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52
Minimizing total variation flow
 Differential and Integral Equations
, 2001
"... (Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect ..."
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Cited by 43 (6 self)
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(Submitted by: Jerry Goldstein) Abstract. We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t →∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts. 1. Introduction. Let Ω be a bounded set in R N with Lipschitzcontinuous boundary ∂Ω. We are interested in the problem ∂u Du = div(
Numerical Solution Of Reservoir Flow Models Based On Large Time Step Operator Splitting Algorithms
 FILTRATION IN POROUS MEDIA AND INDUSTRIAL APPLICATIONS, LECTURE NOTES IN MATHEMATICS
, 1999
"... During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to revie ..."
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Cited by 25 (14 self)
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During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multiphase flow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic twophase flow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convectiondiffusion equation. Both theory and applications are discussed. The general background for the reservoir flow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct ...
Monotone Difference Approximations of BV Solutions to Degenerate ConvectionDiffusion Equations
 SIAM J. Numer. Anal
, 1998
"... We consider consistent, conservativeform, monotone difference schemes for nonlinear convectiondiffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we ch ..."
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Cited by 21 (11 self)
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We consider consistent, conservativeform, monotone difference schemes for nonlinear convectiondiffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [31] states that these socalled BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present cont...
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 use th ..."
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Cited by 21 (7 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.
Semiflows generated by Lipschitz perturbations of nondensely defined linear operators
, 2003
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A Posteriori Error Estimates for Variable TimeStep Discretizations of Nonlinear Evolution Equations
"... We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error ..."
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Cited by 19 (2 self)
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We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive timesteps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.
Analysis of total variation flow and its finite element approximations
 M2AN MATH. MODEL. NUMER. ANAL
, 2002
"... We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish wellposedness of the problem by the energy method. The main idea of our approach is t ..."
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Cited by 14 (2 self)
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We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish wellposedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [19] and the prescribed mean curvature flow [15]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h, k → 0, and to the total variation gradient flow problem as h, k, ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h²). In particular, it is shown that all error bounds depend on 1/ε only in some lower polynomial order for small ε.
Some Qualitative Properties for the Total Variational Flow
"... We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also s ..."
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Cited by 13 (0 self)
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We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also studied. It is shown that the proles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely dierent to the case of the problem associated to the pLaplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations. Key words: Total variation ow, nonlinear parabolic equations, asymptotic behaviour, eigenvalue type problem, propagation of the support. AMS (MOS) subject classication: 35K65, 35K55. 1 Introduction Let be a ...
On a nonlocal aggregation model with nonlinear diffusion
, 2008
"... Abstract. We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the wellposedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative ..."
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Cited by 11 (0 self)
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Abstract. We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the wellposedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops L ∞ xnorm blowup in finite time. 1. Introduction and