Luis Ángel Caffarelli

Nationality: Argentine–American

Born: 1948

PhD 1972 (University of Buenos Aires)

Supervisor:

Calixto Calderón

Affiliations:

University of Minnesota 1973-1983

Courant Institute, NYU 1980-1982

University of Chicago 1983-1986

IAS Princeton 1986-1996

Courant Institute, NYU 1994-1997

University of Texas at Austin 1997-today

For his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge-Ampère equation.

Research fields

Free boundary problems

Fluid dynamics (Navier-Stokes)

Stefan problem

Nonlinear PDE

Monge-Ampere equation

etc.

Focus of today

Optimal mass transport (OMT)

Question (Monge 1781): Cheapest way to transport one mass distribution to another?

\mu_0

\mu_0

\mu_1

\mu_1

Mathematical formulation:

T:\Omega\to\Omega\quad\text{s.t.}\quad T_*\mu_0 = \mu_1

T:\Omega\to\Omega\quad\text{s.t.}\quad T_*\mu_0 = \mu_1

Additional requirement: $T$ should minimize

\displaystyle E(T) =

\displaystyle E(T) =

\displaystyle\int_\Omega |T(\mathbf x)-\mathbf x|d \mathbf x

\displaystyle\int_\Omega |T(\mathbf x)-\mathbf x|d \mathbf x

Example of transport map

\mu_0 = d\mathbf x

\mu_0 = d\mathbf x

\text{Domain}\; \Omega = \mathbb T^2

\text{Domain}\; \Omega = \mathbb T^2

\mu_1 =

\mu_1 =

T:\Omega\to\Omega, \; T_*\mu_0 = \mu_1

T:\Omega\to\Omega, \; T_*\mu_0 = \mu_1

Transport map

Polar decomposition of matrices

A \in \mathbb{R}^{n\times n}

A \in \mathbb{R}^{n\times n}

\Rightarrow \quad A = PQ

\Rightarrow \quad A = PQ

P=P^\top, \; P\geq 0

P=P^\top, \; P\geq 0

Q^\top Q = I

Q^\top Q = I

\Sigma_0 \in \mathrm{P}(n) = \{ \Sigma\in\mathbb{R}^{n\times n} \mid \Sigma = \Sigma^\top, \Sigma>0 \}

\Sigma_0 \in \mathrm{P}(n) = \{ \Sigma\in\mathbb{R}^{n\times n} \mid \Sigma = \Sigma^\top, \Sigma>0 \}

A \in \mathrm{GL}(n)

A \in \mathrm{GL}(n)

acts on:

A\cdot \Sigma_0 = A\Sigma_0 A^\top

A\cdot \Sigma_0 = A\Sigma_0 A^\top

via:

Bundle structure

GL(n)

GL(n)

P(n)\simeq GL(n)/O(n,\Sigma_0)

P(n)\simeq GL(n)/O(n,\Sigma_0)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\pi(A)=A\Sigma_0 A^\top

\pi(A)=A\Sigma_0 A^\top

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K

K

Bundle structure

GL(n)

GL(n)

P(n)\simeq GL(n)/O(n,\Sigma_0)

P(n)\simeq GL(n)/O(n,\Sigma_0)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\pi(A)=A\Sigma_0 A^\top

\pi(A)=A\Sigma_0 A^\top

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

d^2(\Sigma_0,\Sigma_1) = \lVert I - P \rVert^2_{F,\Sigma_0}

d^2(\Sigma_0,\Sigma_1) = \lVert I - P \rVert^2_{F,\Sigma_0}

Bundle structure

GL(n)

GL(n)

P(n)\simeq GL(n)/O(n,\Sigma_0)

P(n)\simeq GL(n)/O(n,\Sigma_0)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\pi(P)=P\Sigma_0 P = \Sigma_1

\pi(P)=P\Sigma_0 P = \Sigma_1

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

"Square root" equation:

P

P

Bundle structure

GL(n)

GL(n)

P(n)\simeq GL(n)/O(n,\Sigma_0)

P(n)\simeq GL(n)/O(n,\Sigma_0)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

A = PQ

A = PQ

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Factorization theorem:

A

A

P

P

Bundle structure

GL(n)

GL(n)

P(n)\simeq GL(n)/O(n,\Sigma_0)

P(n)\simeq GL(n)/O(n,\Sigma_0)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\dot B = \Omega B, \; B(0) = A

\dot B = \Omega B, \; B(0) = A

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Vertical gradient flow:

A

A

\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})

\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})

Bundle structure

GL(n)

GL(n)

P(n)

P(n)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\dot B = \Omega B, \; B(0) = A

\dot B = \Omega B, \; B(0) = A

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Vertical gradient flow:

