Z-mappings
for mathematicians
Flavien Léger (inria)
Joint work with Alfred Galichon (NYU)
Introduction
Nonlinear and nonvariational equations
$$Q(u) = 0$$
Common structure to many problems in:
network flows, optimal transport, optimal control, interface dynamics, submodular functions...
Outline
1. Generalities
2. Theory and an algorithm
3. Examples
1. Generalities
We work on a network \((X,A)\) or domain \(\Omega\subset\mathbb{R}^d\).
$$Q(u) = 0$$
\(u\colon X\to\mathbb{R}\) or \(\Omega\to\mathbb{R}\).
Notation:
We write \(Q_x(u)\) for \(Q(u)(x)\)
We sometimes write \(u_x\) for \(u(x)\).
Z-mapping
\(Q\) is a Z-mapping if for all \(u,\tilde u\), for all \(x\),
D E F I N I T I O N
Introduced by Rheinboldt '70, gross substitutes in economics
Definitions
-
isotone if for all \(u\), \(\tilde u\),
\(u\le \tilde u \implies Q(u)\le Q(\tilde u) \) -
inverse isotone if for all \(u\), \({\tilde u}\),
\[Q(u)\le Q(\tilde u) \implies u\le \tilde u\]
Remark: If \(Q\) is inverse isotone then it is injective.
An M-mapping is an inverse isotone Z-mapping.
D E F I N I T I O N
We say that a mapping \(Q\) is
Example: network flow
\[-\mathrm{div} M(u)=f\]
\(M\colon\mathbb{R}^X\to\mathbb{R}^A\) such that:
\(M_{xy}(u)\) increasing in \(u_y\) decreasing in \(u_x\).
Ex: \(M_{xy}(u)=u_y-u_x\),
\(M_{xy}(u)=e^{u_y-u_x}\)
\(Q(u)=-\mathrm{div} M(u) - f\) is a Z-mapping.
\(\longrightarrow -\Delta u = f\)
\(\longrightarrow\) Entropic transport
"flow mapping"
Example: network flow (PDE)
\[-\mathrm{div} \Big(M(x,u(x),\nabla u(x))\Big)=f\]
\(M \colon\Omega\times\mathbb{R}\times \mathbb{R}^d\to\mathbb{R}^d\) such that:
\(M(x,z,p)\) is increasing in \(z\) and monotone in \(p\): $$\langle M(x,z,\tilde p)-M(x,z,p), \tilde p - p\rangle \ge 0$$
\(Q(u)=-\mathrm{div} M(u) - f\) is a Z-mapping.
quasilinear PDE
Example: optimal control
\[\min_{(a_t)}\mathbb{E}\sum_{t=0}^{T}c(x_t,a_t) \]
subject to:
\(T\) = stopping time
(Shortest path, stochastic shortest path, Markov Decision Process)
\(=:u(x)\)
Dynamic programming principle:
\(=:T_x(u)\)
\[u(x)=\min_{a}\Big[c(x,a)+\sum_{y\in X}u(y)p(y|x,a)\Big]\]
Example: optimal control
(Shortest path, stochastic shortest path, Markov Decision Process)
Solve \(u=T(u)\) with \(u\le \tilde u\implies T(u)\le T(\tilde u)\)
\(Q(u)\coloneqq u - T(u)\) is a Z-mapping.
\[T_x(u) = \min_{a}\Big[c(x,a)+\sum_{y\in X}u(y)p(y|x,a)\Big]\]
Example: optimal control (PDE)
\[u(x) = \inf_{v} \int_0^T L(x_t,v_t)\,dt\]
subject to: \(u = 0\) on \(\partial\Omega\)
Then \(Q_x(u)= H(x,-\nabla u(x))\) is a Z-mapping.
Ex (Eikonal):
see also: viscosity solutions
2. Theory and an algorithm
In this section: Z-mapping and \(Q_x(u)\) isotone in \(u_x\)Domain is discrete \((X,A)\).
D E F I N I T I O N
\(u\) is a subsolution if
\[Q(u) \le 0\]
\(u\) is a supersolution if
\[Q(u) \ge 0.\]
Jacobi
D E F I N I T I O N
The Jacobi transform of \(u\) is
\[u^*(x) = \sup\{s\in\R : Q_x(s,u_{-x})\le 0\}.\]
Jacobi algorithm
\[u_{n+1} = u_n^*\]
\(Q_x(u^*_x,u_{-x})=0\)
A L G O R I T H M
Properties of Jacobi
Let \(Q\) be a continuous Z-mapping.
