Speaker: Yves Lucet
Topic: Recent Advances in Computational Convex Analysis

Abstract:

Computational convex analysis focuses on the computation of convex operators that routinely appear in convex analysis e.g. Legendre-Fenchel conjugate, Moreau envelope, etc. It has found applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton–Jacobi equations and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the fast Fourier transform), multifractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.

Starting from known algorithm and their numerical implementation in the CCA numerical library, we will demonstrate what fast algorithms are available, their strengths and their limitations, before explaining the challenges to extend them to piecewise linear-quadratic functions in higher dimension. The basic tools are convex analysis, computational geometry, and parametric optimization.