31 research outputs found
On global minimizers of quadratic functions with cubic regularization
In this paper, we analyze some theoretical properties of the problem of
minimizing a quadratic function with a cubic regularization term, arising in
many methods for unconstrained and constrained optimization that have been
proposed in the last years. First we show that, given any stationary point that
is not a global solution, it is possible to compute, in closed form, a new
point with a smaller objective function value. Then, we prove that a global
minimizer can be obtained by computing a finite number of stationary points.
Finally, we extend these results to the case where stationary conditions are
approximately satisfied, discussing some possible algorithmic applications.Comment: Optimization Letters (2018
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Schnelle Löser für Partielle Differentialgleichungen
This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds
Parallel inexact Newton-Krylov and quasi-Newton solvers for nonlinear elasticity
In this work, we address the implementation and performance of inexact
Newton-Krylov and quasi-Newton algorithms, more specifically the BFGS method,
for the solution of the nonlinear elasticity equations, and compare them to a
standard Newton-Krylov method. This is done through a systematic analysis of
the performance of the solvers with respect to the problem size, the magnitude
of the data and the number of processors in both almost incompressible and
incompressible mechanics. We consider three test cases: Cook's membrane
(static, almost incompressible), a twist test (static, incompressible) and a
cardiac model (complex material, time dependent, almost incompressible). Our
results suggest that quasi-Newton methods should be preferred for compressible
mechanics, whereas inexact Newton-Krylov methods should be preferred for
incompressible problems. We show that these claims are also backed up by the
convergence analysis of the methods. In any case, all methods present adequate
performance, and provide a significant speed-up over the standard Newton-Krylov
method, with a CPU time reduction exceeding 50% in the best cases