The Least Gradient Flow, the Total Variation Flow, and the Inverse Mean Curvature Flow: Applications, Theory, and Numerical Approximations

Xiaobing Feng, Department of Mathematics, The University of Tennessee

The least gradient (LG) flow, the total variation (TV) flow, and the inverse mean curvature (IMC) flow arise from diverse applications in geometry (nonparametric minimal surfaces), image processing (image denosing), and the general relativity (isoperimetric inequality for black holes), respectively. However, their mathematical descriptions and the difficulties are closely related. This "close" relation can then be exploited when comes to develop efficient numerical methods for approximating these flows. In this talk, I will first explain the origins/applications of these flows, then describe the "connection" between them, and finally discuss some recent developments on PDE analysis and numerical analysis of the LG, TV and IMC flows. Numerical experiment results will also be presented in the talk.