A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science

In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented

Internal space renormalization group for the Luttinger model

The absence of fermionic, asymptotical one-particle states in the Luttinger model raises the suspicion that the interactions are actually strong at the vicinity of the Fermi points. The functional internal space renormalization group method, a systematical scheme for the computation of effective coupling strengths off the Fermi points, is applied to shed some light on this issue. A simple truncation of the evolution equation shows that the theory is indeed strongly coupled at the Fermi points for arbitrarily small value of the bare coupling constant and the removal of the cutoff is blocked by Landau poles. A peculiar feature of the normal ordering scheme is pointed out to explain the absence of these effects in the bosonized solution.Comment: 18 page

Functional Callan-Symanzik equation

We describe a functional method to obtain the exact evolution equation of the effective action with a parameter of the bare theory. When this parameter happens to be the bare mass of the scalar field, we find a functional generalization of the Callan-Symanzik equations. Another possibility is when this parameter is the Planck constant and controls the amplitude of the fluctuations. We show the similarity of these equations with the Wilsonian renormalization group flows and also recover the usual one loop effective action.Comment: 17 pages, to appear in Annals of Physic