Abstract We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math. , to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.
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Connecting HamiltonJacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Nonconvex Priors.
Many imaging problems can be formulated as inverse problems expressed as finitedimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multitime HamiltonJacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multitime) HamiltonJacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of nonconvex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging minplus algebra techniques that have been originally developed for solving certain HamiltonJacobi partial differential equations arising in optimal control. Note that connections between viscous HamiltonJacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and logconcave priors have been highlighted in Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and HamiltonJacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain nonlogconcave priors using an analogue of minplus algebra techniques.
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 Award ID(s):
 1820821
 NSFPAR ID:
 10313187
 Date Published:
 Journal Name:
 Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer,
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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