If \(Q(u)\le 0\) then
\[Q(u^*)\le 0 \quad\text{and}\quad u^* \ge u\]
P R O P O S I T I O N
Useful for algorithm or showing existence of solutions:
if \(Q(u_0)\le 0\), then
\[Q(u_n)\le 0, \quad u_{n+1}\ge u_n\]
"method of subsolutions", "Perron's method"...
Let \(Q\) be a continuous Z-mapping. Then
\[u\le \tilde u \implies u^*\le \tilde{u}^*\]
P R O P O S I T I O N
Useful when sandwich \(v_0\le u_0\le w_0\), then
\[v_n\le u_n\le w_n\]
\(v_n\): subsolution, thus increasing to a solution
\(w_n\): supersolution, thus decreasing to a solution
Properties of Jacobi
Rheinbolt ’70
M-mappings
Suppose that a Z-mapping \(Q\) satisfies:
\[u\mapsto \psi(Q(u)) \text{ strictly isotone}\]
with \(\psi\) isotone and more "strictness" assumptions
Then \(Q\) is an M-mapping.
Recall that M-mapping = Z-mapping and comparison
\[Q(u)\le Q(\tilde u)\implies u\le \tilde u\]
T H E O R E M
L., Galichon '2022
ex: \(\sum_xQ_x(u)\)
Berry, Gandhi, Haile '2013; Chen, Choo, Galichon, Weber '2021
M-mappings
Suppose that a Z-mapping \(Q\) satisfies:
\[\lambda\mapsto Q(e_\lambda u) \text{ strictly isotone}\]
with \(\lambda\mapsto e_\lambda u\) isotone and more "strictness" assumptions
Then \(Q\) is an M-mapping.
T H E O R E M
L., Galichon '2022
Example: generated Jacobian equations
\((X,\mu)\) and \((Y,\nu)\)
\(G\colon X\times Y\times \mathbb{R}\to\mathbb{R}\) such that \(G(x,y,z)\) decreasing in \(z\).
Ex: \(G(x,y,z) = \langle x,y\rangle - z\).
Given \(v\colon Y\to\mathbb{R}\), let
$$T_v(x)=\argmax_y G(x,y,v(y))$$
Goal: find \(v\) such that
$$T_{v\#}\mu=\nu$$
3. Examples
3.1. Generated Jacobian equations
\(Q(u) = \nu-T_{v\#}\mu\) is a Z-mapping
\((X,\mu)\) and \((Y,\nu)\), \(G\colon X\times Y\times \mathbb{R}\to\mathbb{R}\) with \(G(x,y,z)\) decreasing in \(z\).
Given \(v\colon Y\to\mathbb{R}\), let
$$T_v(x)=\argmax_y G(x,y,v(y))$$
Goal: find \(v\) such that
$$T_{v\#}\mu=\nu$$
3. Examples
Particular case: Optimal transport
\(G(x,y,z)=-c(x,y)-z\)
Rem:discontinuity \(\to\) need regularization.
Real auction, or replace min by softmin: Entropic OT. Jacobi is Sinkhorn.
\[T_{v\#}\mu=\nu\]
Algorithm: Jacobi = Bertsekas' (naive) auction algorithm
$$T_v(x)=\argmax_y -c(x,y)-v(y)$$
PDEs: Monge–Ampère
3. Examples
“Particular case”: Gale–Shapley
\(G(x,y,z)=\alpha(x,y)-\iota_{\{\gamma(x,y)\ge z\}}\)
\[T_{v\#}\mu=\nu\]
Algorithm: Jacobi ≈ Gale–Shapley algorithm
$$T_v(x)=\argmax_{\{y : \gamma(x,y)\ge v(y)\}} \alpha(x,y)$$
(Gale–Shapley ’62)
3.2. Mean curvature motion
mean curvature motion = \(L^2\) gradient flow of
\[P(u)=\frac12\int_{\mathbb{R}^d} \lvert\nabla u\rvert\,dx\]
(\(u=1_K\)). Heat content:
\[P_\varepsilon(u)=-\frac{1}{\varepsilon}\iint G_\varepsilon(x-y) u(x) (1-u(y))\,dxdy + \int \chi(u(x))\,dx\]
submodular, nonconvex
\(Q(u)=DP(u)\) is a Z-mapping.
3. Examples
Algorithm: Jacobi = MBO
(Merriman–Bence–Osher ’92)