65K10 Optimization and variational techniques [See also 49Mxx, 93B40]
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The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.
A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.
A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications.