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Developmental Partial Differential Equations Nastassia Pouradier Duteil Rutgers University - Camden Kinet Young Researchers Workshop November 30, 2016 Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 1 / 32

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SLIDE 1

Developmental Partial Differential Equations

Nastassia Pouradier Duteil

Rutgers University - Camden

Kinet Young Researchers’ Workshop November 30, 2016

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 1 / 32

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SLIDE 2

Outline

1

Motivation: A description of oogenesis

2

The heat equation on time-varying manifolds

3

A “Lie bracket” between transport and heat

4

Control of growth via a signal

5

Future Directions

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 2 / 32

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SLIDE 3

Motivation: A description of oogenesis

Drosophila oogenesis

Figure: Drosophila melanogaster oogenesis

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 3 / 32

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SLIDE 4

Motivation: A description of oogenesis

Morphogens

Morphogen

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

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SLIDE 5

Motivation: A description of oogenesis

Morphogens

Morphogen

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

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SLIDE 6

Motivation: A description of oogenesis

Morphogens

Morphogen

Figure: “French flag model”

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

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SLIDE 7

Motivation: A description of oogenesis

Morphogens

Morphogen

Figure: “French flag model” Figure: Morphlogies of Drosophila eggshells and Gurken patterning

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 4 / 32

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SLIDE 8

Motivation: A description of oogenesis

Mechanism of Gurken diffusion and internalization

Figure: Gurken diffusion from oocyte nucleus in the perivitelline space and internalization into the follicle cells

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 5 / 32

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SLIDE 9

Motivation: A description of oogenesis

Mechanism of Gurken diffusion and internalization

Figure: Gurken in Drosophila willistoni

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 6 / 32

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SLIDE 10

The heat equation on time-varying manifolds

1

Motivation: A description of oogenesis

2

The heat equation on time-varying manifolds

3

A “Lie bracket” between transport and heat

4

Control of growth via a signal

5

Future Directions

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 7 / 32

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SLIDE 11

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 12

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane v[·] : Pc(Rd) → Lip(Rd, Rd) Growth vector field x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 13

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane v[·] : Pc(Rd) → Lip(Rd, Rd) Growth vector field x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 14

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane v[·] : Pc(Rd) → Lip(Rd, Rd) Growth vector field x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 15

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane v[·] : Pc(Rd) → Lip(Rd, Rd) Growth vector field x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 16

The heat equation on time-varying manifolds

General model

Mt: time-varying compact manifold of dimension n embedded in Rd = Rn+1 Organism’s membrane v[·] : Pc(Rd) → Lip(Rd, Rd) Growth vector field µt ∈ P(Mt): probability measure on Mt (also, µt ∈ Pc(Rd)) Morphogen diffusing in intercellular space x M0 Mt xt φ u w φ#u φ#w

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 8 / 32

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SLIDE 17

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

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SLIDE 18

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

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SLIDE 19

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2 ∆t: Laplace-Beltrami operator on Mt

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

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SLIDE 20

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2 ∆t: Laplace-Beltrami operator on Mt

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 9 / 32

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SLIDE 21

The heat equation on time-varying manifolds

Wasserstein distance: Monge transportation problem

How do you best move a pile of sand to fill up a given hole of the same total volume?

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

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SLIDE 22

The heat equation on time-varying manifolds

Wasserstein distance: Monge transportation problem

Monge’s problem (1781) Given µ, ν ∈ P(X) and c : X × X → R+ a Borel-measurable function, Minimize

  • X

c(x, T(x))dµ(x) among all transport maps T : X → X s.t. T#µ = ν.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

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SLIDE 23

The heat equation on time-varying manifolds

Wasserstein distance: Monge transportation problem

Kantorovich’s formulation (1940’s) Given µ, ν ∈ P(X) and c : X × X → R+ a Borel-measurable function, Minimize

  • X×X

c(x, y)dγ(x, y) where γ ∈ Π(µ, ν) := {ρ ∈ P(X × X) | π1#ρ = µ, π2#ρ = ν}.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 10 / 32

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SLIDE 24

The heat equation on time-varying manifolds

p-Wasserstein distance

Wp(µ, ν) = inf

γ∈Π(µ,ν)

  • Rn×Rn |x − y|pdγ(x, y)

1/p

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 11 / 32

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SLIDE 25

The heat equation on time-varying manifolds

p-Wasserstein distance

Wp(µ, ν) = inf

γ∈Π(µ,ν)

  • Rn×Rn |x − y|pdγ(x, y)

1/p

Figure: Two measures with different L1 and W1 distances (respectively O(1) and O(δ)).

