Lagrange Multiplier Approach to Variational Problems and Applications

Ito, Kazufumi; Kunisch, Karl

doi:10.1137/1.9780898718614

Cited by 497 publications

(528 citation statements)

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“…According to the superlinear convergence theorem for Newton differentiable mappings [13,14] it suffices to verify that the inverses of the generalized gradients G F (y, u, p) are uniformly bounded in a neighborhood of (y c , u c , p c ), which was achieved in Lemma 4. Combining the results of Section 3.1 and this section we showed that the solutions of the regularized problem (P c ) converge in the sense of Proposition 3.8 and that each regularized problem with a fixed value of c can be solved with superlinear rate.…”

Section: Semi-smooth Newton Method: Wellposedness and Convergencementioning

confidence: 99%

“…Solving (1.7) numerically by Newton-type methods is impeded by the lack of C 1 regularity of the sgn c operator. In Section 4 it will be shown that semi-smooth Newton methods are applicable to (1.7) [14]. This requires the verification of Newton differentiability, which is quite standard by now, as well as well-posedness of the Newton-step and uniform boundedness of the inverse of the generalized derivatives, which is more delicate to verify.…”

Section: Introduction Problem Statement Regularizationmentioning

confidence: 99%

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Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Kunisch

Wachsmuth

2011

ESAIM: COCV

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Abstract.In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.Mathematics Subject Classification. 49K20, 47J20, 49M15.

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Section: Semi-smooth Newton Method: Wellposedness and Convergencementioning

confidence: 99%

Section: Introduction Problem Statement Regularizationmentioning

confidence: 99%

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Kunisch

Wachsmuth

2011

ESAIM: COCV

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“…Various types of methods exist in the literature and, in this paper, we choose two of them which have proved to be efficient for such problems (see e.g. [27]), namely the Augmented Lagrangian method and the Bermúdez-Moreno method. Their definitions and derivations are different, but interestingly the obtained structure of the algorithms is the same.…”

Section: Treating the Velocity Inequality With Two Duality Methodsmentioning

confidence: 99%

Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case

Fernández-Nieto

Gallardo

Vigneaux

2018

Journal of Computational Physics

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This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated to the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez-Moreno) to discretize the problem. To be able to accurately simulate the stopping behaviour of the avalanche, new schemes need to be designed, involving the classical notion of wellbalancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.

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“…In the context of infinite dimensional problems, a lot of effort has been put into developing efficient numerical schemes for the solution of quadratic minimization problems bound to side constraints, see e.g., [15,16] and the references therein. Since it is known that the system of equations of the penalized discrete problem becomes ill-conditioned as ε → 0, it is natural to ask for other, more robust methods for specific applications.…”

Section: Minimization Problem (M)mentioning

confidence: 99%

Numerical quadratic energy minimization bound to convex constraints in thin-film micromagnetics

2012

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We analyze the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2983-2991). We introduce an appropriate functional analytic framework and prove well-posedness of the model in that setting. The scheme for the numerical approximation of solutions consists of two ingredients: The energy space is discretized in a conforming way using Raviart-Thomas finite elements; the non-linear but convex side constraint is treated with a penalty method. This strategy yields a convergent sequence of approximations as discretization and penalty parameter vanish. The proof generalizes to a large class of minimization problems and is of interest beyond the scope of thin-film micromagnetics.Mathematics Subject Classification (2010) 65K05 · 65K15 · 49M20 1 Introduction and abstract setting

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Lagrange Multiplier Approach to Variational Problems and Applications

Cited by 497 publications

References 0 publications

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case

Numerical quadratic energy minimization bound to convex constraints in thin-film micromagnetics

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