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95
A Modified ForwardBackward Splitting Method For Maximal Monotone Mappings
 SIAM J. Control Optim
, 1998
"... We consider the forwardbackward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for mon ..."
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Cited by 94 (0 self)
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We consider the forwardbackward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is monotone and (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.
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 66 (9 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(
Existence and stability for FokkerPlanck equations with logconcave reference measure
, 2007
"... We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability pr ..."
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Cited by 41 (12 self)
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We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a FokkerPlanck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.
On The Minimizing Property Of A Second Order Dissipative System In Hilbert Spaces
 SIAM J. CONTROL AND OPTIMIZATION
, 1998
"... We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below then the solution trajectories are minimizing ..."
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Cited by 39 (4 self)
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We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below then the solution trajectories are minimizing for it, and converge weakly towards a minimizer of Phi if there exists one; this convergence is strong when Phi is even or when the optimal set has a nonempty interior. We introduce a second order proximallike iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We find conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.
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 ..."
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Cited by 39 (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.
Existence and uniqueness of nonnegative solutions to the stochastic porous media equation
, 2007
"... ..."
Fast/slow Diffusion And Collapsing Sandpiles
 J.Diff.Equat
, 1996
"... . We regard the limit as p ! 1 of the flow governed by the pLaplacian as providing a simplistic model for the "collapse of an initially unstable sandpile." Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain case ..."
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Cited by 29 (3 self)
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. We regard the limit as p ! 1 of the flow governed by the pLaplacian as providing a simplistic model for the "collapse of an initially unstable sandpile." Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases. In particular we note that such models entail "instantaneous " mass transfer governed by MongeKantorovich theory. 1. Introduction We study in this paper a "fast/slow diffusion" PDE problem, which can be very loosely interpreted as modelling the collapse of a sandpile from an initially unstable configuration. The mathematical issue is to understand the behavior of the solution u p of the initial value problem ae u p;t \Gamma \Delta p u p = 0 in R n \Theta (0; 1) u p = g on R n \Theta ft = 0g (1.1) in the "infinitely fast/infinitely slow diffusion" limit p ! 1. Here 1 p ! 1, \Delta p u p = div(jDu p j p\Gamma2 Du p ) is the pLaplacian, and Du p denotes the gradient of u p with respect...
Crystalline mean curvature flow of convex sets
, 2004
"... We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R N. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalize ..."
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Cited by 25 (7 self)
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We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R N. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φcurvature flow in the sense of AlmgrenTaylorWang. As a byproduct, it turns out that the flat φcurvature flow starting from a compact convex set is unique.
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 23 (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 ...
Generation theory for semigroups of holomorphic mappings in Banach spaces
 Abstr. Appl. Anal
, 1996
"... Abstract. We study nonlinear semigroups ofholomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog ofthe Hille exponential formula. We then apply our resul ..."
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Cited by 22 (15 self)
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Abstract. We study nonlinear semigroups ofholomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog ofthe Hille exponential formula. We then apply our results to the null point theory ofsemiplus complete vector fields. We study the structure ofnull point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of the implicit function theorem and a discussion of some open problems are also included.