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 11 / 32

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SLIDE 26

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2 ∆t: Laplace-Beltrami operator on Mt

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 12 / 32

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SLIDE 27

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2 ∆t: Laplace-Beltrami operator on Mt

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 12 / 32

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SLIDE 28

The heat equation on time-varying manifolds

Laplace-Beltrami operator

Laplace-Beltrami operator: generalization of the Laplacian on Riemannian manifolds. ∆f := ∇ · ∇f Let (xi)i∈{1,...,n} be a coordinate system on Mt and gt be the metric tensor of Mt. Let f ∈ C∞(Mt). ∆tf = 1

  • |gt|

n

  • i=1

∂ ∂xi (

  • |gt|

n

  • j=1

gij

t

∂ ∂xj f )

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 13 / 32

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SLIDE 29

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) where v: a Lipschitz function w.r.t. the Wasserstein distance W2 ∆t: Laplace-Beltrami operator on Mt

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 14 / 32

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SLIDE 30

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) Theorem (Piccoli, Pouradier Duteil, Rossi) There exists a unique solution to Equation (1).

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 14 / 32

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SLIDE 31

The heat equation on time-varying manifolds

Coupling of diffusion and manifold evolution

Evolution of µt by the combined transport and diffusion: Transport-diffusion PDE ∂tµt + ∇ · (v[µt]µt) = ∆tµt (1) Theorem (Piccoli, Pouradier Duteil, Rossi) There exists a unique solution to Equation (1). Weak formulation For all f ∈ C ∞(R, Rd) ∂t

  • Rd f dµt −
  • Rd(∇f · v[µt])dµt =
  • Mt

∆tf dµt. (2)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 14 / 32

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SLIDE 32

The heat equation on time-varying manifolds

Proof of existence

Sketch of proof (Existence). Introduce a discrete scheme that alternates time steps of transport and diffusion, and prove that it admits a convergent subsequence Prove that the limit is a solution to the PDE (1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 15 / 32

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SLIDE 33

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0. n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 34

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 35

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 36

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 37

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 38

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 39

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 40

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 41

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 42

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 43

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 44

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

slide-45
SLIDE 45

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

slide-46
SLIDE 46

The heat equation on time-varying manifolds

Proof of existence and uniqueness: Discrete scheme

Scheme S Define τn = tn := 2−nT. Let µn(0) := µ0.

1 Let φtn

ltn be the flow of v[µn(ltn)] and consider φtn ltn#µn(ltn).

2 Let µn((l + 1)tn) = e∆(l+1)tnτn(φtn

ltn#µn(ltn)).

n etnv φtn

0 #µn(0)

φtn

tn#µn(tn)

e∆tnτn etnv e∆2tnτn

µn(0) µn(tn) µn(2tn)

n + 1

µn+1(0) µn+1(tn+1) µn+1(2tn+1) µn+1(3tn+1) µn+1(4tn+1)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 16 / 32

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SLIDE 47

A “Lie bracket” between transport and heat

1

Motivation: A description of oogenesis

2

The heat equation on time-varying manifolds

3

A “Lie bracket” between transport and heat

4

Control of growth via a signal

5

Future Directions

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 17 / 32

slide-48
SLIDE 48

A “Lie bracket” between transport and heat

Lie bracket: intuitive example

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 18 / 32

slide-49
SLIDE 49

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

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SLIDE 50

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

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SLIDE 51

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

slide-52
SLIDE 52

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

slide-53
SLIDE 53

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

slide-54
SLIDE 54

A “Lie bracket” between transport and heat

Non commutativity of transport and heat

Definition: “Lie bracket” between transport and heat [∆, v]µ := lim

t=τ→0

Φ−t#

  • eτ∆t(Φt#µ)
  • − eτ∆0µ

tτ with Φt#: push-forward via the flow generated by v eτ∆t: semigroup generated by ∆t at time τ.

µ

φt φt#µ eτ∆t

eτ∆t(φt#µ)

φ−t

φ−t#(eτ∆t(φt#µ))

eτ∆0

eτ∆0(µ)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 19 / 32

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SLIDE 55

A “Lie bracket” between transport and heat

Simple example: Transport of S1

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2

Figure: Transport of S1 by v(x, y) := (x − 1, 2y). At t = 0.25, the resulting ellipse is centered at (1 − e0.25, 0).

