Abstract

Optimization Problems and Inverse Problems are nowadays very popular with several applications in engineering and data analysis. Typically, inverse problems are ill-posed and admit infinite solutions. Analogously, non convex objective functions may admit multiple local extremal points drastically making their identification difficult or even impossible.

A common strategy to overcome this problem is to add regularization terms to the objective function. The aim of the regularization term is to gain convexity to improve the identifiability of local minimizers and improving the well-conditioning of the problem. A very effective and challenging choice of regularizers are based on L1-norms (compressed sensing, LASSO) or Total variation of the optimization design parameters. The use of such regularization strategies is hampered by the difficulty in finding efficient and robust numerical solution algorithms.

In this Seminar we will introduce the problem of regularization in a very broad sense and describe a Legendre duality based approach to the L1-norm and TV regularization. We will discuss numerical solution approaches based on the Dynamic Monge Kantorovich (DMK) scheme, originally developed for the numerical solution of the L1 Optimal Transport problem. We will discuss a few equivalent approaches based on extending the results presented by Bouchitte’ and Buttazzo for the L1 Optimal Transport problem, for which DMK-like gradient flows can be envisaged leading to convergent numerical schemes.

We will show a few interesting applications to classical examples, such as 1-Harmonic functions, 1-D signal TV-Denoising, and compressed sensing of graph Laplacian partial eigenproblem.

The video of the seminar will appear shortly afterwards in this Mediaspace channel.