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THE RECONSTRUCTION OF THE EARLY UNIVERSE: GRAVITATION AND OPTIMAL TRANSPORTATION, A REVIEW Yann BRENIER CNRS-CMLS, Ecole Polytechnique, Palaiseau Optimal Transport in the Applied Sciences, RICAM-JKU, Linz, 8-12 Dec 2014 Yann Brenier (CNRS,

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THE RECONSTRUCTION OF THE EARLY UNIVERSE: GRAVITATION AND OPTIMAL TRANSPORTATION, A REVIEW

Yann BRENIER

CNRS-CMLS, Ecole Polytechnique, Palaiseau

Optimal Transport in the Applied Sciences, RICAM-JKU, Linz, 8-12 Dec 2014

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 1 / 25

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RECONSTRUCTION OF THE EARLY UNIVERSE

Following Peebles 1989, Frisch and coauthors (Nature 417) 2002, have managed to reconstruct the history of the universe from the

  • bservable distribution of galaxies.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 2 / 25

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THE SEMI-NEWTONIAN GRAVITATIONAL MODEL OF THE EARLY UNIVERSE (Zeldovich, Peebles...)

The trajectory t → X(t, a) ∈ T3 = R3/Z3 of each "particle"(*) obeys 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, 1 + t △ ϕ = ρ =

  • T3 δ(x − X(t, a))da

where ρ(t, x) and ϕ(t, x) respectively denote the density field (supposed to be of unit average) and the gravitational potential.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 3 / 25

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THE SEMI-NEWTONIAN GRAVITATIONAL MODEL OF THE EARLY UNIVERSE (Zeldovich, Peebles...)

The trajectory t → X(t, a) ∈ T3 = R3/Z3 of each "particle"(*) obeys 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, 1 + t △ ϕ = ρ =

  • T3 δ(x − X(t, a))da

where ρ(t, x) and ϕ(t, x) respectively denote the density field (supposed to be of unit average) and the gravitational potential. General relativity is taken into account only through the terms in red which take into account Big Bang effects.

(*) for computations, a "particle" roughly corresponds to a ...cluster of galaxies!

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 3 / 25

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THE INITIAL CONSTRAINTS

Because of the Big-Bang terms, the equations 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X(t, a)) = 0, 1 + t △ ϕ = ρ =

  • δ(x − X(t, a))da

are degenerate at t = 0. At initial time, we get a continuum of

  • particles. Their density is uniform and their initial velocity is

"slaved" by the gravitational potential:

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25

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THE INITIAL CONSTRAINTS

Because of the Big-Bang terms, the equations 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X(t, a)) = 0, 1 + t △ ϕ = ρ =

  • δ(x − X(t, a))da

are degenerate at t = 0. At initial time, we get a continuum of

  • particles. Their density is uniform and their initial velocity is

"slaved" by the gravitational potential: ρ0(x) = 1, X0(a) = a, dX0 dt (a) = −∇ϕ0(a)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25

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THE INITIAL CONSTRAINTS

Because of the Big-Bang terms, the equations 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X(t, a)) = 0, 1 + t △ ϕ = ρ =

  • δ(x − X(t, a))da

are degenerate at t = 0. At initial time, we get a continuum of

  • particles. Their density is uniform and their initial velocity is

"slaved" by the gravitational potential: ρ0(x) = 1, X0(a) = a, dX0 dt (a) = −∇ϕ0(a) which is totally different from usual N-body Newton’s gravitation.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25

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ZELDOVICH APPROXIMATION

A very simple approximate solution was proposed by Zeldovich ∼ 1970 for the semi-newtonian model 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, ρ =

  • δ(x − X(t, a))da = 1 + t △ ϕ

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 5 / 25

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ZELDOVICH APPROXIMATION

A very simple approximate solution was proposed by Zeldovich ∼ 1970 for the semi-newtonian model 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, ρ =

  • δ(x − X(t, a))da = 1 + t △ ϕ

→: X(t, a) = a − t∇ϕ0(a), △ϕ0(x) = lim

t→0

ρ(t, x) − 1 t Each particle just travels with a constant velocity due to the initial density fluctuation, until a collision ocurs, which is somewhat reminiscent of Lucretius’ (99-55 BC) "DE RERUM NATURA".

