6,081 research outputs found
Directional differentiability for elliptic quasi-variational inequalities of obstacle type
The directional differentiability of the solution map of obstacle type
quasi-variational inequalities (QVIs) with respect to perturbations on the
forcing term is studied. The classical result of Mignot is then extended to
the quasi-variational case under assumptions that allow multiple solutions of
the QVI. The proof involves selection procedures for the solution set and
represents the directional derivative as the limit of a monotonic sequence of
directional derivatives associated to specific variational inequalities.
Additionally, estimates on the coincidence set and several simplifications
under higher regularity are studied. The theory is illustrated by a detailed
study of an application to thermoforming comprising of modelling, analysis
and some numerical experiments
Directional differentiability for elliptic quasi-variational inequalities of obstacle type
The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments
Existence, iteration procedures and directional differentiability for parabolic QVIs
We study parabolic quasi-variational inequalities (QVIs) of obstacle type.
Under appropriate assumptions on the obstacle mapping, we prove the existence
of solutions of such QVIs by two methods: one by time discretisation through
elliptic QVIs and the second by iteration through parabolic variational
inequalities (VIs). Using these results, we show the directional
differentiability (in a certain sense) of the solution map which takes the
source term of a parabolic QVI into the set of solutions, and we relate this
result to the contingent derivative of the aforementioned map. We finish with
an example where the obstacle mapping is given by the inverse of a parabolic
differential operator.Comment: 41 page
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the -Laplacian
In this paper we propose a multigrid optimization algorithm (MG/OPT) for the
numerical solution of a class of quasilinear variational inequalities of the
second kind. This approach is enabled by the fact that the solution of the
variational inequality is given by the minimizer of a nonsmooth energy
functional, involving the -Laplace operator. We propose a Huber
regularization of the functional and a finite element discretization for the
problem. Further, we analyze the regularity of the discretized energy
functional, and we are able to prove that its Jacobian is slantly
differentiable. This regularity property is useful to analyze the convergence
of the MG/OPT algorithm. In fact, we demostrate that the algorithm is globally
convergent by using a mean value theorem for semismooth functions. Finally, we
apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow
of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several experiments
are carried out to show the efficiency of the proposed algorithm when solving
this kind of fluid mechanics problems
Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities
Stochastic variational inequalities (SVI) model a large class of equilibrium
problems subject to data uncertainty, and are closely related to stochastic
optimization problems. The SVI solution is usually estimated by a solution to a
sample average approximation (SAA) problem. This paper considers the normal map
formulation of an SVI, and proposes a method to build asymptotically exact
confidence regions and confidence intervals for the solution of the normal map
formulation, based on the asymptotic distribution of SAA solutions. The
confidence regions are single ellipsoids with high probability. We also discuss
the computation of simultaneous and individual confidence intervals
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