58,868 research outputs found
Entropic Wasserstein Gradient Flows
This article details a novel numerical scheme to approximate gradient flows
for optimal transport (i.e. Wasserstein) metrics. These flows have proved
useful to tackle theoretically and numerically non-linear diffusion equations
that model for instance porous media or crowd evolutions. These gradient flows
define a suitable notion of weak solutions for these evolutions and they can be
approximated in a stable way using discrete flows. These discrete flows are
implicit Euler time stepping according to the Wasserstein metric. A bottleneck
of these approaches is the high computational load induced by the resolution of
each step. Indeed, this corresponds to the resolution of a convex optimization
problem involving a Wasserstein distance to the previous iterate. Following
several recent works on the approximation of Wasserstein distances, we consider
a discrete flow induced by an entropic regularization of the transportation
coupling. This entropic regularization allows one to trade the initial
Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to
deal with numerically. We show how KL proximal schemes, and in particular
Dykstra's algorithm, can be used to compute each step of the regularized flow.
The resulting algorithm is both fast, parallelizable and versatile, because it
only requires multiplications by a Gibbs kernel. On Euclidean domains
discretized on an uniform grid, this corresponds to a linear filtering (for
instance a Gaussian filtering when is the squared Euclidean distance) which
can be computed in nearly linear time. On more general domains, such as
(possibly non-convex) shapes or on manifolds discretized by a triangular mesh,
following a recently proposed numerical scheme for optimal transport, this
Gibbs kernel multiplication is approximated by a short-time heat diffusion
Efficient sum-of-exponentials approximations for the heat kernel and their applications
In this paper, we show that efficient separated sum-of-exponentials
approximations can be constructed for the heat kernel in any dimension. In one
space dimension, the heat kernel admits an approximation involving a number of
terms that is of the order for any x\in\bbR and
, where is the desired precision. In all
higher dimensions, the corresponding heat kernel admits an approximation
involving only terms for fixed accuracy
. These approximations can be used to accelerate integral
equation-based methods for boundary value problems governed by the heat
equation in complex geometry. The resulting algorithms are nearly optimal. For
points in the spatial discretization and time steps, the cost is
in terms of both memory and CPU time for
fixed accuracy . The algorithms can be parallelized in a
straightforward manner. Several numerical examples are presented to illustrate
the accuracy and stability of these approximations.Comment: 23 pages, 5 figures, 3 table
A characteristic particle method for traffic flow simulations on highway networks
A characteristic particle method for the simulation of first order
macroscopic traffic models on road networks is presented. The approach is based
on the method "particleclaw", which solves scalar one dimensional hyperbolic
conservations laws exactly, except for a small error right around shocks. The
method is generalized to nonlinear network flows, where particle approximations
on the edges are suitably coupled together at the network nodes. It is
demonstrated in numerical examples that the resulting particle method can
approximate traffic jams accurately, while only devoting a few degrees of
freedom to each edge of the network.Comment: 15 pages, 5 figures. Accepted to the proceedings of the Sixth
International Workshop Meshfree Methods for PDE 201
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