Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Glimpse into the work of Alessio Figalli
Klas Modin
Seminar series about Fields Medalists 2018
Alessio Figalli
Nationality: Italian
Born: 1984
Master's degree 2006 (SNS Pisa)
PhD 2007 (SNS Pisa and ENS Lyon)
Supervisors:
Luigi Ambrosio (SNS Pisa)
Cedric Villani (ENS Lyon)
Affiliations:
CNRS 2007
École polytechnique 2008-2009
University of Texas in Austin 2009-2016
ETH Zürich 2016-today
Awarded the Fields Medal for his contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry, and probability
Research fields
Optimal mass transport (OMT)
Fluid dynamics (Euler equations, etc)
Weak KAM theory
Geometric measure theory
Stochastic analysis
Random matrices
etc.
Very productive 120+ papers, 2 books, 25 lecture notes, ...
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 c(T(\mathbf x),\mathbf x)d \mathbf x
\displaystyle\int_\Omega |T(\mathbf x)-\mathbf x|^2d \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
Fundamental result for \(L^2\) OMT
Theorem (Brenier 1991):
\(\mu_0,\mu_1\) prob measures on \(\mathbb R^n\) with finite second moment, \(\mu_0=\rho_0d \mathbf x\)
Then \((*)\) has unique solution \[ T = \nabla P\]
where \(P:\mathbb R^n \to \mathbb R\) is convex
If \(\mu_1 = \rho_1 d\mathbf x\) then (formally) \(P\) solves the Monge-Ampere equation \[ \det(D^2P) = \frac{\rho_0}{\rho_1\circ\nabla P} \iff \text{MA}(P) = \frac{\rho_0}{\rho_1\circ\nabla P} \]
\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\mathbb R^n} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)
Sobolev result for \(L^2\) OMT
Theorem (De Philippis and Figalli 2013):
If \(\Omega\subset \mathbb R^n\) is convex and bounded, \(P:\Omega\to\mathbb R\) is convex, and \[ 0 < \lambda \leq \det(D^2 P) \leq \Lambda < \infty \]
then \(P \in W^{2,1}_{\text{loc}}(\Omega)\)
Corollary
The solution \( T = \nabla P\) to \((*)\) belong to \(W^{1,1}_{\text{loc}}(\Omega)\)
\displaystyle \min_{T_*d\mathbf x=\rho d\mathbf x} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mathbf x\qquad (*)
Incompressible hydrodynamics
\displaystyle\frac{\partial \mathbf v}{\partial t} + \mathbf v\cdot \nabla \mathbf v = -\nabla p\qquad (*)
Domain \(\Omega\subset \mathbb R^3\), velocity field \(\mathbf v = \mathbf v(t,\mathbf x)\)
Discovery by Arnold 1966:
Solutions to \((*)\) correspond to geodesics on the infinite-dimensional manifold \(\mathrm{SDiff}(\Omega)\) with Riemannian \(L^2\)-metric \[\langle V,V\rangle_{S} = \int_\Omega |V|^2 d \mathbf x \] where \[\mathbf v = V\circ S^{-1} \]
\displaystyle \nabla \cdot\mathbf v = 0
pressure
OMT \(\leftrightarrow\) hydrodynamics
\displaystyle \pi: F\mapsto F_*d\mathbf{x}
(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
Atmospheric hydrodynamics
\displaystyle\frac{\partial \mathbf v}{\partial t} + \mathbf v\cdot \nabla \mathbf v = -\nabla p + \begin{pmatrix}v_2 \\ -v_1 \\ 0\end{pmatrix} + \mathbf{g}
\displaystyle \nabla \cdot\mathbf v = 0
Semi-geostrophic wind:
Coriolis force
\displaystyle \mathbf v^g = \begin{pmatrix}\partial_2 p \\ -\partial_1 p \end{pmatrix}
Aim: equation for \(\mathbf v^g\)
Semigeostrophic equation (SGE)
\displaystyle\frac{\partial \nabla p}{\partial t} + (\mathbf v\cdot \nabla) \nabla p = -J\nabla p - \mathbf v
\displaystyle \nabla \cdot\mathbf v = 0
where
Now 2D: \(\mathbf v = (v_1,v_2)\) and replace l.h.s. by semi-geostrophic wind
J = \begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}
Notice: no time-derivative on \(\mathbf v\)
As written, SGE is hard to analyze!