A

A

\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})

\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})

Bundle structure

GL(n)

GL(n)

P(n)

P(n)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\dot \Sigma = 2I - \Sigma_1^{-1}\Sigma - \Sigma\Sigma_1^{-1}

\dot \Sigma = 2I - \Sigma_1^{-1}\Sigma - \Sigma\Sigma_1^{-1}

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Horizontal gradient flow:

\Sigma(t)

\Sigma(t)

\displaystyle H_{\Sigma_1}(\Sigma) = -\frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}\Sigma) + \frac{1}{2}\log\det(\Sigma_1^{-1}\Sigma)

\displaystyle H_{\Sigma_1}(\Sigma) = -\frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}\Sigma) + \frac{1}{2}\log\det(\Sigma_1^{-1}\Sigma)

Relative entropy:

Bundle structure

GL(n)

GL(n)

P(n)

P(n)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V

\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Horizontal gradient flow:

P(t)

P(t)

\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)

\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)

Lifted gradient flow on $K$ for

Bundle structure

GL(n)

GL(n)

P(n)

P(n)

I

I

\Sigma_0

\Sigma_0

\Sigma_1

\Sigma_1

\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V

\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)

\dot\Sigma = S\Sigma + \Sigma S

\dot\Sigma = S\Sigma + \Sigma S

K = P(n)

K = P(n)

Horizontal gradient flow:

P(t)

P(t)

Hessian of $F(P)$ strictly positive on $K$ $\Rightarrow$ unique limit!

Nice, but...

...what about Monge-Ampere and Caffarelli?!

?

Linear optimal mass transport

$\mathrm{P}(n) \iff$ multivariate Gaussians with zero mean

\displaystyle \rho_\Sigma(x) = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(\Sigma)}}\mathrm{exp}(-\frac{1}{2}x^\top \Sigma^{-1}x)

\displaystyle \rho_\Sigma(x) = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(\Sigma)}}\mathrm{exp}(-\frac{1}{2}x^\top \Sigma^{-1}x)

\Sigma \in P(n)

\Sigma \in P(n)

$\Rightarrow$ transport map $T:\mathbb{R}^n \to \mathbb{R}^n$ linear, $T(x) = A x$

T_*\rho_\Sigma = \rho_{A\Sigma A^{\top}}

T_*\rho_\Sigma = \rho_{A\Sigma A^{\top}}

$\Rightarrow$

\displaystyle d^2(A\Sigma_0A^\top, \Sigma_1) = \int_{\mathbb{R}^n}|T_*\rho_{\Sigma_0} - \rho_{\Sigma_1}|^2dx

\displaystyle d^2(A\Sigma_0A^\top, \Sigma_1) = \int_{\mathbb{R}^n}|T_*\rho_{\Sigma_0} - \rho_{\Sigma_1}|^2dx

$L^2$ instead of $L^1$ cost

Fundamental result for $L^2$ OMT

Theorem (Brenier 1987):

$\mu_0,\mu_1$ prob measures on $\Omega\subset\mathbb R^n$ (open, bounded) with $\mu_i=\rho_id \mathbf x$
Then $(*)$ has unique solution $T = \nabla \psi$

where $\psi:\Omega \to \mathbb R$ is convex

$\psi$ solves the weak Monge-Ampere equation $\int_\Omega \eta \rho_1 dx = \int_\Omega (\eta\circ\nabla\psi) \rho_0 dx,\qquad \forall \eta \in C(\Omega)$

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$ (strong MA)

Fundamental result for $L^2$ OMT

Theorem (Brenier 1991):

$T \in L^2(\Omega,\mathbb{R}^n)$ such that $T_*\mu_0 = \mu_1$

Exist unique $S:\Omega\to\Omega$ with $S_*\mu_0 = \mu_0$ and convex $\psi$ such that $T = (\nabla \psi) \circ S$

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$ (strong MA)

\mathrm{Id}

\mathrm{Id}

\mu_0

\mu_0

\mu_1

\mu_1

\pi(T)=T_*\mu_0

\pi(T)=T_*\mu_0

\mathrm{Hor}

\mathrm{Hor}

$T$

$\nabla\psi$

$S$

Fundamental result for $L^2$ OMT

Theorem (Brenier 1991):

$T \in L^2(\Omega,\mathbb{R}^n)$ such that $T_*\mu_0 = \mu_1$

Exist unique $S:\Omega\to\Omega$ with $S_*\mu_0 = \mu_0$ and convex $\psi$ such that $T = (\nabla \psi) \circ S$

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$ (strong MA)

Subdifferential: $\partial \psi(x) = \{u \in\mathbb{R}^n\mid \forall y, \; \psi(y) \geq \psi(x) + u\cdot (y-x) \}$

