Augmented Lagrangian Active Set Methods for Obstacle Problems

Kärkkäinen, Tommi; Kunisch, Karl; Tarvainen, P.

doi:10.1023/b:jota.0000006687.57272.b6

Cited by 75 publications

(77 citation statements)

References 19 publications

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“…We formalize this in the next proposition. The proof here is a slight modification and simplification of the more general treatment in [19], Theorem 5.3.1, along the lines of the convergence analyses in different problem domains, as given in [42][43][44]. Proof.…”

Section: General Prototype-based Clustering and Its Convergencementioning

confidence: 97%

Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering

2017

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Abstract:Clustering is an unsupervised machine learning and pattern recognition method. In general, in addition to revealing hidden groups of similar observations and clusters, their number needs to be determined. Internal clustering validation indices estimate this number without any external information. The purpose of this article is to evaluate, empirically, characteristics of a representative set of internal clustering validation indices with many datasets. The prototype-based clustering framework includes multiple, classical and robust, statistical estimates of cluster location so that the overall setting of the paper is novel. General observations on the quality of validation indices and on the behavior of different variants of clustering algorithms will be given.

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Section: General Prototype-based Clustering and Its Convergencementioning

confidence: 97%

Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering

2017

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“…So, if N dof denotes the dimension of the finite elements space, in the case without early retirement we have to solve the linear system with N dof unknowns 14) at each time step, where matrix M h does not depend on time. Notice that the dependence on time of the PDE operator is restricted to the velocity and this term has been discretized by characteristics method, which combined with the previously described Crank-Nicolson scheme leads to a time-independent matrix.…”

Section: Lagrange-galerkin Discretizationmentioning

confidence: 99%

“…Both papers are applied to general (possibly degenerated) convection-diffusion-reaction equations under certain assumptions. Moreover, the non-linearities associated to the inequality constraints in the complementarity formulation due to early retirement are treated by means of the recently introduced augmented Lagrangian active set method [14]. Among the different possible alternatives (projected relaxation, penalization, other duality techniques...) this method results to be efficient and robust (see, for example, [4] for a comparison with a duality algorithm in an Asian option pricing problem of American style).…”

mentioning

confidence: 99%

Mathematical Analysis and Numerical Methods for Pricing Pension Plans Allowing Early Retirement

Calvo-Garrido¹,

Pascucci²,

Vázquez³

2013

SIAM J. Appl. Math.

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Abstract. In this paper, we address the mathematical analysis and numerical solution of a model for pricing a defined benefit pension plan. More precisely, the benefits received by the member of the plan depend on the average salary and early retirement is allowed. Thus, the mathematical model is posed as an obstacle problem associated to a Kolmogorov equation in the time region where the salary is being averaged. Previously to the initial averaging date, a nonhomogeneous one factor Black-Scholes equation is posed. After stating the model, existence and regularity of solutions are studied. Moreover, appropriate numerical methods based on a Lagrange-Galerkin discretization and an augmented Lagrangian active set method are proposed. Finally, some numerical examples illustrate the performance of the numerical techniques and the properties of the solution and the free boundary.

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“…The problem describes the elastoplastic torsion of a cylindrical bar with quadratical cross section [24,25,30]. As the parameter C becomes smaller, the problem becomes more difficult to solve.…”

Section: Test Casesmentioning

confidence: 99%

“…As it turns out, with an appropriate grid sequencing, the convergence rate of the semismooth Newton method can be made to be nearly independent of the number of unknowns of the system using either the Fischer-Burmeister function or the minimum function. This is a tremendous improvement over the approaches developed in [24,38], in which the number of semismooth Newton iterations nearly doubles when the grid is refined by a factor of 2. Using the two-level restricted Schwarz preconditioner with sufficient overlap, the number of linear iterations also becomes nearly independent of the number of unknowns of the system.…”

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confidence: 96%

Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

Yang

Cai

2010

J Sci Comput

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We develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of NewtonKrylov-Schwarz methods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems. We present numerical results for several large scale calculations obtained on machines with hundreds of processors.

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Augmented Lagrangian Active Set Methods for Obstacle Problems

Cited by 75 publications

References 19 publications

Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering

Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering

Mathematical Analysis and Numerical Methods for Pricing Pension Plans Allowing Early Retirement

Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

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