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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT)

Introduction to Interval and Taylor Model Methods

Markus Neher

KIT – The Research University in the Helmholtz Association

www.kit.edu

Outline

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

1 Enclosure methods 2 Interval analysis 3 Taylor models 4 Interval and Taylor model methods for ODEs

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Enclosure Methods

Enclosure Methods

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Also called: guaranteed methods, rigorous methods, validated methods, verified methods, . . . Aim: Compute guaranteed bounds for the solution of a problem, including

Discretization errors (ODEs, PDEs, optimization), Truncation errors (Newton’s method, summation), Roundoff errors.

Used for

Modelling of uncertain data, Sensitivity analysis, Bounding of roundoff errors.

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval Arithmetic

Interval Arithmetic

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Compact real intervals (Warmus 1956, Sunaga 1958, Moore&Yang 1959, Moore 1962, 1966): IR = {x = [x, x] | x ≤ x} (x, x ∈ R). Width: w(x) = x − x. Hausdorff distance: q(x, y) = q [x, x], [y, y] = max x − y

• , |x − y|
• .

Interval vectors, interval matrices:

Analogously. x = (xi) ∈ IRn: w(x) =

n

max

i=1 w(xi).

Interval Arithmetic

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Basic arithmetic operations: x ◦ y := {x ◦ y | x ∈ x, y ∈ y},

• ∈ {+, −, ∗, /}

(0 ∈ y for /). x + y = [x + y, x + y], x − y = [x − y, x − y], x ∗ y = [min{xy, xy, xy, xy, }, max{xy, xy, xy, xy, }], x / y = x ∗ [1 / y, 1 / y]. No inverses: [0, 1] − [0, 1] = [−1, 1], [1, 2]/[1, 2] = 1 2, 2

• .

Floating point IA: x y =

• x

y, x y ], etc.

Ranges and Inclusion Functions

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Range of f : D → E: Rg (f, D) := {f(x) | x ∈ D}. Inclusion function F : IR → IR of f : D ⊆ R → R: F(x) ⊇ Rg (f, x) for all x ⊆ D. Examples:

Rational functions: For f : x → f(x), replace each occurrence of x by x, evaluate using interval arithmetic: f(x) = x 1 + x ⇒ F [0, 1] = [0, 1] 1 + [0, 1] = [0, 1] [1, 2] = [0, 1]. Elementary functions: For f ∈ {exp, ln, cos, arcsin, tanh, . . . } let F(x) := Rg (f, x): exp [−1, 1] =

• e−1,

e

• ,

sin [1, 2] =

• sin 1, 1
• .

IA: Dependency

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Multiple instances of the same variable are considered independent. f(x) = x 1 + x = 1 − 1 1 + x , x = [0, 1]:

x 1 + x = [0, 1] [1, 2] = [0, 1], 1 − 1 1 + x = 1 − 1 [1, 2] = 1 − 1 2, 1 =

• 0, 1

2 = Rg (f, x).

Theorem

Let f : D ⊂ Rm → R be Lipschitz-continuous and let x ⊆ x0 ⊆ D. Then q

• F(x), Rg (f, x)

≤ γw(x), γ ≥ 0.

Ranges and Inclusion Functions

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Centered form: f(x) = f(c) + g(x, c)(x − c) ⇒ Rg (f, x) ⊆ F(c) + G(x, c)(x − c). Mean value form: Rg (f, x) ⊆ f(c) + F ′(x)(x − c), c ∈ x. f(x) = x 1 + x , x = [0, 1], c = 1 2 : f 1 2

• + F ′([0, 1])
• [0, 1] − 1

2

• =
• − 1

6, 5 6

• ⊃
• 0, 1

2

• = Rg (f, x) .

Theorem

Let f : D ⊂ Rm → R be a sufficiently smooth function and be represented in the centered form. Furthermore, let x ⊆ x0 ⊆ D. Then q

• f(x), Rg (f, x)

≤ γ

• w(x)

2, γ ≥ 0.

IA: Wrapping Effect

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Enclosing non-intervals by intervals causes overestimation. Example: f : (x, y) → √ 2 2 (x + y, y − x) (Rotation) Interval evaluation of f on [−1, 1], [−1, 1]

• :

−2 −1 1 2 −2 −1 1 2 x y −2 −1 1 2 −2 −1 1 2 x y

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor Models

Taylor Models

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor model Tn,f = (pn,f , in,f ) of order n of f : x → R (Berz and Hoffstätter 1994, Makino and Berz 1996): ∀x ∈ x : f(x) ∈ pn,f (x − x0) + in,f Rg (p, i) =

• p(x) + ξ | x ∈ x, ξ ∈ i

⊂ Rm. Example: x = [−1, 1], x0 = 0: ex = 1 + x + 1 2x2 + 1 6x3 + 1 24x4eξ, x, ξ ∈ x, T3,ex = 1 + x + 1

2x2 + 1 6x3 + [0.015, 0.114],

x ∈ x.

