Results 1  10
of
344
Image Sequence Analysis via Partial Differential Equations
, 1999
"... This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, ..."
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Cited by 44 (3 self)
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This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and viceversa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtaine...
Error bounds for monotone approximation schemes for HamiltonJacobiBellman equations
, 2007
"... We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic HamiltonJacobiBellman equations by introducing a new notion of consistency. Our results are robust and general – they improve and extend earlier results by Kry ..."
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Cited by 36 (2 self)
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We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic HamiltonJacobiBellman equations by introducing a new notion of consistency. Our results are robust and general – they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank–Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 33 (9 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
Semismooth Newton methods for operator equations in function spaces
, 2000
"... We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our resul ..."
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Cited by 29 (3 self)
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We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and fforder semismoothness from finitedimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finitedimensional Csubdifferential to our infinitedimensional framework. We apply these semismoothness results to develop a Newtonlike method for nonsmooth operator equations and prove its local qsuperlinear convergence to regular solutions. If the underlying operator is fforder semismoothness, convergence of qorder 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...
Adaptive Multilevel  Methods for Obstacle Problems
, 1992
"... We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by ..."
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Cited by 26 (3 self)
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We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned cgiterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semilocal and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.
Residual Type A Posteriori Error Estimates For Elliptic Obstacle Problems
 Numer. Math
"... . A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positiv ..."
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Cited by 22 (9 self)
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. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Key words. a posteriori error estimates, residual, elliptic obstacle, positivity preserving interpolation. 1991 Mathematics Subject Classification. 65N15, 65N30; 41A05, 41A29, 41A36 1 Introduction A posteriori error estimates are computable quantities in terms of the discrete solution and data, which are instrumental for adaptive mesh refinement (and coarsening), error control, and equidistribution of the computational effort. Since the seminal pa...
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 21 (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.
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 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.
Stochastic mathematical programs with equilibrium constraints, modeling and . . .
 SCHOOL OF INDUSTRIAL AND SYSTEM ENGINEERING, GEORGIA INSTITUTE OF TECHNOLOGY
, 2005
"... In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We sho ..."
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Cited by 20 (5 self)
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In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized KarushKuhnTucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally we present some preliminary numerical test results.