NicolasDelanoue-S´ebastienLagrange R to R Aguaranteednumericalmethodtoclassifysmoothmappingsfrom

(1)

Introduction to classification Stable mappings of the plane and their singularities

Interval analysis and mappings fromR²toR². Computing the Apparent Contour Conjecture, Application and Conclusion

A guaranteed numerical method to classify smooth mappings from R ² to R ²

Nicolas Delanoue - S´ ebastien Lagrange

LARIS - Universit´e d’Angers - France

SMART 2014

1st Small Symposium on Set-Membership: Applications, Reliability and Theory The University of Manchester, Aerospace Research Institute, Manchester, UK

http://www.umari.manchester.ac.uk/aboutus/events/

Nicolas Delanoue - S´ebastien Lagrange LARIS - Universit´e d’Angers - FranceA guaranteed numerical method to classify smooth mappings fromR²toR²

(2)

Outline

1

Introduction to classification Objects, Equivalence, Invariants Discretization - Portrait of a map

2

Stable mappings of the plane and their singularities Stable maps

Whitney Theorem - Normal forms

Compact simply connected with boundary

3

Interval analysis and mappings from R

²

to R

²

.

4

Computing the Apparent Contour

5

Conjecture, Application and Conclusion Conclusion

(3)

Objects, Equivalence, Invariants The one dimentional case Discretization - Portrait of a map

Objects

The set of square matrices of order n (denoted by M

_n

)

Equivalence

We can defined an equivalence relation ∼ between elements of M

_n

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

(4)

Objects

The set of square matrices of order n (denoted by M

_n

) Equivalence

We can defined an equivalence relation ∼ between elements of M

_n

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

(5)

Objects

The set of square matrices of order n (denoted by M

_n

) Equivalence

We can defined an equivalence relation ∼ between elements of M

_n

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

(6)

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

_n

, A 6∼ B and sp(A) = sp(B )

Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

(7)

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

_n

, A 6∼ B and sp(A) = sp(B ) Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

(8)

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

_n

, A 6∼ B and sp(A) = sp(B ) Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

(9)

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Eigenvalues,

Jordanisation

Real bilinear A ∼ B ⇔ ∃U, V ∈ SO , A = UBV Singularvalues forms

Smooth maps f ∼ f

⁰

if there exist diffeomorphic ? changes of variables (g , h) on X and Y such that f = g ◦ f

⁰

◦ h

(10)

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Eigenvalues, Jordanisation

Real bilinear A ∼ B ⇔ ∃U, V ∈ SO , A = UBV Singularvalues forms

Smooth maps f ∼ f

⁰

if there exist diffeomorphic ? changes of variables (g , h) on X and Y such that f = g ◦ f

⁰

◦ h

(11)

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Eigenvalues, Jordanisation

Real bilinear A ∼ B ⇔ ∃U, V ∈ SO , A = UBV Singularvalues forms

Smooth maps f ∼ f

⁰

if there exist diffeomorphic ? changes of variables (g , h) on X and Y such that f = g ◦ f

⁰

◦ h

(12)

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

⁻¹

Eigenvalues, Jordanisation

Real bilinear A ∼ B ⇔ ∃U, V ∈ SO , A = UBV Singularvalues forms

Smooth maps f ∼ f

⁰

if there exist diffeomorphic ? changes of variables (g , h) on X and Y such that f = g ◦ f

⁰

◦ h

(13)

Global picture

One wants a global “picture” of a given map which does not depend on a choice of system of coordinates neither on the source set nor on the target set.

Definition - Equivalence

Let f and f

⁰

be two smooth maps. Then f ∼ f

⁰

if there exists diffeomorphisms g : X → X

⁰

and h : Y

⁰

→ Y such that the diagram

X −−−−→

^f

Y



 y

^g

x



^h

X

⁰ ^f

0

−−−−→ Y

⁰

commutes.

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Global picture

One wants a global “picture” of a given map which does not depend on a choice of system of coordinates neither on the source set nor on the target set.

