421 research outputs found
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
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
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