Abstract. We present a new class of iterative schemes for large scale setvalued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn-Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach.
A variety of models for the membrane-mediated interaction of particles in
lipid membranes, mostly well-established in theoretical physics, is reviewed
from a mathematical perspective. We provide mathematically consistent
formulations in a variational framework, relate apparently different modelling
approaches in terms of successive approximation, and investigate existence and
uniqueness. Numerical computations illustrate that the new variational
formulations are directly accessible to effective numerical methods
The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional, and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere. Lemma 5.3. Let the functional J and the subspace decomposition satisfy either the assumptions of Lemma 5.1 or of Lemma 5.2. Assume that the operator M : dom J → dom J , M − Id : dom J → V satisfies the sufficient descent condition J (w) − J (M(w)) ≥ ε J (w) − J (M ex (w)) for a fixed ε > 0. Then M satisfies the assumptions of Theorem 4.1. Proof. By continuity of J on dom J , for any w ∈ dom J there exists aw ∈ dom J ∩ w + V with J (w) = (1 − ε)J (w) + εJ (M ex (w)). Now we setM(w) =w. Then we have J (M(w)) ≤ (1 − ε)J (w) + εJ (M ex (w)) = J (M(w)) ≤ J (w), which is the required monotonicity. Continuity of J and of J • M ex (Lemma 5.1) imply continuity of J •M. Furthermore J (M(w)) = J (w) implies J (M ex (w)) = J (w) which shows stability. 6 end 7 Computev 0 = B 0 r 0 8 Output: v ν = Π Wν J k=0 J l=k+1 P l v k
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