Definition - Equivalence

Let f and f

⁰

be two smooth maps. Then f ∼ f

⁰

if there exists diffeomorphisms g : X → X

⁰

and h : Y

⁰

→ Y such that the diagram

X −−−−→

^f

Y



 y

^g

x



^h

X

⁰ ^f

0

−−−−→ Y

⁰

commutes.

(15)

Example

R −−−−→

^2x+6

R



 y

^g

x



^h

R −−−−→

^x+1

R

(16)

Example

R −−−−→

^2x+6

R



 y

^x+2

x



^2y

R −−−−→

^x+1

R

(17)

Examples

1

f

1

(x) = x

²

, f

2

(x) = ax

²

+ bx + c, a 6= 0 f

₁

∼ f

₂

2

f

₁

(x) = x

²

+ 1, f

₂

(x) = x + 1, f

1

6∼ f

2

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Examples

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Examples

(20)

Proposition

Suppose that f ∼ f

⁰

with

x

1 f

−−−−→ y

1



 y

^g

x



^h

x

₂ ^f

0

−−−−→ y

₂

then f

⁻¹

({y

₁

}) is homeomorphic to f

⁰⁻¹

({y

₂

}).

(21)

Proposition

Suppose that f ∼ f

⁰

with

x

₁

−−−−→

^f

y

₁



 y

^g

x



^h

x

₂ ^f

0

−−−−→ y

₂

then rank df

x1

= rank df

_x⁰₂

. Proof

Chain rule, df = dh · df

⁰

· dg

(22)

Definition

Let us defined by S

_f

the set of critical points of f : S

_f

= {x ∈ X | df (x) is singular }.

Corollary

f ∼ f

⁰

⇒ S

_f

' S

_f⁰

where ' means homeomorphic.

i.e. the topology of the critical points set is an invariant.

(23)

This is not a strong enough invariant,

there exists smooth maps f , f

⁰

: [0, 1] → [0, 1] such that S

_f

' S

_f⁰

and f 6∼ f

⁰

.

Figure : Singularity theory.

(24)

This is not a strong enough invariant,

there exists smooth maps f , f

⁰

: [0, 1] → [0, 1] such that S

_f

' S

_f⁰

and f 6∼ f

⁰

.

Figure : Topology of X .

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

Definition - Abstract simplicial complex

Let N be a finite set of symbols {(a

⁰

), (a

¹

), . . . , (a

ⁿ

)}

An abstract simplicial complex K is a subset of the powerset of N satisfying : σ ∈ K ⇒ ∀σ

₀

⊂ σ, σ

₀

∈ K

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K =

(a

⁰

), (a

¹

), (a

²

), (a

³

), (a

⁴

), (a

⁰

, a

¹

), (a

¹

, a

²

), (a

⁰

, a

²

), (a

³

, a

⁴

),

(a

⁰

, a

¹

, a

²

)

This will be denoted by a

⁰

a

¹

a

²

+ a

³

a

⁴

(35)

Definition

Given abstract simplicial complexes K and L, a simplicial map F : K

⁰

→ L

⁰

is a map with the following property :

(a

⁰

, a

¹

, . . . , a

ⁿ

) ∈ K ⇒ (F (a

⁰

), F (a

¹

), . . . , F (a

ⁿ

)) ∈ L.