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 5 / 25

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1d Zeldovich solutions with sticky collisions

horizontal : space /vertical : time

0.5 1 1.5 2 2.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 6 / 25

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DE RERUM NATURA LIBER SECUNDUS 216 − 224 LUCRETIUS (99 − 55BC) Quod nisi declinare solerent (corpora), omnia deorsum imbris uti guttae caderent per inane profundum ...Ita nihil umquam natura creasset. But if (corpora) were not in the habit of deviating, they would all fall straight down through the depths of the void, like drops of rain... In that case, nature would never have produced anything.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 7 / 25

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RECONSTRUCTING THE EARLY UNIVERSE?

It is plausible (Peebles 1989) to reconstruct the early universe from the only knowledge of the observed density field ρ(T, x) .

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 8 / 25

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RECONSTRUCTING THE EARLY UNIVERSE?

It is plausible (Peebles 1989) to reconstruct the early universe from the only knowledge of the observed density field ρ(T, x) . As a matter of fact, the only initial condition we need to recover is ρ′

0(x) = lim t↓0

ρ(t, x) − 1 t = △ϕ0(x) which is supposed to be a random field of very small amplitude according to the quantum theory of the VERY early universe.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 8 / 25

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MASS CONCENTRATIONS IN THE OBSERVED DISTRIBUTION OF MASS IN THE UNIVERSE

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 9 / 25

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THE MONGE-AMPERE-KANTOROVICH PROBLEM

Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ(T, x)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25

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THE MONGE-AMPERE-KANTOROVICH PROBLEM

Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ(T, x) if the Zeldovich map X(T, a) = a − T∇ϕ0(a) has convex potential.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25

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THE MONGE-AMPERE-KANTOROVICH PROBLEM

Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ(T, x) if the Zeldovich map X(T, a) = a − T∇ϕ0(a) has convex potential. Frisch and collaborators designed an effective computational method, based on Bertsekas’ algorithm, for about 104 particles.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

The Monge approach to the problem has two drawbacks:

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude;

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude; ii) it rules out any collision effect. (As well known in optimal transport theory, collisions are possible only at the final time T.)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude; ii) it rules out any collision effect. (As well known in optimal transport theory, collisions are possible only at the final time T.) So, I have suggested a slightly different approach to the problem (Confluentes Math. 2011), still based on optimal transport theory, leaving several analytic and computational issues largely open.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25

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MONGE-AMPERE GRAVITATION (MAG)

In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ(t, x) = det(I + tD2ϕ(t, x)) instead of ρ(t, x) = 1 + t △ ϕ(t, x)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25

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MONGE-AMPERE GRAVITATION (MAG)

In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ(t, x) = det(I + tD2ϕ(t, x)) instead of ρ(t, x) = 1 + t △ ϕ(t, x) i) This is exact in 1d;

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25

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MONGE-AMPERE GRAVITATION (MAG)

In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ(t, x) = det(I + tD2ϕ(t, x)) instead of ρ(t, x) = 1 + t △ ϕ(t, x) i) This is exact in 1d; ii) asymptotically correct at early times and for weak fields;

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25

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SLIDE 25

MONGE-AMPERE GRAVITATION (MAG)

In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ(t, x) = det(I + tD2ϕ(t, x)) instead of ρ(t, x) = 1 + t △ ϕ(t, x) i) This is exact in 1d; ii) asymptotically correct at early times and for weak fields; iii) makes Zeldovich approximation exact (will be shown);

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25

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MONGE-AMPERE GRAVITATION (MAG)

In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ(t, x) = det(I + tD2ϕ(t, x)) instead of ρ(t, x) = 1 + t △ ϕ(t, x) i) This is exact in 1d; ii) asymptotically correct at early times and for weak fields; iii) makes Zeldovich approximation exact (will be shown); iv) might be as good as the Poisson equation as an approximation to the Einstein equations (to be investigated!).

