9 research outputs found
A strong comparison principle for the generalized Dirichlet problem for Monge-Ampere
We prove a strong form of the comparison principle for the elliptic
Monge-Ampere equation, with a Dirichlet boundary condition interpreted in the
viscosity sense. This comparison principle is valid when the equation admits a
Lipschitz continuous weak solution. The result is tight, as demonstrated by
examples in which the strong comparison principle fails in the absence of
Lipschitz continuity. This form of comparison principle closes a significant
gap in the convergence analysis of many existing numerical methods for the
Monge-Ampere equation. An important corollary is that any consistent, monotone,
stable approximation of the Dirichlet problem for the Monge-Ampere equation
will converge to the viscosity solution
Numerical Optimal Transport from 1D to 2D using a Non-local Monge-Amp\`ere Equation
We consider the numerical solution of the optimal transport problem between
densities that are supported on sets of unequal dimension. Recent work by
McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type
equation. We provide a new level set framework for interpreting this non-linear
PDE. We also propose a novel discretisation that combines carefully constructed
monotone finite difference schemes with a variable-support discrete version of
the Dirac delta function. The resulting method is consistent and monotone.
These new techniques are described and implemented in the setting of 1D to 2D
transport, but can easily be generalised to higher dimensions. Several
challenging computational tests validate the new numerical method
On the Reduction in Accuracy of Finite Difference Schemes on Manifolds without Boundary
We investigate error bounds for numerical solutions of divergence structure
linear elliptic PDEs on compact manifolds without boundary. Our focus is on a
class of monotone finite difference approximations, which provide a strong form
of stability that guarantees the existence of a bounded solution. In many
settings including the Dirichlet problem, it is easy to show that the resulting
solution error is proportional to the formal consistency error of the scheme.
We make the surprising observation that this need not be true for PDEs posed on
compact manifolds without boundary. By carefully constructing barrier
functions, we prove that the solution error achieved by a scheme with
consistency error is bounded by
in dimension . We also provide a specific
example where this predicted convergence rate is observed numerically. Using
these error bounds, we further design a family of provably convergent
approximations to the solution gradient.Comment: 28 pages, 7 figure
A Convergent Quadrature Based Method For The Monge-Amp\`ere Equation
We introduce an integral representation of the Monge-Amp\`ere equation, which
leads to a new finite difference method based upon numerical quadrature. The
resulting scheme is monotone and fits immediately into existing convergence
proofs for the Monge-Amp\`ere equation with either Dirichlet or optimal
transport boundary conditions. The use of higher-order quadrature schemes
allows for substantial reduction in the component of the error that depends on
the angular resolution of the finite difference stencil. This, in turn, allows
for significant improvements in both stencil width and formal truncation error.
We present two different implementations of this method. The first exploits the
spectral accuracy of the trapezoid rule on uniform angular discretizations to
allow for computation on a nearest-neighbors finite difference stencil over a
large range of grid refinements. The second uses higher-order quadrature to
produce superlinear convergence while simultaneously utilizing narrower
stencils than other monotone methods. Computational results are presented in
two dimensions for problems of various regularity.Comment: 25 pages, 6 figure