(36)

Example - Simplicial map

K = a

0

a

1

+ a

1

a

2

+ a

2

a

3

, L = b

0

b

1

+ b

1

b

2

b

b b

F : a

⁰

7→ b

⁰

a

¹

7→ b

¹

a

²

7→ b

²

a

³

7→ b

¹

(37)

Example - NOT a Simplicial map

K = a

0

a

1

+ a

1

a

2

+ a

2

a

3

, L = b

0

b

1

+ b

1

b

2

b

b b

F : a

⁰

7→ b

⁰

a

¹

7→ b

¹

a

²

7→ b

²

a

³

7→ b

⁰

(38)

Example - Simplicial map

K = a

0

a

1

a

2

+ a

1

a

2

a

3

, L = b

0

b

1

b

2

b

F : a

⁰

7→ b

⁰

a

¹

7→ b

¹

a

²

7→ b

²

a

³

7→ b

⁰

(39)

Definition

Let f and f

⁰

be continous maps. Then f and f

⁰

are topologically conjugate if there exists homeomorphism g : X → X

⁰

and h : Y → Y

⁰

such that the diagram

X −−−−→

^f

Y



 y

^g

x



^h

X

⁰ ^f

0

−−−−→ Y

⁰

commutes.

Proposition

f ∼ f

⁰

⇒ f ∼

₀

f

⁰

(40)

Definition

Let f be a smooth map and F a simplicial map, F is a portrait of f if

f ∼

₀

F

(41)

Example - Simplicial map The simplicial map

b

b b

is a portrait of [−4, 3] 3 x 7→ x

²

− 1 ∈ R

-4 -3 -2 -1 1 2 3

(42)

Introduction Stable maps

Whitney Theorem - Normal forms Compact simply connected with boundary

Proposition

For every closed subset A of R

ⁿ

, there exists a smooth real valued function f such that

A = f

⁻¹

({0})

We are not going to consider all cases . . .

(43)

Proposition

For every closed subset A of R

ⁿ

, there exists a smooth real valued function f such that

A = f

⁻¹

({0}) We are not going to consider all cases . . .

(44)

Definition

Let f be a smooth map, f is stable if their exists a nbrd N

_f

such that

∀f

⁰

∈ N

_f

, f

⁰

∼ f

Examples

1

g : x 7→ x

²

is stable,

2

f

0

: x 7→ x

³

is not stable, since with f : x 7→ x(x

²

− ), 6= 0 ⇒ f

6∼ f

₀

.

(45)

Whitney Theorem - 1955

Let X and Y be 2-dimentional manifolds and f be generic. The critical point set S

_f

is a regular curve. With p ∈ S

_f

, one has

T

p

S

_f

⊕ ker df

p

= T

p

X or T

p

S

_f

= ker df

p

Geometric representation

1

if T

p

S

f

⊕ ker df

p

= T

p

X ,

2

if T

p

S

_f

= ker df

p

,

(46)

Geometric representation

1

T

p

S

f

⊕ ker df

p

= T

p

X : fold point

2

T

p

S

p

= ker df

p

: cusp point

(47)

Geometric representation

1

T

p

S

f

⊕ ker df

p

= T

p

X : fold point

2

T

p

S

p

= ker df

p

: cusp point

(48)

Normal forms

1

If T

_p

S

_f

⊕ ker df

_p

= T

_p

X , then there exists a nbrd N

_p

such that

f |N

_p

∼ (x, y ) 7→ (x, y

²

).

2

If T

_p

S

_f

= ker df

_p

, then there exists a nbrd N

_p

such that f |N

_p

∼ (x, y) 7→ (x , xy + y

³

).

(49)

Definition

Let f a smooth map from X → R

²

with X a simply connected compact subset of R

²

with smooth boundary ∂X . The apparent contour of f is

f (S

f

∪ ∂X )

The topology of the Apparent contour is an invariant.

(50)

Definition

Let f a smooth map from X → R

²

with X a simply connected compact subset of R

²

with smooth boundary ∂X . The apparent contour of f is

f (S

f

∪ ∂X )

The topology of the Apparent contour is an invariant.

(51)

Definition

Let f a smooth map from X → R

²

with X a simply connected compact subset of R

²

with smooth boundary ∂X . The apparent contour of f is

f (S

f

∪ ∂X )

The topology of the Apparent contour is an invariant.

(52)

Definition

Let f a smooth map from X → R

²

with X a simply connected compact subset of R

²

with smooth boundary ∂X . The apparent contour of f is

f (S

f

∪ ∂X )

The topology of the Apparent contour is an invariant.

