A semi-smooth Newton and Primal–Dual Active Set method for Non-Smooth Contact Dynamics

<abstract><p>In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.</p></abstract>

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Mesh independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems

The ANZIAM Journal ◽

10.1017/s1446181100012657 ◽

2007 ◽

Vol 49 (1) ◽

pp. 1-38 ◽

Cited By ~ 9

Author(s):

M. Hintermüller

Keyword(s):

Optimal Control ◽

Numerical Test ◽

Control Problems ◽

Active Set ◽

Active Set Method ◽

Constrained Optimal Control ◽

Mixed Control ◽

Mesh Independence ◽

Primal Dual ◽

Control State

A class of mixed control-state constrained optimal control problems for elliptic partial differential equations arising, for example, in Lavrentiev-type regularized state constrained optimal control is considered. Its numerical solution is obtained via a primal-dual activeset method, which is equivalent to a class of semi-smooth Newton methods. The locally superlinear convergence of the active-set method in function space is established, and its mesh independence is proved. The paper contains a report on numerical test runs including a comparison with a short-step path-following interior-point method and a coarse-to-fine mesh sweep, that is, a nested iteration technique, for accelerating the overall solution process. Finally, convergence and regularity properties of the regularized problems with respect to a vanishing Lavrentiev parameter are considered. 2000 Mathematics subject classification: primary 65K05; secondary 90C33.

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Inexact primal–dual active set method for solving elastodynamic frictional contact problems

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2020.11.017 ◽

2021 ◽

Vol 82 ◽

pp. 36-59

Author(s):

Stéphane Abide ◽

Mikaël Barboteu ◽

Soufiane Cherkaoui ◽

David Danan ◽

Serge Dumont

Keyword(s):

Frictional Contact ◽

Contact Problems ◽

Active Set ◽

Active Set Method ◽

Primal Dual

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A primal-dual active set method for solving multi-rigid-body dynamic contact problems

Mathematics and Mechanics of Solids ◽

10.1177/1081286517733505 ◽

2017 ◽

Vol 23 (3) ◽

pp. 489-503 ◽

Cited By ~ 1

Author(s):

Mikaël Barboteu ◽

Serge Dumont

Keyword(s):

Local Level ◽

Mathematical Problem ◽

Local Treatment ◽

Contact Problems ◽

Dynamic Contact ◽

Contact Dynamics ◽

Active Set ◽

Contact Conditions ◽

Type Method ◽

Primal Dual

In this work, an active set type method is considered in order to solve a mathematical problem that describes the frictionless dynamic contact of a multi-body rigid system, the so-called nonsmooth contact dynamics (NSCD) problem. Our aim, here, is to present the local treatment of contact conditions by an active set type method dedicated to NSCD and to carry out a comparison with the various well-known methods based on the bipotential theory and the augmented Lagrangian theory. After presenting the mechanical formulation of the NSCD and the resolution of the global problem concerning the equations of motion, we focus on the local level devoted to the resolution of the contact law. Then we detail the numerical treatment of the contact conditions within the framework of the primal-dual active set strategy. Finally, numerical experiments are presented to establish the efficiency of the proposed method by considering the comparison with the other numerical methods.

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