scholarly journals Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

2013 ◽  
Vol 33 (4) ◽  
pp. 1126-1155 ◽  
Author(s):  
L. Blank ◽  
H. Garcke ◽  
L. Sarbu ◽  
V. Styles
Keyword(s):  
Systems Analysis ◽  
Active Set ◽  
Active Set Method ◽  
Primal Dual

scholarly journals Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets

2021 ◽  
Vol 30 (1) ◽  
pp. 90-115
Author(s):  
Yiyuan Qian ◽  
◽  
Haiming Song ◽  
Xiaoshen Wang ◽  
Kai Zhang ◽  
...  
Keyword(s):  
Free Boundary ◽  
Variable Coefficients ◽  
Euler Method ◽  
The Finite Element Method ◽  
Variational Theory ◽  
Active Set ◽  
Bounded Linear ◽  
Active Set Method ◽  
Backward Euler ◽  
Primal Dual

<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>

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

2007 ◽  
Vol 49 (1) ◽  
pp. 1-38 ◽  
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.

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

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

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

2021 ◽  
Vol 387 ◽  
pp. 114153
Author(s):  
Stéphane Abide ◽  
Mikaël Barboteu ◽  
Soufiane Cherkaoui ◽  
Serge Dumont
Keyword(s):  
Contact Dynamics ◽  
Active Set ◽  
Active Set Method ◽  
Primal Dual ◽  
Smooth Contact

Primal-Dual Active Set Method for American Lookback Put Option Pricing

2017 ◽  
Vol 7 (3) ◽  
pp. 603-614 ◽  
Author(s):  
Haiming Song ◽  
Xiaoshen Wang ◽  
Kai Zhang ◽  
Qi Zhang
Keyword(s):  
Variational Inequality ◽  
Free Boundary ◽  
Standard Technique ◽  
Main Concern ◽  
Pricing Model ◽  
Active Set ◽  
Active Set Method ◽  
Lookback Options ◽  
Main Challenge ◽  
Primal Dual

AbstractThe pricing model for American lookback options can be characterised as a two-dimensional free boundary problem. The main challenge in this problem is the free boundary, which is also the main concern for financial investors. We use a standard technique to reduce the pricing model to a one-dimensional linear complementarity problem on a bounded domain and obtain a corresponding variational inequality. The inequality is discretised by finite differences and finite elements in the temporal and spatial directions, respectively. By enforcing inequality constraints related to the options using Lagrange multipliers, the discretised variational inequality is reformulated as a set of semi-smooth equations, which are solved by a primal-dual active set method. One of the major advantages of our algorithm is that we can obtain the option values and the free boundary simultaneously, and numerical simulations show that our approach is as efficient as some other methods.

scholarly journals The Primal-Dual Active Set Method for a Class of Nonlinear Problems with T-Monotone Operators

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiahui He ◽  
Peng Yang
Keyword(s):  
Variational Inequality ◽  
Nonlinear Problems ◽  
Monotone Operators ◽  
Variational Inequality Problems ◽  
Active Set ◽  
Active Set Method ◽  
Algebraic Setting ◽  
Convergence Results ◽  
The Family ◽  
Primal Dual

The family of primal-dual active set methods is drawing more attention in scientific and engineering applications due to its effectiveness and robustness for variational inequality problems. In this work, we introduce and study a primal-dual active set method for the solution of the variational inequality problems with T-monotone operators. We show that the sequence generated by the proposed method globally and monotonously converges to the unique solution of the variational inequality problem. Moreover, the convergence rate of the proposed scheme is analyzed under the framework of the algebraic setting; i.e., the established convergence results show that the iteration number of the methods is bounded by the number of the unknowns. Finally, numerical results show that the efficiency can be achieved by the primal-dual active set method.

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