Taylor Models

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Minimized dependency: pn,f is processed symbolically to order n. Higher order terms are enclosed into the interval of the result. Minimized wrapping effect: TMs can represent non-convex sets. Example:

T2,f =

• x1

2 + x2

1 + x2

• ,

xi ∈ [−1, 1]. −1

1 2 x1 x2

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval and Taylor Model Methods for IVPs

Initial Value Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IVP: u′ = f(t, u), u(t0) = u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ Rm, tend > t0

Initial Value Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IVP: u′ = f(t, u), u(t0) = u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ Rm, tend > t0

Initial Value Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IVP: u′ = f(t, u), u(t0) = u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ Rm, tend > t0

Initial Value Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IVP: u′ = f(t, u), u(t0) = u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ Rm, tend > t0

Verified Initial Value Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

IVP: u′ = f(t, u), u(t0) = u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ Rm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Interval Method for IVP with Uncertainty

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(t, u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend] f : R × Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0

Taylor Method for IVPs

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Autonomous IVP: u′ = f(u), u(t0) = u0, where f : D ⊂ Rm → Rm, f ∈ Cn(D), u0 ∈ D Taylor method: u(t) =

n

∑

k=0

(t − t0)k k! u(k)(t0) + Rn Automatic (recursive) computation of Taylor coefficients: u(0) = f [0](u) = u, u(1) = f [1](u) = f(u), 1 k!u(k) = f [k](u) = 1 k

• ∂f [k−1]

∂u f

• (u) for k ≥ 2

Moore’s enclosure method

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Interval IVP: u′ = f(u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend], where f : D ⊂ Rm → Rm, f ∈ Cn(D), u0 ⊂ D Interval iteration: For j = 1, 2, . . . : A priori enclosure: vj ⊇ u(t) for all t ∈ [tj−1, tj] ("Alg. I") Truncation error:

zj := hn+1

j

f [n+1](vj),

hj = tj − tj−1

u(tj) ∈ uj := uj−1 +

n

∑

k=1

hk

j f [k](uj−1) + zj

("Algorithm II")

Predictor-Corrector Scheme

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

t0 t1 t2 u0 u1 u2 v1 v2 t u

A Priori Enclosures

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Constant a priori enclosure by Picard iteration: Find hj, vj such that uj−1 + [0, hj]f(vj) ⊆ vj Step size restrictions: Explicit Euler steps A priori enclosures using Picard iterations:

Interval polynomials: Lohner 1988, Corliss & Rihm 1996, Makino 1998, Nedialkov & Jackson 2001 Arbitrary interval functions: Rauh, Auer & Hofer 2005

Alternative a priori bounds: Neumaier 1994, N. 1999, N. 2007, Alexandre dit Sandretto & Chapoutot 2016

Refinement step

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

The iteration uj = uj−1 +

n

∑

k=1

hk

j f [k](uj−1) + zj

is width increasing: w(uj) = w(uj−1) +

n

∑

k=1

hk

j w

• f [k](uj−1)

+ w(zj) → Reduce overestimation by improved evaluation of rhs

Modifications of Algorithm II

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Local coordinate systems (Moore 1965, Eijgenraam 1981, Lohner 1988) Zonotopes (Kühn 1998) Hermite-Obreshkov-Method (Nedialkov & Jackson 1998) Implicit methods (Rihm 1998) Runge-Kutta-Methods (Petras & Hartmann 1999, Bouissou & Martel 2006, Alexandre dit Sandretto & Chapoutot 2016)

Dependency Reduction: Direct Interval Method

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Apply mean value form to f [k](uj−1): For fixed uj−1 ∈ uj−1,

• f [k](uj−1) | uj−1 ∈ uj−1
• ⊆ f [k](

uj−1) + J

• f [k](uj−1)

(uj−1 − uj−1), where J

• f [k]

is the Jacobian of f [k] Let I denote the identity matrix and let Sj−1 := I +

n

∑

k=1

hk

0J

• f [k](uj−1)
• ,

zj = hn+1 f [n](vj) Then u(tj; u0) ∈ uj := uj−1 +

n

∑

k=1

hk

j f [k](

uj−1) + zj + Sj−1(uj−1 − uj−1)

Wrapping Effect in Global Error Propagation

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Wrapping effect: Sj−1(uj−1 − uj−1) may overestimate S = {Sj−1(uj−1 − uj−1) | Sj−1 ∈ Sj−1, uj−1 ∈ uj−1} → propagate S as a parallelepiped

• u0 := m(u0), B0r0 = u0 −

u0, B0 = I; for some nonsingular Bj−1:

• uj

=

• uj−1 +

n

∑

k=1

hk

j−1f [k](

uj−1) + m(zj), uj =

• uj−1 +

n

∑

k=1

hk

j−1f [k](

uj−1) + zj + (Sj−1Bj−1)rj−1,         

• uj: approximate point solution for the central IVP

zj: local error; rj: global error

Global Error Propagation

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Global error: rj =

• B−1

j

(Sj−1Bj−1)

• rj−1 + B−1

j

(zj − m(zj)) Direct method: Bj = I Pep method (Eijgenraam, Lohner): Bj = m(Sj−1Bj−1) QR method (Lohner): m(Sj−1Bj−1) = QjRj, Bj := Qj Blunting method (Berz, Makino): Bj = m(Sj−1Bj−1) + εQj, ε > 0 + = ⊂

Wrapping Effect

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Direct method: Pep method: QR method:

Taylor Model Methods for ODEs

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Taylor expansion of solution w.r.t. time and initial values → minimized dependency problem Enclosure sets for flow can be non-convex → minimized wrapping effect Computation of Taylor coefficients by Picard iteration: Parameters describing initial set treated symbolically Interval remainder bounds by fixed point iteration (Makino 1998)

Example: Quadratic Problem

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

u′ = v, u(0) ∈ [0.95, 1.05], v′ = u2, v(0) ∈ [−1.05, −0.95].

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 2
• 1.5
• 1
• 0.5

0.5 1

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5

COSY Infinity AWA

Applications

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Dagstuhl Seminar 16491 Introduction to Interval and Taylor Model Methods Dec 5, 2016

Solar system dynamics, orbits of NEOs (Berz et al.) Space flight simulation (Armellin & Di Lizia) Parametric ODEs in chemistry, biology, engineering (Stadtherr, Lin & Enszer) Control problems in engineering (Rauh, Auer et al.) → other talks