Numerical solution of the second boundary value problem for the Elliptic Monge-Amp ere equation

This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampere equation. The boundary conditions correspond to the optimal transportation of measures supported on two domains, where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit and non-local. These boundary conditions are reformulated as a nonlinear Hamilton-Jacobi PDE on the boundary. This formulation allows us to extend the convergent, wide stencil Monge-Ampere solvers proposed by Froese and Oberman to this problem. Several non-trivial computational examples demonstrate that the method is robust and fast.

[1]  Philippe Delanoë.  Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator , 1991 .

[2]  J. Frédéric Bonnans,et al.  Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation , 2003, SIAM J. Numer. Anal..

[3]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[4]  Boualem Khouider,et al.  An efficient numerical algorithm for the L2 optimal transport problem with applications to image processing , 2010, 1009.6039.

[5]  John Norbury,et al.  Generalised Lagrangian solutions for atmospheric and oceanic flows , 1991 .

[6]  Adam M. Oberman,et al.  The Dirichlet problem for the convex envelope , 2010, 1007.0773.

[7]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[8]  Adam M. Oberman.  A convergent monotone difference scheme for motion of level sets by mean curvature , 2004, Numerische Mathematik.

[9]  Jean-David Benamou,et al.  Mixed L2-Wasserstein Optimal Mapping Between Prescribed Density Functions , 2001 .

[10]  G. Loeper,et al.  Numerical Analysis/Partial Differential Equations Numerical solution of the Monge-Ampère equation by a Newton's algorithm , 2005 .

[11]  U. Frisch,et al.  A reconstruction of the initial conditions of the Universe by optimal mass transportation , 2001, Nature.

[12]  Gian Luca Delzanno,et al.  Robust, multidimensional mesh-motion based on Monge-Kantorovich equidistribution , 2011, J. Comput. Phys..

[13]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[14]  Bertrand Maury,et al.  Handling congestion in crowd motion modeling , 2011, Networks Heterog. Media.

[15]  L. Rüschendorf,et al.  Numerical and analytical results for the transportation problem of Monge-Kantorovich , 2000 .

[16]  Adam M. Oberman.  The convex envelope is the solution of a nonlinear obstacle problem , 2007 .

[17]  M. Cullen,et al.  An Extended Lagrangian Theory of Semi-Geostrophic Frontogenesis , 1984 .

[18]  Pierre-Louis Lions,et al.  THE Neumann problem for equations of Monge-Ampère type , 1986 .

[19]  Adam M. Oberman,et al.  Convergent Difference Schemes for Degenerate Elliptic and Parabolic Equations: Hamilton-Jacobi Equations and Free Boundary Problems , 2006, SIAM J. Numer. Anal..

[20]  J. F. Williams,et al.  MOVING MESH GENERATION USING THE PARABOLIC MONGE–AMPÈRE EQUATION∗ , 2008 .

[21]  Adam M. Oberman,et al.  Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher , 2010, SIAM J. Numer. Anal..

[22]  Dimitri P. Bertsekas,et al.  Auction algorithms for network flow problems: A tutorial introduction , 1992, Comput. Optim. Appl..

[23]  Adam M. Oberman,et al.  Exact semi-geostrophic flows in an elliptical ocean basin , 2004 .

[24]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[25]  Brittany D. Froese.  A Numerical Method for the Elliptic Monge-Ampère Equation with Transport Boundary Conditions , 2011, SIAM J. Sci. Comput..

[26]  W. Wasow,et al.  On the Approximation of Linear Elliptic Differential Equations by Difference Equations with Positive Coefficients , 1952 .

[27]  A. Lachapelle,et al.  COMPUTATION OF MEAN FIELD EQUILIBRIA IN ECONOMICS , 2010 .

[28]  J. Urbas.  On the second boundary value problem for equations of Monge-Ampère type. , 1997 .

[29]  Dimitri P. Bertsekas,et al.  Convex Analysis and Optimization , 2003 .

[30]  D. Kinderlehrer,et al.  Approximation of Parabolic Equations Using the Wasserstein Metric , 1999 .

[31]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[32]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[33]  Allen Tannenbaum,et al.  On the Computation of Optimal Transport Maps Using Gradient Flows and Multiresolution Analysis , 2008, Recent Advances in Learning and Control.

[34]  Allen R. Tannenbaum,et al.  An Efficient Numerical Method for the Solution of the L2 Optimal Mass Transfer Problem , 2010, SIAM J. Sci. Comput..

[35]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1991 .

[36]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[37]  J. A. Carrillo,et al.  Numerical Simulation of Diffusive and Aggregation Phenomena in Nonlinear Continuity Equations by Evolving Diffeomorphisms , 2009, SIAM J. Sci. Comput..

[38]  L. Caffarelli.  Boundary regularity of maps with convex potentials , 1992 .

[39]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[40]  V. Oliker,et al.  Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem , 2003 .

[41]  L. Caffarelli.  Boundary regularity of maps with convex potentials – II , 1996 .