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

Main observation: the MONGE-AMPERE GRAVITATIONAL MODEL 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, det(I + tD2ϕ) = ρ =

  • δ(x − X(t, a))da

can be derived from a principle of least action,

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 13 / 25

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THE MONGE-AMPERE GRAVITATION MODEL

Main observation: the MONGE-AMPERE GRAVITATIONAL MODEL 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, det(I + tD2ϕ) = ρ =

  • δ(x − X(t, a))da

can be derived from a principle of least action, set on the Hilbert space (H, || · ||) of all L2 maps a → X(a) on T3, that crucially involves the subset S ⊂ H of all Lebesgue-measure preserving maps and the squared distance function to this set.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 13 / 25

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THE MAG LEAST ACTION PRINCIPLE

The MAG action, for a curve t → X(t) ∈ H just reads t1

t0

t3/2||dX dt ||2 + 3t−1/2Q[X(t)] dt, Q[X] = 1 2dist2(X, S) where S ⊂ H is the set of all Lebesgue-measure preserving maps.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 14 / 25

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THE MAG LEAST ACTION PRINCIPLE

The MAG action, for a curve t → X(t) ∈ H just reads t1

t0

t3/2||dX dt ||2 + 3t−1/2Q[X(t)] dt, Q[X] = 1 2dist2(X, S) where S ⊂ H is the set of all Lebesgue-measure preserving maps. Using OPTIMAL TRANSPORT THEORY, we may recover the Monge-Ampère gravitation (MAG) equations:

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 14 / 25

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THE MAG LEAST ACTION PRINCIPLE

The MAG action, for a curve t → X(t) ∈ H just reads t1

t0

t3/2||dX dt ||2 + 3t−1/2Q[X(t)] dt, Q[X] = 1 2dist2(X, S) where S ⊂ H is the set of all Lebesgue-measure preserving maps. Using OPTIMAL TRANSPORT THEORY, we may recover the Monge-Ampère gravitation (MAG) equations: (Exercise!) 2t 3 d2X dt2 + dX dt + ∇ϕ(t, X) = 0, det(I + tD2ϕ) = ρ =

  • δ(x − X(t, a))da

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 14 / 25

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SPECIAL STRUCTURE OF THE MAG ACTION

The squared distance to the closed bounded subset S ⊂ H is differentiable everywhere in H except on a "meager" set N and Q[X] = 1 2dist2(X, S) = 1 2||∇Q[X]||2, ∀X ∈ H \ N

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 15 / 25

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SPECIAL STRUCTURE OF THE MAG ACTION

The squared distance to the closed bounded subset S ⊂ H is differentiable everywhere in H except on a "meager" set N and Q[X] = 1 2dist2(X, S) = 1 2||∇Q[X]||2, ∀X ∈ H \ N So, for curves t → X(t) that stay out of N for almost every time, t1

t0

t3/2||dX dt ||2 + 3t−1/2Q[X] dt = t1

t0

t3/2||dX dt ||2 + 3 2t−1/2||∇Q[X||2 dt

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 15 / 25

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SPECIAL STRUCTURE OF THE MAG ACTION

