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The Flow of ODEs
Fabian Immler & Christoph Traut ITP 2016
λ → ∀
=
I s a b e l l e
β α
Introduction
Motivation
◮ Lorenz attractor, chaos
1 / 14
The Flow of ODEs Fabian Immler & Christoph Traut ITP 2016 e l - - PowerPoint PPT Presentation
The Flow of ODEs Fabian Immler & Christoph Traut ITP 2016 e l - - PowerPoint PPT Presentation
The Flow of ODEs Fabian Immler & Christoph Traut ITP 2016 e l l e b a s I = Introduction Motivation Lorenz attractor, chaos 1 / 14 Introduction Motivation Lorenz attractor, chaos Tuckers
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof)
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program
Contribution
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program
Contribution
◮ formalization of flow:
general theory for dependence on initial conditions
1 / 14
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Introduction
Motivation
◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program
Contribution
◮ formalization of flow:
general theory for dependence on initial conditions
◮ use existing verified ODE-solver [Immler, TACAS 2015]:
bounds on variational equation
1 / 14
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Structure
Flow Dependence on Initial Condition Numerics
2 / 14
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Structure
Flow Dependence on Initial Condition Numerics
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE) t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems
◮ formalize flow ϕ(x0, t):
solution w.r.t. initial condition t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems
◮ formalize flow ϕ(x0, t):
solution w.r.t. initial condition
◮ formalize dependence on initial
condition t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems
◮ formalize flow ϕ(x0, t):
solution w.r.t. initial condition
◮ formalize dependence on initial
condition
◮ qualitative: continuous
t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
SLIDE 17
The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems
◮ formalize flow ϕ(x0, t):
solution w.r.t. initial condition
◮ formalize dependence on initial
condition
◮ qualitative: continuous ◮ quantitative: differentiable
t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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The Flow of ODEs
◮ ordinary differential equation
(ODE)
◮ [Immler, H¨
- lzl @ ITP 2012]:
initial value problems
◮ formalize flow ϕ(x0, t):
solution w.r.t. initial condition
◮ formalize dependence on initial
condition
◮ qualitative: continuous ◮ quantitative: differentiable
t ∈ R x ∈ Rn ˙ x(t) = f (x(t))
3 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
4 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
◮ continuous ϕ 4 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
◮ continuous ϕ ◮ differentiable ϕ 4 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
◮ continuous ϕ ◮ differentiable ϕ
◮ technicalities demand “a firm and
extensive background in the principles of real analysis.”
4 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
◮ continuous ϕ ◮ differentiable ϕ
◮ technicalities demand “a firm and
extensive background in the principles of real analysis.”
◮ proofs in chapter 17 4 / 14
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Formalization
◮ continuity and differentiability are
“natural” properties (chapter 7):
◮ continuous ϕ ◮ differentiable ϕ
◮ technicalities demand “a firm and
extensive background in the principles of real analysis.”
◮ proofs in chapter 17
◮ interface to the rest of the theory
that hides technical constructions
4 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X) t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
◮ maximal existence interval ex-ivl
t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
◮ maximal existence interval ex-ivl
◮ t∗ ∈ ex-ivl(x1) ◮ t∗ ∈ ex-ivl(x2)
t∗ x1 x2 ˙ x(t) = f (x(t))
5 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
◮ maximal existence interval ex-ivl
◮ t∗ ∈ ex-ivl(x1) ◮ t∗ ∈ ex-ivl(x2)
Theorem (flow solves IVP)
For t ∈ ex-ivl(x0): t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
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The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
◮ maximal existence interval ex-ivl
◮ t∗ ∈ ex-ivl(x1) ◮ t∗ ∈ ex-ivl(x2)
Theorem (flow solves IVP)
For t ∈ ex-ivl(x0):
◮ ˙
ϕ(x0, t) = f (ϕ(x0, t)) t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
SLIDE 31
The Interface: ex-ivl and ϕ
◮ locally Lipschitz continuous
f : Rn → Rn (on open set X)
◮ ϕ(x0, t) :=
“unique solution of IVP ˙ x(t) = f (x(t)) ∧ x(0) = x0”
◮ maximal existence interval ex-ivl
◮ t∗ ∈ ex-ivl(x1) ◮ t∗ ∈ ex-ivl(x2)