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 20 / 32

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SLIDE 56

A “Lie bracket” between transport and heat

Simple example: Discrete scheme

π/2 π 3π/2 2π 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0.099 0.1 Initial signal Diffusion-growth Growth-diffusion

Figure: Iterative diffusion and transport

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 21 / 32

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SLIDE 57

A “Lie bracket” between transport and heat

Simple example: Convergence of the bracket

Figure: Convergence of the numerical approximations of the bracket to the theoretical expression for the initial signals µ0(θ) = 0.1dθ.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 22 / 32

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SLIDE 58

A “Lie bracket” between transport and heat

Simple example: Convergence of the bracket

Figure: Convergence of the numerical approximations of the bracket to the theoretical expression for the initial signal µ0(θ) = 0.1(cos(θ) + 1)dθ.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 22 / 32

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SLIDE 59

Control of growth via a signal

1

Motivation: A description of oogenesis

2

The heat equation on time-varying manifolds

3

A “Lie bracket” between transport and heat

4

Control of growth via a signal

5

Future Directions

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 23 / 32

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SLIDE 60

Control of growth via a signal

Control of manifold evolution

Complete coupling of signal s and manifold r with control of s at a point.      ∂tr = s, ∂ts = ∆rs, s(t, θ = 0) = u(t). (2) where r(t, θ): radius of the cell; s(t, θ): growing signal (solving the heat equation); ∆r: Laplace-Beltrami operator (depending on r); u(t): control (value of s at a given point).

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 24 / 32

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SLIDE 61

Control of growth via a signal

Example: S1

           ∂tr = s, ∂ts = ∆rs, s(t, θ = 0) = u(t), ∂θs(t, θ = π) = 0. (3) with r(0, θ) = 1 (constant radius) and s(0, θ) = 0 (zero signal). The Laplace-Beltrami operator is: ∆rs = 1 r2 + r2

θ

∂2

θs − rrθ + rθ∂2 θr

(r2 + r2

θ )2 ∂θs

(4)

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 25 / 32

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SLIDE 62

Control of growth via a signal

Simulations: constant control

Figure: Simulations with a constant control u ≡ 1.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 26 / 32

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SLIDE 63

Control of growth via a signal

Simulations: sine control

Figure: Simulations with a sinusoidal control.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 27 / 32

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SLIDE 64

Control of growth via a signal

Simulations: growth of circle

Figure: Simulations with a control u(t) = 0.25 sin( 2π

5 t) for t ∈ [0, 2.5] and

u(t) = 0 for t ∈ [2.5, 10].

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 28 / 32

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SLIDE 65

Control of growth via a signal

Controllability

Exact controllability Find a control u : [0; T] → R such that the unique solution of (3) with ∀θ ∈ [0, 2π], r(t = 0, θ) = r0 and s(t = 0, θ) = 0 satisfies ∀θ ∈ [0, 2π], r(t = T, θ) = r1(θ) and s(t = T, θ) = 0. Exact controllability cannot be obtained in general (e.g. for non-smooth configurations). Hence we relax our goal: Approximate controllability Find a control u : [0; T] → R such that the unique solution of (3) with ∀θ ∈ [0, 2π], r(t = 0, θ) = r0(θ) and s(t = 0, θ) = 0 satisfies r(t = T, ·) − r1(·)L2 < ǫ and s(t = T)L2 < ǫ.

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 29 / 32

slide-66
SLIDE 66

Control of growth via a signal

Approximate controllability

Theorem The system            ∂tr = s, ∂ts = ∆rs, s(t, θ = 0) = u(t), ∂θs(t, θ = π) = 0 is approximately controllable for r on [0, T].

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 30 / 32

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SLIDE 67

Future Directions

Future Directions

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 31 / 32

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SLIDE 68

Future Directions

Future Directions

Reaction-diffusion equations - generalized Wasserstein distance

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 31 / 32

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SLIDE 69

Future Directions

Future Directions

Reaction-diffusion equations - generalized Wasserstein distance Explicit control from one shape to another

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 31 / 32

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SLIDE 70

Future Directions

Future Directions

Reaction-diffusion equations - generalized Wasserstein distance Explicit control from one shape to another Cost of control - optimal control given a number of harmonics in Fourier series

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 31 / 32

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SLIDE 71

Future Directions

Thank you for your attention! Any questions?

Nastassia Pouradier Duteil (Rutgers) Developmental PDEs Kinet, Nov. 30, 2016 32 / 32