The squared distance to the closed bounded subset S ⊂ H is differentiable everywhere in H except on a "meager" set N and Q[X] = 1 2dist2(X, S) = 1 2||∇Q[X]||2, ∀X ∈ H \ N So, for curves t → X(t) that stay out of N for almost every time, t1

t0

t3/2||dX dt ||2 + 3t−1/2Q[X] dt = t1

t0

t3/2||dX dt ||2 + 3 2t−1/2||∇Q[X||2 dt Reorganizing the squares and integrating by part in time, we get t1

t0

t−1/2||tdX dt − ∇Q[X(t)]||2 dt + time boundary term

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 15 / 25

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GRADIENT FLOW SOLUTIONS AS SPECIAL LEAST-ACTION SOLUTIONS

Due to the special structure of the MAG action, we find as particular least action solutions any solution to the GRADIENT FLOW EQUATION tdX dt = ∇Q[X(t)] , Q[X] = 1 2 inf{||X − s||2 ; s ∈ S} which stays out of the "bad set" N for almost every time.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 16 / 25

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GRADIENT FLOW SOLUTIONS AS SPECIAL LEAST-ACTION SOLUTIONS

Due to the special structure of the MAG action, we find as particular least action solutions any solution to the GRADIENT FLOW EQUATION tdX dt = ∇Q[X(t)] , Q[X] = 1 2 inf{||X − s||2 ; s ∈ S} which stays out of the "bad set" N for almost every time. It turns out that Zeldovich solutions are just exact solutions of this gradient flow equation as long as there are no collisions!

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 16 / 25

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GLOBAL SOLUTIONS OF THE GRADIENT FLOW

We have 2Q[X] = dist2(X, S) = ||X||2 − 2 sup

s∈S

[((X, s)) − ||s||2]

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 17 / 25

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SLIDE 38

GLOBAL SOLUTIONS OF THE GRADIENT FLOW

We have 2Q[X] = dist2(X, S) = ||X||2 − 2 sup

s∈S

[((X, s)) − ||s||2] Thus tdX dt = ∇Q[X] = X(t) − ∇R[X] , R[X] = sup

s∈S

[((X, s)) − 1 2||s||2] where R is Lipschitz, convex, differentiable out of the "bad set" N.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 17 / 25

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SLIDE 39

GLOBAL SOLUTIONS OF THE GRADIENT FLOW

We have 2Q[X] = dist2(X, S) = ||X||2 − 2 sup

s∈S

[((X, s)) − ||s||2] Thus tdX dt = ∇Q[X] = X(t) − ∇R[X] , R[X] = sup

s∈S

[((X, s)) − 1 2||s||2] where R is Lipschitz, convex, differentiable out of the "bad set" N. For every t0 > 0, X0 ∈ H, the theory of maximal monotone

  • perators gives a unique solution X ∈ C0([t0, +∞[, H)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 17 / 25

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SLIDE 40

GLOBAL SOLUTIONS OF THE GRADIENT FLOW

We have 2Q[X] = dist2(X, S) = ||X||2 − 2 sup

s∈S

[((X, s)) − ||s||2] Thus tdX dt = ∇Q[X] = X(t) − ∇R[X] , R[X] = sup

s∈S

[((X, s)) − 1 2||s||2] where R is Lipschitz, convex, differentiable out of the "bad set" N. For every t0 > 0, X0 ∈ H, the theory of maximal monotone

  • perators gives a unique solution X ∈ C0([t0, +∞[, H) in the sense

X(t0) = X0, tdX(t + 0) dt = X(t) − ∇R[X(t)] , ∀t ≥ t0

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 17 / 25

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SLIDE 41

GLOBAL SOLUTIONS OF THE GRADIENT FLOW

We have 2Q[X] = dist2(X, S) = ||X||2 − 2 sup

s∈S

[((X, s)) − ||s||2] Thus tdX dt = ∇Q[X] = X(t) − ∇R[X] , R[X] = sup

s∈S

[((X, s)) − 1 2||s||2] where R is Lipschitz, convex, differentiable out of the "bad set" N. For every t0 > 0, X0 ∈ H, the theory of maximal monotone

  • perators gives a unique solution X ∈ C0([t0, +∞[, H) in the sense

X(t0) = X0, tdX(t + 0) dt = X(t) − ∇R[X(t)] , ∀t ≥ t0 where ∇R[X] is the unique element with minimal norm of ∂R at X.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 17 / 25

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SLIDE 42

THE MODIFIED MAG ACTION

The gradient flow equation, in the sense of maximal monotone

  • perators, takes into account collisions. (In 1d, we get "sticky"

collisions.)

