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15
Multigrid methods for obstacle problems
"... Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which ..."
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Cited by 15 (3 self)
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Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1.
Convergence rate analysis of a multiplicative Schwarz method for variational inequalities
 SIAM J. Numer. Anal
, 2003
"... Abstract. This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general theory is particularl ..."
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Cited by 9 (2 self)
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Abstract. This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general theory is particularly applied to some obstacle problems, which yields a linear convergence for the corresponding Schwarz overlapping domain decomposition method of one and two levels. Numerical experiments are presented to confirm the convergence estimate derived in this paper.
A nonlinear multigrid method for total variation minimization from image restoration
 J. Sci. Comput
"... Abstract Image restoration has been an active research topic and variational formulations are particularly effective in high quality recovery. Although there exist many modelling and theoretical results, available iterative solvers are not yet robust in solving such modeling equations. Recent attemp ..."
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Cited by 7 (3 self)
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Abstract Image restoration has been an active research topic and variational formulations are particularly effective in high quality recovery. Although there exist many modelling and theoretical results, available iterative solvers are not yet robust in solving such modeling equations. Recent attempts on developing optimisation multigrid methods have been based on first order conditions. Different from this idea, this paper proposes to use piecewise linear function spanned subspace correction to design a multilevel method for directly solving the total variation minimisation. Our method appears to be more robust than the primaldual
Adaptive Finite Element Methods For Variational Inequalities: Theory And Applications In Finance
, 2007
"... We consider variational inequalities (VIs) in a bounded open domain Ω ⊂ Rd with a piecewise smooth obstacle constraint. To solve VIs, we formulate a fullydiscrete adaptive algorithm by using the backward Euler method for time discretization and the continuous piecewise linear finite element method ..."
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Cited by 3 (0 self)
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We consider variational inequalities (VIs) in a bounded open domain Ω ⊂ Rd with a piecewise smooth obstacle constraint. To solve VIs, we formulate a fullydiscrete adaptive algorithm by using the backward Euler method for time discretization and the continuous piecewise linear finite element method for space discretization. The outline of this thesis is the following. Firstly, we introduce the elliptic and parabolic variational inequalities in Hilbert spaces and briefly review general existence and uniqueness results (Chapter 1). Then we focus on a simple but important example of VI, namely the obstacle problem (Chapter 2). One interesting application of the obstacle problem is the Americantype option pricing problem in finance. We review the classical model as well as some recent advances in option pricing (Chapter 3). These models result in VIs with integrodifferential operators. Secondly, we introduce two classical numerical methods in scientific computing: the finite element method for elliptic partial differential equations (PDEs) and the Euler method for ordinary different equations (ODEs). Then we combine these two
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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Cited by 2 (1 self)
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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
Nonsmooth Newton methods for setvalued saddle point problems
 SIAM J. Numer. Anal
"... Abstract. We present a new class of iterative schemes for large scale set– valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton–type methods for the nonlinear Schur c ..."
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Abstract. We present a new class of iterative schemes for large scale set– valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton–type methods for the nonlinear Schur complement or as Uzawa–type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach. 1.
9 Are New Energy Sources Within Reach?
"... A regularized continuation method is developed for a twodimensional inverse medium scattering problem in nearfield optics, which reconstructs the scatterer of an inhomogeneous medium deposited on a homogeneous substrate from data accessible through photon scanning tunneling microscopy (PSTM) exper ..."
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A regularized continuation method is developed for a twodimensional inverse medium scattering problem in nearfield optics, which reconstructs the scatterer of an inhomogeneous medium deposited on a homogeneous substrate from data accessible through photon scanning tunneling microscopy (PSTM) experiments. In addition to the illposedness of inverse scattering problems, two difficulties arise because of the layered background medium and limited aperture data. Numerical experiments are included to illustrate the robust behavior of the method. Related topics will also be discussed. This is a joint work with Peijun Li of University of Michigan.
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
Twolevel domain decomposition algorithm for a nonlinear inverse DOT problem
"... Diffuse optical tomography is to find the value of optical coefficients in a tissue using near infrared lights, which is usually modelled by an optimization problem being composed of two steps: the forward solver to compute the photon density function and the inverse solver to update the coefficien ..."
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Diffuse optical tomography is to find the value of optical coefficients in a tissue using near infrared lights, which is usually modelled by an optimization problem being composed of two steps: the forward solver to compute the photon density function and the inverse solver to update the coefficients based on the forward solver. Since the resulting problem is mathematically nonlinear illposed inverse problem and numerically largescale computational problem demanding high quality image, it is highly desirable to reduce the amount of computation needed. In this paper, domain decomposition method is adopted to decrease the computation complexity of the problem. Among many methods of domain decomposition techniques, two level multiplicative overlapping domain decomposition method and two level space decomposition method are used to the forward and inverse solver, respectively. The convergence and computational cost of each method are described. And the efficiency of using combined two methods is verified by the implementation of reconstructing the absorption coefficient on square domain and thin domain. 1.