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.

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 $u_x$.

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 $p$: $$\langle M(x,z,\tilde p)-M(x,z,p), \tilde p - p\rangle \ge 0$$

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)

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.

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
    $$u^*(x) = \sup\{s\in\R : Q_x(s,u_{-x})\le 0\}.$$

$Q_x(u^*_x,u_{-x})=0$

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

Recall that M-mapping = Z-mapping and comparison

$$Q(u)\le Q(\tilde u)\implies u\le \tilde u$$

M-mappings

T H E O R E M

L., Galichon '2022

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

$G(x,y,z)=-c(x,y)-z$

$$T_{v\#}\mu=\nu$$

$$T_v(x)=\argmax_y -c(x,y)-v(y)$$

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)

Thank you!