Two-scale methods for convex envelopes

Li, Wenbo; Nochetto, Ricardo

doi:10.1090/mcom/3521

Two-scale methods for convex envelopes
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by Wenbo Li and Ricardo H. Nochetto;

Math. Comp. 91 (2022), 111-139

DOI: https://doi.org/10.1090/mcom/3521

Published electronically: October 13, 2021

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Abstract:

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780].

References

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271–283. MR 1115933, DOI 10.3233/ASY-1991-4305
Sören Bartels, Linear convergence in the approximation of rank-one convex envelopes, M2AN Math. Model. Numer. Anal. 38 (2004), no. 5, 811–820. MR 2104430, DOI 10.1051/m2an:2004040
Olivier Bokanowski, Stefania Maroso, and Hasnaa Zidani, Some convergence results for Howard’s algorithm, SIAM J. Numer. Anal. 47 (2009), no. 4, 3001–3026. MR 2551155, DOI 10.1137/08073041X
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 19–27. MR 864651, DOI 10.4171/RMI/23
Guido De Philippis and Alessio Figalli, Optimal regularity of the convex envelope, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4407–4422. MR 3324933, DOI 10.1090/S0002-9947-2014-06306-X
Georg Dolzmann, Numerical computation of rank-one convex envelopes, SIAM J. Numer. Anal. 36 (1999), no. 5, 1621–1635. MR 1706747, DOI 10.1137/S0036142997325581
G. Dolzmann and N. J. Walkington, Estimates for numerical approximations of rank one convex envelopes, Numer. Math. 85 (2000), no. 4, 647–663. MR 1771783, DOI 10.1007/PL00005395
Xiaobing Feng and Max Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids, SIAM J. Numer. Anal. 55 (2017), no. 2, 691–712. MR 3623696, DOI 10.1137/16M1061709
Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, vol. 89, Birkhäuser/Springer, [Cham], 2016. Second edition [of MR1829162]. MR 3560611, DOI 10.1007/978-3-319-43374-5
M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3, 865–888 (2003). MR 1972219, DOI 10.1137/S1052623401383558
Max Jensen and Iain Smears, On the notion of boundary conditions in comparison principles for viscosity solutions, Hamilton-Jacobi-Bellman equations, Radon Ser. Comput. Appl. Math., vol. 21, De Gruyter, Berlin, 2018, pp. 143–154. MR 3823876
Wenbo Li and Ricardo H. Nochetto, Optimal pointwise error estimates for two-scale methods for the Monge-Ampère equation, SIAM J. Numer. Anal. 56 (2018), no. 3, 1915–1941. MR 3819162, DOI 10.1137/18M1165670
R. H. Nochetto, D. Ntogkas, and W. Zhang, Two-scale method for the Monge-Ampère equation: convergence to the viscosity solution, Math. Comp. 88 (2019), no. 316, 637–664. MR 3882279, DOI 10.1090/mcom/3353
R. H. Nochetto, D. Ntogkas, and W. Zhang, Two-scale method for the Monge-Ampère equation: pointwise error estimates, IMA J. Numer. Anal. 39 (2019), no. 3, 1085–1109. MR 3984051, DOI 10.1093/imanum/dry026
Ricardo H. Nochetto and Wujun Zhang, Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form, Found. Comput. Math. 18 (2018), no. 3, 537–593. MR 3807356, DOI 10.1007/s10208-017-9347-y
Ricardo H. Nochetto and Wujun Zhang, Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Ampère equation, Numer. Math. 141 (2019), no. 1, 253–288. MR 3903208, DOI 10.1007/s00211-018-0988-9
Adam M. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689–1694. MR 2286077, DOI 10.1090/S0002-9939-07-08887-9
Adam M. Oberman, Computing the convex envelope using a nonlinear partial differential equation, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 759–780. MR 2413037, DOI 10.1142/S0218202508002851
Adam M. Oberman and Yuanlong Ruan, A partial differential equation for the rank one convex envelope, Arch. Ration. Mech. Anal. 224 (2017), no. 3, 955–984. MR 3621814, DOI 10.1007/s00205-017-1092-5
Adam M. Oberman and Luis Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5871–5886. MR 2817413, DOI 10.1090/S0002-9947-2011-05240-2
Martin L. Puterman and Shelby L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res. 4 (1979), no. 1, 60–69. MR 543609, DOI 10.1287/moor.4.1.60
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. MR 274683, DOI 10.1515/9781400873173
Iain Smears and Endre Süli, Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients, SIAM J. Numer. Anal. 52 (2014), no. 2, 993–1016. MR 3196952, DOI 10.1137/130909536
Neil S. Trudinger and John I. E. Urbas, On second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc. 30 (1984), no. 3, 321–334. MR 766792, DOI 10.1017/S0004972700002069
Michael Ulbrich, Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, MOS-SIAM Series on Optimization, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2011. MR 2839219, DOI 10.1137/1.9781611970692
Shawn W. Walker, FELICITY: A MATLAB/C++ toolbox for developing finite element methods and simulation modeling, SIAM J. Sci. Comput. 40 (2018), no. 2, C234–C257. MR 3775142, DOI 10.1137/17M1128745
S. W. Walker, FELICITY: Finite ELement Implementation and Computational Interface Tool for You, http://www.mathworks.com/matlabcentral/fileexchange/31141-felicity.