Theorem (flow solves IVP)
For t ∈ ex-ivl(x0):
◮ ˙
ϕ(x0, t) = f (ϕ(x0, t))
◮ ϕ(x0, 0) = x0
t x0 ϕ(x0, t) ˙ x(t) = f (x(t))
5 / 14
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Flow property
t t + s x
Theorem (Flow property)
(t ∈ ex-ivl(x) ∧ s ∈ ex-ivl(ϕ(x, t))) = ⇒ ϕ(x, t + s) = ϕ(ϕ(x, t), s)
6 / 14
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Structure
Flow Dependence on Initial Condition Numerics
7 / 14
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Structure
Flow Dependence on Initial Condition Numerics
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
continuous-on [0; a] g = ⇒ ∀t. 0 ≤ g(t) ≤ C + K · t g(s) ds = ⇒ ∀t ∈ [0; a]. g(t) ≤ C · eK·t
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
◮ exponential sensitivity
O(et) x1 x2 t ∈ ex-ivl(x1) ∩ ex-ivl(x2) = ⇒ |ϕ(x1, t) − ϕ(x2, t)| ∈ O(et)
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
◮ exponential sensitivity ◮ same existence interval in neighborhood
t∗ x1 x2
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
◮ exponential sensitivity ◮ same existence interval in neighborhood
t∗ x1
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
◮ exponential sensitivity ◮ same existence interval in neighborhood ◮ continuous ϕ at (x1, t∗)
t∗ x1 ∀ε > 0. ∃δ. ϕ(Uδ(x1, t∗)) ⊆ Uε(ϕ(x1, t∗))
8 / 14
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Technical Lemmas
◮ Gr¨
- nwall lemma
◮ exponential sensitivity ◮ same existence interval in neighborhood ◮ continuous ϕ at (x1, t∗) ◮ continuity w.r.t. right-hand side of ODE
˙ x(t) = f (x(t)); ˙ x(t) = g(x(t)) |f − g| < ε = ⇒ |ϕf (x1, t) − ϕg(x1, t)| ∈ O(et)
8 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1 u(t) ϕ(x0, t) t x0
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) derivative of f : R → R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
Theorem (derivative of flow)
∂ϕ ∂x (x0, t) = u(t)
9 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with f ′(x) : R
Variational Equation (R)
- ˙
u(t) = f ′(ϕ(x0, t)) · u(t) u(0) = 1
10 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with Df |x : Rn×n
Variational Equation (Rn)
- ˙
u(t) = Df |ϕ(x0,t)·u(t) u(0) = 1L
10 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with Df |x : Rn×n
Variational Equation (Rn)
- ˙
u(t) = Df |ϕ(x0,t)·u(t) u(0) = 1L
requires: normed vector space of linear functions
◮ mathematics in Isabelle/HOL is type class based
◮ topological, metric, vector, normed spaces are type classes 10 / 14
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Differentiability
ODE ˙ x(t) = f (x(t)) with Df |x : Rn×n
Variational Equation (Rn)
- ˙
u(t) = Df |ϕ(x0,t)·u(t) u(0) = 1L
requires: normed vector space of linear functions
◮ mathematics in Isabelle/HOL is type class based
◮ topological, metric, vector, normed spaces are type classes ◮ type of (bounded/continuous) linear functions 10 / 14
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Structure
Flow Dependence on Initial Condition Numerics
11 / 14
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Structure
Flow Dependence on Initial Condition Numerics
12 / 14
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Numerics
◮ encode derivative of flow as linear ODE
12 / 14
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Numerics
◮ encode derivative of flow as linear ODE ◮ [Immler, NFM2013/TACAS 2015]:
verified numerical enclosures for solutions of ODEs
12 / 14
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Numerics
◮ encode derivative of flow as linear ODE ◮ [Immler, NFM2013/TACAS 2015]:
verified numerical enclosures for solutions of ODEs
◮ van der Pol equations:
˙ x = y ˙ y = (1 − x2)y − x (x0, y0) = (1.25, 2.27)
12 / 14
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- 13 / 14
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- 13 / 14
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- 13 / 14
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- 13 / 14
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Conclusion
◮ clean interface: flow ϕ, ex-ivl
14 / 14
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Conclusion
◮ clean interface: flow ϕ, ex-ivl ◮ hides tedious technical constructions
14 / 14
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Conclusion
◮ clean interface: flow ϕ, ex-ivl ◮ hides tedious technical constructions ◮ employ existing verified algorithm for numerical enclosures
14 / 14
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Conclusion
◮ clean interface: flow ϕ, ex-ivl ◮ hides tedious technical constructions ◮ employ existing verified algorithm for numerical enclosures ◮ general theory with concrete application:
◮ Lorenz attractor ◮ step towards formal verification of Tucker’s proof 14 / 14