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31
A Saddle Point Approach to the Computation of Harmonic Maps
, 2006
"... In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and where the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic ..."
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In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and where the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. Minimization problems of the form studied here arise for example in liquid crystal and superconductor simulations. The solution is characterized by a nonlinear saddle point problem, and we show that the corresponding linearized problem is well–posed near the exact solution. The main result of this paper is to establish a corresponding result for a proper finite element discretization of the harmonic map problem. Iterative schemes for the discrete nonlinear saddle point problems are investigated. Some mesh independent preconditioners for the iterative methods are also proposed.
A LINE SEARCH MULTIGRID METHOD FOR LARGESCALE NONLINEAR OPTIMIZATION
, 2009
"... We present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems. At each iteration on each level, the algorithm computes either a “direct search” direction on the current level or a “recursive search” direction from coar ..."
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We present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems. At each iteration on each level, the algorithm computes either a “direct search” direction on the current level or a “recursive search” direction from coarser level models. Introducing a new condition that must be satisfied by a backtracking line search procedure, the “recursive search” direction is guaranteed to be a descent direction. Global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels. Using a limited memory BFGS quasiNewton method to produce the “direct search” direction, preliminary numerical experiments show that our line search multigrid approach is promising.
Some New Domain Decomposition and Multigrid Methods for Variational Inequalities
, 2002
"... this paper, we use I h as the linear Lagrangian interpolation operator which uses the function values at the hlevel nodes. In addition, we also need a nonlinear interpolation operator I H : S h 7! SH . Assume that n0 are all the interior nodes for TH and let ! i be the support for the ..."
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this paper, we use I h as the linear Lagrangian interpolation operator which uses the function values at the hlevel nodes. In addition, we also need a nonlinear interpolation operator I H : S h 7! SH . Assume that n0 are all the interior nodes for TH and let ! i be the support for the nodal basis function of the coarse mesh at x 0 . The nodal values for I H v for any v 2 S h is de ned as (I H v)(x 0 ) = min x2! i v(x), c.f [13]. This operator satis es H v v; 8v 2 S h ; and I H v 0; 8v 0; v 2 S h : (15) Moreover, it has the following monotonicity property v I v; 8h 1 h 2 h; 8v 2 S h : (16) As I v equals v at least at one point in ! i , it is thus true that for any u; v 2 S h kI u I v (u v)k 0 c d H ju vj 1 ; jI vj 1 c d jvj 1 ; (17) where d indicates the dimension of the physical domain i.e
Wavelets and Optical Flow Motion Estimation
"... Abstract. This article describes the implementation of a simple waveletbased opticalflow motion estimator dedicated to continuous motions such as fluid flows. The wavelet representation of the unknown velocity field is considered. This scalespace representation, associated to a simple gradientbas ..."
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Abstract. This article describes the implementation of a simple waveletbased opticalflow motion estimator dedicated to continuous motions such as fluid flows. The wavelet representation of the unknown velocity field is considered. This scalespace representation, associated to a simple gradientbased optimization algorithm, sets up a welldefined multiresolution framework for the optical flow estimation. Moreover, a very simple closure mechanism, approaching locally the solution by highorder polynomials, is provided by truncating the wavelet basis at fine scales. Accuracy and efficiency of the proposed method is evaluated on image sequences of turbulent fluid flows.
An Additive Schwarz Method for the Constrained Minimization of Functionals in Reflexive Banach Spaces
"... Summary. In this paper, we show that the additive Schwarz method proposed in [3] to solve oneobstacle problems converges in a much more general framework. We prove that this method can be applied to the minimization of functionals over a general enough convex set in a reflexive Banach space. In the ..."
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Summary. In this paper, we show that the additive Schwarz method proposed in [3] to solve oneobstacle problems converges in a much more general framework. We prove that this method can be applied to the minimization of functionals over a general enough convex set in a reflexive Banach space. In the Sobolev spaces, the proposed method is an additive Schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals. Also, we show that the one, twolevel variants of the method in the finite element space converge, and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters. 1
A Multilevel Algorithm for Simultaneously Denoising and Deblurring Images
"... In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the totalvariation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved, fast ..."
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In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the totalvariation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved, fast algorithms for the combined denoising and deblurring model remain to be a challenge. Recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem. The aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring. Our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator. A fast algorithm can then be obtained for directly solving the variational model. Supporting numerical experiments on gray scale images are presented. AMS classifications. 68U10, 65F10, 65K10.
Multilevel and Adaptive Methods for Some Nonlinear Optimization Problems
, 2005
"... ∗ Signatures are on file in the Graduate School. In this thesis, we propose new multilevel and adaptive methods for solving nonlinear nonconvex optimization problems without relying on the linearization. We focus on two particular applications, that come from the fields of quantization and materia ..."
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∗ Signatures are on file in the Graduate School. In this thesis, we propose new multilevel and adaptive methods for solving nonlinear nonconvex optimization problems without relying on the linearization. We focus on two particular applications, that come from the fields of quantization and materials science. For the first problem, a multilevel quantization scheme is developed, that possesses a uniform convergence independent of the problem size. This is the first multilevel quantization scheme in the literature with a rigorous proof of uniform convergence with respect to the grid size and the number of grid levels for nonconstant densities. The proposed scheme can be generalized to higher dimensions, and both scalar and vector versions demonstrate significant speedup comparing to the traditional Lloyd method. We also provide some new characterizations for the convergence of the Lloyd iteration and other possible acceleration techniques including Newtonlike methods. For the second optimization problem, this thesis presents a novel algorithm aimed at automating phase diagram construc
MULTILEVEL SCHWARZ METHOD FOR THE MINIMIZATION WITH CONSTRAINTS OF NONQUADRATIC FUNCTIONALS
"... We succinctly present the results in [2] and [3] on the convergence rate of a multilevel method for the constrained minimization of nonquadratic functionals. The main goal of this paper is to check up the dependence of this convergence rate on the mesh and overlapping parameters by numerical tests c ..."
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We succinctly present the results in [2] and [3] on the convergence rate of a multilevel method for the constrained minimization of nonquadratic functionals. The main goal of this paper is to check up the dependence of this convergence rate on the mesh and overlapping parameters by numerical tests concerning the solution of the twoobstacle problem of a nonlinear elastic membrane. AMS subject classification: 65N55, 65N30, 65J15 1
AIMS ’ Journals Volume X, Number 0X, XX 200X pp. X–XX A TWOLEVEL DOMAIN DECOMPOSITION METHOD FOR IMAGE RESTORATION
"... (Communicated by the associate editor name) Abstract. Image restoration has drawn much attention in recent years and a surge of research has been done on variational models and their numerical studies. However, there remains an urgent need to develop fast and robust methods for solving the minimizat ..."
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(Communicated by the associate editor name) Abstract. Image restoration has drawn much attention in recent years and a surge of research has been done on variational models and their numerical studies. However, there remains an urgent need to develop fast and robust methods for solving the minimization problems and the underlying nonlinear PDEs to process images of moderate to large size. This paper aims to propose a twolevel domain decomposition method, which consists of an overlapping domain decomposition technique and a coarse mesh correction, for directly solving the total variational minimization problems. The iterative algorithm leads to a system of small size and better conditioning in each subspace, and is accelerated with a piecewise linear coarse mesh correction. Various numerical experiments and comparisons demonstrate that the proposed method is fast and robust particularly for images of large size. 1. Introduction. Image