B. pr,new + Ir,new := S(r + J) pℓ,new(x) := QS−1x + c Iℓ,new := [0, 0] ∈ IRn,n Factorization for [ti+1, ti+2]: (pℓ,new + Iℓ,new) ◦ (pr,new + Ir,new)

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A solver verifyode is implemented in MATLAB/INTLAB. Plan : available in 2018 (middle to end of year) Up to now it includes shrink wrapping, blunting, and preconditioning (QR-,parallelepiped method). Here we show results for I) Van der Pol equation II) Motion of Asteroid 1997 XF11 in solar system Results are compared with: AWA developed by Lohner [7] COSY-VI developed by Berz and his group [2] RIOT developed by Eble [5] in his PhD thesis

Remark: RIOT uses shrink wrapping as specified by Makino and Berz. ⇒ also RIOT may not produce correct, verified results.

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I) Van der Pol equation y′

1

= y2 y′

2

= (1 − y2

1 )y2 − y1

interval initial conditions at t0 := 0: y1(0) ∈ [2.999, 3.001] y2(0) ∈ [−3.001, −2.999] end of integration te := 10 Taylor order m := 10

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2 4 6 8 10 t

  • 3
  • 2
  • 1

1 2 3 4 y Component y1(t) 2 4 6 8 10 t

  • 4
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  • 2
  • 1

1 2 3 y Component y2(t)

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AWA COSY-VI RIOT verifyode method 4 None/Blunting QR/Blunting sw QR [y1]

  • 5.983
  • 6.288E-1
  • 5.245
  • 7.020E-1
  • 6.044

229E-1

  • 6.044

230E-1

  • 6.044

230E-1

  • 6.044

229E-1

[y2]

  • 2.627

40

  • 2.593

662

  • 2.630

7

  • 2.630

7

  • 2.628

40

  • 2.630

7

d([y1]) 3.048E-2 1.775E-1 1.849E-2 1.856E-2 1.860E-2 1.850E-2 d([y2]) 1.331E-2 6.893E-2 7.219E-3 7.253E-3 1.218E-2 7.298E-3 time [s] 0.46 20.79 3.64 53.66 24.10 11.48

Remark: AWA can integrate upto te = 10 only with method 4: 0: interval 1: parallelepiped 2: QR-decomposition 3: intersection of 0 and 1 4: intersection of 0 and 2

Florian Bünger Taylor models 40 / 45

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

II) Motion of Asteroid 1997 XF11 in solar system y′

1 = y4

y′

4 = −γ y1 d

d := (y2

1 + y2 2 + y2 3 )3/2

y′

2 = y5

y′

5 = −γ y2 d

γ := 0.9986 y′

3 = y6

y′

6 = −γ y3 d

y1, y2, y3: (x, y, z)-coord. of XF11 in solar system with center = sun. The components y3+i = y′

i ,

i = 1, 2, 3, are their velocities. interval initial conditions at t0 := 0 :

y1(0) ∈ −1.77269098191512 ± 0.5 · 10−7 y2(0) ∈ 0.1487214852342955 ± 0.5 · 10−7 y3(0) ∈ −0.07928350462244194 ± 0.5 · 10−7 y4(0) ∈ 0.2372031916516237 ± 0.5 · 10−6 y5(0) ∈ 0.612524538758628 ± 0.5 · 10−6 y6(0) ∈ 0.04583217572165624 ± 0.5 · 10−6

Florian Bünger Taylor models 41 / 45

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

AWA t0 te

  • rder p

step size enclosure method error tolerances 5.5π 11π 18 h0 = 0.0001 4 ǫabs = 1E-16 ǫerr = 1E-16 COSY-VI t0 te

  • rder n

step size local error tolerance 5.5π 11π 10 h0 = 0.1 hmin = 0.001 hmax = 1 1E-11 preconditioning shrink wrapping weighted order None/QR On/Blunting None RIOT t0 te

  • rder n

sparsity tolerance step size control 5.5π 11π 10 1E-20 AUTO h0 = 0.1 hmin = 0.001 local error tolerance

  • rder check

bounder 1E-11 TotalDegree LDB verifyode t0 te

  • rder n

sparsity tolerance step size control 5.5π 11π 10 1E-20 h0 = 0.1 hmin = 0.001 local error tolerance bounder shrink wrapping relative ǫ-inflation 1E-11 NAIVE On 1E-1

Table: Parameter settings test case II)

Florian Bünger Taylor models 42 / 45

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

AWA COSY-VI verifyode None/On QR/Blunting sw QR years 2.75 2.75 2.75 2.75 2.75 [y1]

  • 5.670 980

1 439E-1

  • 5.670 240

2 180E-1

  • 5.670 966

1 453E-1

  • 5.670 990

1 429E-1

  • 5.670 997

1 423E-1

[y2] 1.838 740

32

1.838 747

24

1.838 739

2

1.838 739

2

1.838 739

3

[y3]

  • 1.318 213

63E-1

  • 1.318134

343E-1

  • 1.318 214

63E-1

  • 1.318 213

64E-1

  • 1.318 215

62E-1

[y4]

  • 5.867 503

52E-1

  • 5.867444

612E-1

  • 5.867 506

50E-1

  • 5.867 508

48E-1

  • 5.867 509

47E-1

[y5] 4.998 817

6 838E-2 5.001 823 4.993 833E-2

4.998 851

6 810E-2

4.998 782

6 873E-2

4.998 704

6 951E-2

[y6]

