TY - JOUR

T1 - A convergence framework for optimal transport on the sphere

AU - Hamfeldt, Brittany Froese

AU - Turnquist, Axel G.R.

N1 - Funding Information:
Brittany Froese Hamfeldt was partially supported by NSF DMS-1619807 and NSF DMS-1751996. Axel G. R. Turnquist was partially supported by an NSF GRFP.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/7

Y1 - 2022/7

N2 - We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge–Ampère type equations on the plane. Existing techniques for proving convergence do not immediately apply because the PDE lacks both a comparison principle and a unique solution, which makes it difficult to produce a stable, well-posed scheme. By augmenting the discretization with an additional term that constrains the solution gradient, we obtain a strong form of stability. A modification of the Barles–Souganidis convergence framework then establishes convergence to the mean-zero solution of the original PDE.

AB - We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge–Ampère type equations on the plane. Existing techniques for proving convergence do not immediately apply because the PDE lacks both a comparison principle and a unique solution, which makes it difficult to produce a stable, well-posed scheme. By augmenting the discretization with an additional term that constrains the solution gradient, we obtain a strong form of stability. A modification of the Barles–Souganidis convergence framework then establishes convergence to the mean-zero solution of the original PDE.

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U2 - 10.1007/s00211-022-01292-1

DO - 10.1007/s00211-022-01292-1

M3 - Article

AN - SCOPUS:85131330110

VL - 151

SP - 627

EP - 657

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -