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14
Semismooth Newton methods for variational inequalities of the first kind
, 2003
"... Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of discreti ..."
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Cited by 27 (9 self)
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Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local superlinear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L ∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.
Distributed optimal control of the CahnHilliard system including the case of a doubleobstacle homogeneous free energy density
 SIAM J. Control Optim
"... Abstract. In this paper we study the distributed optimal control for the Cahn–Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the doubleobstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequali ..."
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Cited by 8 (0 self)
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Abstract. In this paper we study the distributed optimal control for the Cahn–Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the doubleobstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau–Yosida approximation. Corresponding firstorder optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fréchet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive firstorder optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of Cstationarity. Key words. Cahn–Hilliard system, doubleobstacle potential, mathematical programming with equilibrium constraints, distributed optimal control, Yosida regularization, Cstationarity
Truncated nonsmooth Newton multigrid methaods for convex minimization problems
 In Proc. of DD18, submitted. cited in [29
"... Summary. We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level and a linear coarse correction. Suitable postprocessing guarantees global ..."
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Cited by 5 (2 self)
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Summary. We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level and a linear coarse correction. Suitable postprocessing guarantees global convergence even in the case of a single multigrid step for each linear subproblem. Numerical examples show that the overall efficiency is comparable to multigrid for similar linear problems. 1
On the use of policy iteration as an easy way of pricing American options
 SIAM Journal on Financial Mathematics
"... Abstract. In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [10], is an extremely simple generic algorithm for solving linear complementarity problems resulting from the finite difference and finite element approximation of American options. We show ..."
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Cited by 2 (1 self)
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Abstract. In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [10], is an extremely simple generic algorithm for solving linear complementarity problems resulting from the finite difference and finite element approximation of American options. We show that, in general, O(N) is an upper and lower bound on the number of iterations needed to solve a discrete LCP of size N. If embedded in a class of standard discretisations with M time steps, the overall complexity of American option pricing is indeed only O(N(M +N)), and, therefore, for M ∼ N, identical to the pricing of European options, which is O(MN). We also discuss the numerical properties and robustness with respect to model parameters in relation to penalty and projected relaxation methods.
Multilevel algorithms for largescale interior point methods in bound constrained optimization
, 2006
"... We develop and compare multilevel algorithms for solving bound constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and i ..."
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We develop and compare multilevel algorithms for solving bound constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain good initial guesses (“hot starts”) for each nonlinear iteration. A minimal surface problem is used to illustrate the various approaches.
A POLYNOMIAL CHAOS APPROACH TO STOCHASTIC VARIATIONAL INEQUALITIES
"... Abstract. We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show ..."
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Abstract. We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approach and suggest similar error estimates as for linear elliptic problems. AMS classification: 65K15, 65N30, 65N35
TIME DISCRETIZATIONS OF ANISOTROPIC ALLEN–CAHN EQUATIONS
"... Abstract. We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stabilit ..."
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Abstract. We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the doublewell potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinchoff under anisotropic mean curvature flow is no longer frame invariant, but depends on the orientation of the initial configuration. 1.
Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids
"... In this paper, we develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes. Adaptive finite element methods (AFEMs) based on local mesh refinement are an important and efficient approach when the solution ..."
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Cited by 1 (1 self)
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In this paper, we develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes. Adaptive finite element methods (AFEMs) based on local mesh refinement are an important and efficient approach when the solution is
A posteriori error estimator competition for conforming obstacle problems
 NUMER. METHODS PARTIAL DIFFERENTIAL EQ
, 2012
"... This article on the a posteriori error analysis of the obstacle problem with affine obstacles and Courant finite elements compares five classes of error estimates for accurate guaranteed error control. To treat interesting computational benchmarks, the first part extends the Braess methodology from ..."
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This article on the a posteriori error analysis of the obstacle problem with affine obstacles and Courant finite elements compares five classes of error estimates for accurate guaranteed error control. To treat interesting computational benchmarks, the first part extends the Braess methodology from 2005 of the resulting a posteriori error control to mixed inhomogeneous boundary conditions. The resulting guaranteed global upper bound involves some auxiliary partial differential equation and leads to four contributions with explicit constants. Their efficiency is examined affirmatively for five benchmark examples.