  • 2.628649

789E-2

  • 2.628441

997E-2

  • 2.628648

790E-2

  • 2.628649

789E-2

  • 2.628657

780E-2

d([y1]) 4.581 125E-5 1.938 527E-4 4.856 704E-5 4.376 849E-5 4.252 269E-5 d([y2]) 6.982 230E-6 2.204 949E-5 6.316 541E-6 5.289 746E-6 4.843 246E-6 d([y3]) 4.919 242E-6 2.081 581E-5 4.863 480E-6 5.016 839E-6 4.565 451E-6 d([y4]) 4.791 893E-6 1.670 766E-5 4.245 541E-6 3.877 312E-6 3.666 749E-6 d([y5]) 1.977 308E-5 7.989 099E-5 2.040 883E-5 1.908 225E-5 1.752 760E-5 d([y6]) 1.393 621E-6 5.554 181E-6 1.414 084E-6 1.397 204E-6 1.218 468E-6 time [s] 3.01 2 454.26 1 702.76 250.66 229.54

Florian Bünger Taylor models 43 / 45

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

AWA COSY-VI verifyode None/On QR/Blunting sw QR years 5.5 5.5 5.5 5.5 5.5 [y1]

  • 9.183 529

92449E-1

  • 9.187 245

8 739E-1

  • 9.187 381

8 597E-1

  • 9.187 383

8 595E-1

[y2]

  • 7.427 962

32 376E-1

  • 7.429 834

30 500E-1

  • 7.429 894

30 445E-1

  • 7.429 903

30 435E-1

[y3] 7.680 225

20470E-3

7.655 686

44 973E-3

7.654 751

45 944E-3

7.654 728

45 967E-3

[y4] 9.139 898

4 469E-1

9.137 633

6 730E-1

9.137 553

6 814E-1

9.137 551

6 816E-1

[y5]

  • 4.042 339

7 165E-1

  • 4.044 378

5 129E-1

  • 4.044 446

5 057E-1

  • 4.044 448

5 056E-1

[y6] 6.035 479

4 566E-2

6.035 094

4 952E-2

6.035 085

4 960E-2

6.035 085

4 961E-02

d([y1]) 8.918 362E-4 1.492 804E-4 1.215 158E-4 1.210 497E-4 d([y2]) 4.412 134E-4 6.645 066E-5 5.500 086E-5 5.312 518E-5 d([y3]) 5.975 463E-5 1.071 215E-5 8.806 009E-6 8.759 719E-6 d([y4]) 5.427 170E-4 9.014 262E-5 7.372 523E-5 7.345 154E-5 d([y5]) 4.824 878E-4 7.501 268E-5 6.095 434E-5 6.071 842E-5 d([y6]) 9.119 810E-6 1.413 859E-6 1.247 326E-6 1.230 127E-6 time [s] 5.26

  • 3 427.04

514.07 418.21

Thanks for your attention!

Florian Bünger Taylor models 44 / 45

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Introduction Definition and arithmetic of Taylor models Algorithm for solving ODEs Accompanying methods Numerical examples

References

◮

Chen, X.: Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models, Dissertation at RWTH Aachen University, 2015, Software: https://flowstar.org/dowloads/

◮

Berz, M., et al.: The COSY INFINITY web page, available from www.bt.pa.msu.edu/index_cosy.htm

◮

Bünger, F.: Shrink wrapping for Taylor models revisited, Numerical Algorithms, pp. 1-17, First Online: 30 Sept. 2017.

◮

Dzetkuliˇ c, T. : Rigorous integration of non-linear ordinary differential equations in chebyshev basis, Numer Algor (2015) 69:183–205, Software: https://sourceforge.net/projects/odeintegrator/

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Eble, I.: Über Taylor-Modelle, Dissertation at Karlsruhe Institute of Technology (2007), Software: Riot, C++-implementation, available from www.math.kit.edu/ianm1/~ingo.eble/de

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Hoefkens, J., Berz, M., Makino, K.: Controlling the Wrapping Effect in the Solution of ODEs for Asteroids, Reliable Computing 8, 21-41 (2003)

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Lohner, R.: Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Dissertation at Karlsruhe Institute of Technology (1988)

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Makino, K.: Rigorous analysis of nonlinear motion in particle accelerators, Dissertation at Michigan State University (1998)

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Makino, K., Berz, M.: Suppression of the wrapping effect by Taylor model - based validated integrators, MSU HEP Report 40910 (2003) (online available from www.bt.pa.msu.edu/pub/)

◮

Makino, K., Berz, M.: Suppression of the Wrapping Effect by Taylor Model-based Verified Integrators: Long-term Stabilization by Shrink Wrapping, International Journal of Differential Equations and Applications 10(4), 385-403 (2005)1

◮

Makino, K., Berz, M.: Suppression of the Wrapping Effect by Taylor Model-based Verified Integrators: Long-term Stabilization by Preconditioning, International Journal of Differential Equations and Applications 10(4), 353-384 (2005) (online available from www.bt.pa.msu.edu/pub/)

◮

Neumaier, A.: Taylor Forms - Use and Limits, Reliable Computing 9, 43-79 (2003)

◮

Neher, Jackson, Nedialkov: On Taylor model based integration of ODEs, SIAM J. NUMER. ANAL., Vol. 45, No. 1, pp. 236-262 (2007)

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Rump, S.M.: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)

1For some unknown reason the article cannot be found in the online archive of the International Journal of

Differential Equations and Applications but it is available from the authors’ web page. Florian Bünger Taylor models 45 / 45