(53)

Definition

Let f a smooth map from X → R

²

with X a simply connected compact subset of R

²

with smooth boundary ∂X . The apparent contour of f is

f (S

f

∪ ∂X )

The topology of the Apparent contour is an invariant.

(54)

Theorem (Global properties of generic maps)

Let X be a compact simply connected domain of R

²

with

∂X = Γ

⁻¹

({0}). A generic smooth map f from X to R

²

has the following properties :

1

S

f

is regular curve. Moreover, elements of S are folds and cusp. The set of cusp is discrete.

(55)

Theorem

2

3 singular points do not have the same image,

3

2 singular points having the same image are folds points and they have normal crossing.

(56)

Theorem

4

3 boundary points do not have the same image,

5

2 boundary points having the same image cross normally.

(57)

Theorem

6

3 different points belonging to S

f

∪ ∂X do not have the same image,

7

If a point on the singularity curve and a boundary have the same image, the singular point is a fold and they have normal crossing.

(58)

Theorem

8

if the singularity curve intersects the boundary, then this point is a fold,

9

moreover tangents to the singularity curve and boundary curve are different.

(59)

Cusp Fold - Fold Boundary - Boundary Boundary - Fold

(60)

Proposition

Let f be a smooth generic map from X to R

²

, let us denote by c the map defined by :

c : X → R

²

p 7→ df

p

ξ

p

(1)

where ξ is the vector field defined by ξ

_p

=

∂

2

det df

p

−∂

₁

det df

_p

. If c(p) = 0 and dc

_p

is invertible then p is a simple cusp. This sufficient condition is locally necessary.

(61)

Interval Newton method

c : X → R

²

p 7→ df

_p

ξ

_p

NicolasDelanoue-S´ebastienLagrange R to R Aguaranteednumericalmethodtoclassifysmoothmappingsfrom

A guaranteed numerical method to classify smooth mappings from R 2 to R 2

Nicolas Delanoue - S´ ebastien Lagrange

Outline

Introduction to classification Objects, Equivalence, Invariants Discretization - Portrait of a map

Stable mappings of the plane and their singularities Stable maps

Whitney Theorem - Normal forms

Compact simply connected with boundary

Interval analysis and mappings from R

to R

.

Computing the Apparent Contour

Conjecture, Application and Conclusion Conclusion

Objects

The set of square matrices of order n (denoted by M

)

Equivalence

We can defined an equivalence relation ∼ between elements of M

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

Objects

The set of square matrices of order n (denoted by M

) Equivalence

We can defined an equivalence relation ∼ between elements of M

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

Objects

The set of square matrices of order n (denoted by M

) Equivalence

We can defined an equivalence relation ∼ between elements of M

with

A ∼ B ⇔ ∃P ∈ GL, A = PBP

Invariant

The set of eigenvalues is an invariant because A ∼ B ⇒ sp(A) = sp(B )

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

, A 6∼ B and sp(A) = sp(B )

Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

, A 6∼ B and sp(A) = sp(B ) Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

Invariant

The set of eigenvalues is not a strong enough invariant since

∃A, B ∈ M

, A 6∼ B and sp(A) = sp(B ) Example

A =

0 0 0 0

and B =

0 1 0 0

A really strong invariant

Let us call by J the Jordan method, we have A ∼ B ⇔ J(A) = J(B)

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

Eigenvalues,

Jordanisation

Real bilinear A ∼ B ⇔ ∃U, V ∈ SO , A = UBV Singularvalues forms

Smooth maps f ∼ f

if there exist diffeomorphic ? changes of variables (g , h) on X and Y such that f = g ◦ f

◦ h

Objects Equivalence Invariants

Square matrices A ∼ B ⇔ ∃P ∈ GL, A = PBP

Eigenvalues, Jordanisation

A guaranteed numerical method to classify smooth mappings from R ² to R ²