gpeyre/2013-siims-ot-splitting

N. Papadakis, G. Peyré, E. Oudet. Optimal Transport with Proximal Splitting. SIAM Journal on Imaging Sciences, 7(1), pp. 212–238, 2014.

This Matlab package contains the source code to reproduce the figure of the article:

N. Papadakis, G. Peyré, E. Oudet. Optimal Transport with Proximal Splitting. SIAM Journal on Imaging Sciences, 7(1), pp. 212–238, 2014.

Copyright (c) 2009 Gabriel Peyré

============= Description

This archive contains the Douglas-Rachford (DR) and Primal-Dual (PD) solvers applied to the Benamou-Brenier (BB) problem discretized on a staggered grid.

They can be tested with:

• test_bb_dr.m: DR algorithm.
• test_bb_pd.m: DR algorithm.

============= Principal options

The principal options are the following:

1. chose your test case with

test = 'gaussian';

(you can create your own scenario by defining f0 and f1 the initial and final densities)

2. chose the dimension of the problem:

N=32; P=32; Q=32;

(N and P are the discrete spatial dimensions and Q is the temporal discretization)

3. Parameterization of the solver:

For DR:

mu = 1.98; % should be in ]0,2[ gamma = 1./230.; % should be >0 niter = 1000;

For PD:

sigma=85; niter = 1000; % (increase the maximum number of iteration to have better results)

4. Generalized cost functions:

Minimize \sum_k w_k f_k^\alpha |v_k|^2

alpha= 1; % should be in [0;1];

alpha=1 computes the L2-Wasserstein distance, 0 is for the H^-1 one and intermediate values gives interpolations between the norms

obstacle=zeros(N,P,Q);

Define the points of the 3D volume where the mass can not pass. For instance, setting

obstacle(N/2,P/2,:)=1;

will create an obstacle in the middle of the spatial domain.

============= Exemples of settings:

test = 'gaussian'; N=32; P=32; Q=32;
niter = 200;

test = 'obstacle'; niter = 2000;