$x$

$\psi(x)$

$\psi(x) + u\cdot(y-x)$

$\psi(y)$

$y$

Regularity obstructions

Non-convex domain

Negative curvature domain

[cf. Villani 2009]

"There is no hope for general regularity results
outside the world of nonnegative sectional curvature"

[Villani 2009]

Caffarelli's regularity theory

Theorem (Caffarelli 1992):

$\mu_0,\mu_1$ prob measures on $\Omega\subset\mathbb R^n$ (convex) with $\mu_i=\rho_id \mathbf x$

$\rho_i \in C^{k,\alpha}(\Omega), \quad 0<a\leq \rho_i(x) \leq b < \infty$
Then $(*)$ has unique solution $T = \nabla \psi$

where $\psi\in C^{k+2,\alpha}(\Omega)$ is convex

$\psi$ solves the strong Monge-Ampere equation $\rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

Riemannian structure of OMT

\mathrm{Diff}(\Omega)

\mathrm{Diff}(\Omega)

\mathrm{P}^{\infty}(\Omega)

\mathrm{P}^{\infty}(\Omega)

\mathrm{Id}

\mathrm{Id}

\mu_0

\mu_0

\mu_1

\mu_1

\pi(T)=T_*\mu_0

\pi(T)=T_*\mu_0

Riemannian metric

\displaystyle\mathcal{G}_T(\dot T,\dot T) = \int_{\Omega}\left\vert \dot T \right\vert^2 \mu_0

\displaystyle\mathcal{G}_T(\dot T,\dot T) = \int_{\Omega}\left\vert \dot T \right\vert^2 \mu_0

Induces metric

\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)

\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)

\mathrm{Hor}

\mathrm{Hor}

[Benamou & Brenier (2000), Otto (2001)]

Invariance: $S\in\mathrm{Diff}_{\mu_0}(\Omega)$

\displaystyle\mathcal{G}_T(\dot T,\dot T) = \mathcal{G}_{T\circ S}(\dot T\circ S, \dot T\circ S)

\displaystyle\mathcal{G}_T(\dot T,\dot T) = \mathcal{G}_{T\circ S}(\dot T\circ S, \dot T\circ S)

Exactly $L^2$ -Wasserstein distance

\mathrm{Diff}_{\mu_0}(\Omega)

\mathrm{Diff}_{\mu_0}(\Omega)

OMT $\leftrightarrow$ hydrodynamics

\displaystyle \pi: F\mapsto F_*d\mathbf{x}

\displaystyle \pi: F\mapsto F_*d\mathbf{x}

[Arnold 1966, Hamilton 1982, Caffarelli 1992, Benamou and Brenier 2000, Otto 2001]

\displaystyle \mathrm{Diff}(\Omega)

\displaystyle \mathrm{Diff}(\Omega)

\displaystyle \mathrm{id}

\displaystyle \mathrm{id}

\displaystyle \mathrm{SDiff}(\Omega)

\displaystyle \mathrm{SDiff}(\Omega)

\displaystyle F

\displaystyle F

\displaystyle \nabla P

\displaystyle \nabla P

\displaystyle S

\displaystyle S

\displaystyle \mathcal{P}^\infty(\Omega) \simeq \mathrm{Diff}(\Omega)/\mathrm{SDiff}(\Omega)

\displaystyle \mathcal{P}^\infty(\Omega) \simeq \mathrm{Diff}(\Omega)/\mathrm{SDiff}(\Omega)

\displaystyle d\mathbf{x}

\displaystyle d\mathbf{x}

\displaystyle \rho\, d\mathbf{x}

\displaystyle \rho\, d\mathbf{x}

Brenier's polar

factorization: $F = \nabla P\circ S$

Remember:

$T = \nabla P$ solves OMT problem with $\mu_0 = d\mathbf x$ and $\mu_1 = \rho\,d\mathbf x$

hydrodynamics

OMT

Summary

Slides available at: slides.com/kmodin

(from an applied perspective)

Caffarelli's work enables "safe ground" for geometers to work with smooth optimal transport

Abel Laureate Luis Caffarelli: a glimpse of his work

Klas Modin

Research fields

Optimal mass transport (OMT)

Example of transport map

Polar decomposition of matrices

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Bundle structure

Nice, but...

...what about Monge-Ampere and Caffarelli?!

?

Linear optimal mass transport

Fundamental result for $L^2$ OMT

Fundamental result for $L^2$ OMT

Fundamental result for $L^2$ OMT

Regularity obstructions

Caffarelli's regularity theory

Riemannian structure of OMT

OMT $\leftrightarrow$ hydrodynamics

Summary

(from an applied perspective)