3,489 research outputs found
The Monge-Ampere equation: various forms and numerical methods
We present three novel forms of the Monge-Ampere equation, which is used,
e.g., in image processing and in reconstruction of mass transportation in the
primordial Universe. The central role in this paper is played by our Fourier
integral form, for which we establish positivity and sharp bound properties of
the kernels. This is the basis for the development of a new method for solving
numerically the space-periodic Monge-Ampere problem in an odd-dimensional
space. Convergence is illustrated for a test problem of cosmological type, in
which a Gaussian distribution of matter is assumed in each localised object,
and the right-hand side of the Monge-Ampere equation is a sum of such
distributions.Comment: 24 pages, 2 tables, 5 figures, 32 references. Submitted to J.
Computational Physics. Times of runs added, multiple improvements of the
manuscript implemented
Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity
We investigate the existence of ground state solutions for a class of
nonlinear scalar field equations defined on whole real line, involving a
fractional Laplacian and nonlinearities with Trudinger-Moser critical growth.
We handle the lack of compactness of the associated energy functional due to
the unboundedness of the domain and the presence of a limiting case embedding.Comment: 13 page
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
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