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 18 / 25

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SLIDE 43

THE MODIFIED MAG ACTION

The gradient flow equation, in the sense of maximal monotone

  • perators, takes into account collisions. (In 1d, we get "sticky"

collisions.) So, we introduced in Confl. Math. 2011 the modified MAG action t1

t0

t−1/2||tdX dt − X(t) + ∇R[X(t)]||2 dt

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 18 / 25

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SLIDE 44

THE MODIFIED MAG ACTION

The gradient flow equation, in the sense of maximal monotone

  • perators, takes into account collisions. (In 1d, we get "sticky"

collisions.) So, we introduced in Confl. Math. 2011 the modified MAG action t1

t0

t−1/2||tdX dt − X(t) + ∇R[X(t)]||2 dt where R[X] = sup{((X, s)) − 1 2||s||2, s ∈ S} and S is the subset

  • f all volume-preserving maps.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 18 / 25

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SLIDE 45

NUMERICS

We now show samples of 1D simulations, based on the minimization of the full space and time- discrete version of the action.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 19 / 25

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SLIDE 46

NUMERICS

We now show samples of 1D simulations, based on the minimization of the full space and time- discrete version of the

  • action. We reconstruct initial velocities and compare each

reconstructed solution with the corresponding solution of the IVP

  • btained by a scheme that properly handles sticky collisions.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 19 / 25

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SLIDE 47

NUMERICS

We now show samples of 1D simulations, based on the minimization of the full space and time- discrete version of the

  • action. We reconstruct initial velocities and compare each

reconstructed solution with the corresponding solution of the IVP

  • btained by a scheme that properly handles sticky collisions.

The calculations rely on many (∼ 105) iterations of an elementary sorting algorithm and never require the calculation of ∇R.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 19 / 25

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SLIDE 48

case 1: reconstructed trajectories

horizontal : 51 grid points in x /vertical : 60 grid points in t

0.5 1 1.5 2 2.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 20 / 25

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SLIDE 49

case 1: IVP with reconstructed velocities

horizontal : 51 grid points in x /vertical : 60 grid points in t

0.5 1 1.5 2 2.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 21 / 25

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SLIDE 50

case 2: reconstructed trajectories

horizontal : 51 grid points in x /vertical : 60 grid points in t

0.5 1 1.5 2 2.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 22 / 25

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SLIDE 51

case 2: IVP with reconstructed velocities

horizontal : 51 grid points in x /vertical : 60 grid points in t

0.5 1 1.5 2 2.5

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 23 / 25

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SLIDE 52

OPEN PROBLEMS

Justification of the substitution of the Monge-Ampère equation for the Poisson equation, through a suitable asymptotic analysis

  • f the Einstein equations.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 24 / 25

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SLIDE 53

OPEN PROBLEMS

Justification of the substitution of the Monge-Ampère equation for the Poisson equation, through a suitable asymptotic analysis

  • f the Einstein equations.

Analysis of the modified least action principle and rigorous derivation of a well-posed system of evolution equations (to be done even in the 1D case!).

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 24 / 25

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SLIDE 54

OPEN PROBLEMS

Justification of the substitution of the Monge-Ampère equation for the Poisson equation, through a suitable asymptotic analysis

  • f the Einstein equations.

Analysis of the modified least action principle and rigorous derivation of a well-posed system of evolution equations (to be done even in the 1D case!). Efficient extension of the numerics from 1D to 3D.

Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 24 / 25

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SLIDE 55

REFERENCES

Main: Y. Brenier, Confluentes Mathematici 3 (2011) The EUR problem and the pressure-less Euler-Poisson model

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Mohayaee, Sobolevskii, Mon. Not. R. Astron. Soc. (2003) and references included

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Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 25 / 25