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The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we u ..."
Abstract

Cited by 27 (9 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
H 1 solutions of a class of fourth order nonlinear equations for image processing
 Discrete and Continuous Dynamical Systems, 10(1 and 2), January and
, 2004
"... Abstract. Recently fourth order equations of the form ut = −∇·((G(Jσu))∇∆u) have been proposed for noise reduction and simplification of two dimensional images. The operator G is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator Jσ is ..."
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Cited by 12 (4 self)
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Abstract. Recently fourth order equations of the form ut = −∇·((G(Jσu))∇∆u) have been proposed for noise reduction and simplification of two dimensional images. The operator G is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator Jσ is a standard mollifier. Using ODE methods on Sobolev spaces, we prove existence and uniqueness of solutions of this problem for H1 initial data. 1. Introduction. Image
Convergence Analysis of a Finite Difference Scheme for the Gradient Flow associated with the ROF Model
, 2011
"... We present a convergent analysis of a finite difference scheme for the time dependent partial different equation called gradient flow associated with the RudinOsherFetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of th ..."
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We present a convergent analysis of a finite difference scheme for the time dependent partial different equation called gradient flow associated with the RudinOsherFetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme. An application for image denoising is given. 1
and
, 1999
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov’s method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L1 for entropy solutions. To prove the existence we use t ..."
Abstract
 Add to MetaCart
We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov’s method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L1 for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions. 2001 Academic Press 1.