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Abstract We propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms, are also considered.
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Numerical Optimal Transport from 1D to 2D Using a Non-local Monge-Ampère Equation
Abstract 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-Ampère type equation. We provide a new level-set framework for interpreting this nonlinear 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 they can easily be generalised to higher dimensions. Several challenging computational tests validate the new numerical method.
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- Award ID(s):
- 2308856
- PAR ID:
- 10496037
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- La Matematica
- Volume:
- 3
- Issue:
- 2
- ISSN:
- 2730-9657
- Format(s):
- Medium: X Size: p. 509-535
- Size(s):
- p. 509-535
- Sponsoring Org:
- National Science Foundation
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