Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Abel Laureate Luis Caffarelli: a glimpse of his work
Klas Modin
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_1
Mathematical formulation:
T:\Omega\to\Omega\quad\text{s.t.}\quad T_*\mu_0 = \mu_1
Additional requirement: \(T\) should minimize
\displaystyle E(T) =
\displaystyle\int_\Omega |T(\mathbf x)-\mathbf x|d \mathbf x
Example of transport map
\mu_0 = d\mathbf x
\text{Domain}\; \Omega = \mathbb T^2
\mu_1 =
T:\Omega\to\Omega, \; T_*\mu_0 = \mu_1
Transport map
Polar decomposition of matrices
A \in \mathbb{R}^{n\times n}
\Rightarrow \quad A = PQ
P=P^\top, \; P\geq 0
Q^\top Q = I
\Sigma_0 \in \mathrm{P}(n) = \{ \Sigma\in\mathbb{R}^{n\times n} \mid \Sigma = \Sigma^\top, \Sigma>0 \}
A \in \mathrm{GL}(n)
acts on:
A\cdot \Sigma_0 = A\Sigma_0 A^\top
via:
Bundle structure
GL(n)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
\Sigma_0
\Sigma_1
\pi(A)=A\Sigma_0 A^\top
\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)
\dot\Sigma = S\Sigma + \Sigma S
K
Bundle structure
GL(n)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
\Sigma_0
\Sigma_1
\pi(A)=A\Sigma_0 A^\top
\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)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
d^2(\Sigma_0,\Sigma_1) = \lVert I - P \rVert^2_{F,\Sigma_0}
Bundle structure
GL(n)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
\Sigma_0
\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)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
"Square root" equation:
P
Bundle structure
GL(n)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
\Sigma_0
\Sigma_1
A = PQ
\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)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Factorization theorem:
A
P
Bundle structure
GL(n)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
\Sigma_0
\Sigma_1
\dot B = \Omega B, \; B(0) = 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)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Vertical gradient flow:
A
\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})
Bundle structure
GL(n)
P(n)
I
\Sigma_0
\Sigma_1
\dot B = \Omega B, \; B(0) = 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)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Vertical gradient flow:
A
\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})
Bundle structure
GL(n)
P(n)
I
\Sigma_0
\Sigma_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)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Horizontal gradient flow:
\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)
Relative entropy:
Bundle structure
GL(n)
P(n)
I
\Sigma_0
\Sigma_1
\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)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Horizontal gradient flow:
P(t)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on \(K\) for
Bundle structure
GL(n)
P(n)
I
\Sigma_0
\Sigma_1
\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)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
\dot\Sigma = S\Sigma + \Sigma S
K = P(n)
Horizontal gradient flow:
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)
\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}}
\(\Rightarrow\)
\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 (*)
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 (*)
Note: \(\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0 \) (strong MA)
\mathrm{Id}
\mu_0
\mu_1
\pi(T)=T_*\mu_0
\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 (*)
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 (*)
Riemannian structure of OMT
\mathrm{Diff}(\Omega)
\mathrm{P}^{\infty}(\Omega)
\mathrm{Id}
\mu_0
\mu_1
\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
Induces metric
\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)
\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)
Exactly \(L^2\)-Wasserstein distance
\mathrm{Diff}_{\mu_0}(\Omega)
OMT \(\leftrightarrow\) hydrodynamics
\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{id}
\displaystyle \mathrm{SDiff}(\Omega)
\displaystyle F
\displaystyle \nabla P
\displaystyle S
\displaystyle \mathcal{P}^\infty(\Omega) \simeq \mathrm{Diff}(\Omega)/\mathrm{SDiff}(\Omega)
\displaystyle 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
By Klas Modin
Abel Laureate Luis Caffarelli: a glimpse of his work
Colloquium about Abel Prize winners at the Department of Mathematical Sciences at Chalmers and GU.