Rewrite in simpler form?
(Hoskins and Bretherton 1972)
Brief derivation of dual SGE
\displaystyle \mathbf u = \frac{\partial T}{\partial t}\circ T^{-1} = \frac{\partial \nabla P}{\partial t}\circ \nabla P^*
\displaystyle P(t,\mathbf x) = p(t,\mathbf x) + \frac{|\mathbf x|^2}{2}
Change of variables:
(assume \(P\) convex)
\displaystyle \rho\,d\mathbf x = (\nabla P)_*d\mathbf x
Key: Think of \(T=\nabla P\) as transport map
Apply Arnold's approach:
right translation of \(\partial_t T\) gives vector field
\displaystyle \mathbf u = J(\nabla P^*-\mathbf x)
By construction
\displaystyle \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf u) = 0
Dual SGE
\displaystyle \rho = \det(D^2 P^*) \iff \rho = \text{MA}(P^*)
\displaystyle \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf u) = 0
\displaystyle \rho\,d\mathbf x = (\nabla P)_*d\mathbf x = (\nabla P^*)^*d\mathbf x
\displaystyle \mathbf u = J(\nabla P^*-\mathbf x)
Close relation to vorticity equation
\displaystyle \frac{\partial \omega}{\partial t} + \nabla\cdot(\omega \mathbf u) = 0
\displaystyle \mathbf u = J\nabla \psi
\displaystyle \omega = \Delta\psi
\displaystyle \rho \sim \omega + 1
\displaystyle P^* \sim \psi+\frac{|\mathbf x|^2}{2}
\displaystyle \text{MA}(P^*) = \det(I + D^2\psi)
\displaystyle \approx 1 + \Delta\psi
Problem with existence for SGE
Theorem (Benamou and Brenier 1998):
Dual SGE admits global weak solution
\displaystyle\frac{\partial \nabla p}{\partial t} + (\mathbf v\cdot \nabla) \nabla p = -J\nabla p - \mathbf v
\displaystyle \nabla \cdot\mathbf v = 0
Recall: SGE
Does weak dual SGE solution give solution to SGE?
How is \(p,\mathbf v\) reconstructed from \(\rho,P^*\) ?
\displaystyle p = P -\frac{|\mathbf x|^2}{2}
\displaystyle \mathbf v = - D^2P^*\circ\nabla P\Big(\partial_t\nabla P+ J(\nabla P - \mathbf x) \Big)
Easy!
Problem: no meaning a priori
Existence theorem for SGE
Theorem (Ambrosio, Colombo, De Philippis, Figalli 2012):
\((\rho,P^*)\) weak solution to dual SGE on \(\mathbb T^2\) with initial data fulfilling
- \(P_0^*\) convex
- \( 0 < \lambda \leq \rho_0 \leq \Lambda < \infty \)
Then \[\mathbf v = - D^2P^*\circ\nabla P\Big(\partial_t\nabla P+ J(\nabla P - \mathbf x) \Big)\] is well-defined weakly and \((\mathbf v,p=P -\frac{|\mathbf x|^2}{2}) \) is weak solution to SGE
Proof uses:
- Condition \((*)\) retained through time (\(\rho\) is transported)
- \(\lambda \leq \det(D^2P) \leq \Lambda \Rightarrow P\in W^{2,1}\) by De Philippis and Figalli
\((*)\)
THANKS!
References
-
G. De Philippis, A. Figalli
\(W^{2,1}\) regularity for solutions of the Monge-Ampere equation
Invent. Math. (2013)
-
L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli
Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case
Comm. Partial Differential Equations (2012)
Slides available at: slides.com/kmodin
Glimpse into the work of Alessio Figalli
By Klas Modin
Glimpse into the work of Alessio Figalli
Seminar series about Fields Medalists 2018 at the Department of Mathematical Sciences at Chalmers and GU.
