Invited Session Thu.3.H 0112

Thursday, 15:15 - 16:45 h, Room: H 0112

Cluster 16: Nonlinear programming [...]

Nonlinear multilevel and domain decomposition methods in optimization


Chair: Michal Kocvara



Thursday, 15:15 - 15:40 h, Room: H 0112, Talk 1

Zdenek Dostal
Optimal massively parallel algorithms for large QP/QPQC problems arising in mechanics

Coauthor: Tomas Kozubek


We first review our results in development of optimal algorithms for the minimization of a strictly convex quadratic function subject to separable convex constraints and/or equality constraints. A unique feature of our algorithms is the bound on the rate of convergence in terms of the bounds on the spectrum of the Hessian of the cost function, independent of representation of the constraints. When applied to the class of convex QP or QPQC problems with the spectrum in a given positive interval and a sparse Hessian matrix, the algorithms enjoy optimal complexity.
The efficiency of our algorithms is demonstrated on the solution of contact problems of elasticity with or without friction by our TFETI domain decomposition method. We prove numerical scalability of our algorithms, i.e., their capability to find an approximate solution in a number of matrix-vector multiplications that is independent of the discretization parameter. Both umerical and parallel scalability of the algorithms is documented by the results of numerical experiments with the solution of contact problems with millions unknowns and analysis of industrial problems.



Thursday, 15:45 - 16:10 h, Room: H 0112, Talk 2

James Anthony Turner
Applications of domain decomposition to topology optimization

Coauthors: Michal Kocvara, Daniel Loghin


When modelling structural optimization problems, there is a perpetual need for increasingly accurate conceptual designs, with the number of degrees of freedom used in obtaining solutions continually rising. This impacts heavily on the overall computational effort required by a computer and it
is therefore natural to consider alternative possibilities. One approach is to consider parallel computing and in particular domain decomposition. The first part of this talk will discuss the application of domain decomposition to a typical topology optimization problem via an interior point approach. This method has the potential to be carried out in
parallel and therefore can exploit recent developments in the area. The second part of the talk will focus on a nonlinear reaction diffusion system solved using Newton’s method. Current work considers applying domain decomposition to such a system using a Newton Krylov Schur (NKS) type approach. However, strong local nonlinearities can have a drastic
effect on the global rate of convergence. Our aim is to instead consider a three step procedure that applies Newton’s method locally on subdomains in order to address this issue.



Thursday, 16:15 - 16:40 h, Room: H 0112, Talk 3

Rolf Krause
Inherently nonlinear decomposition and multilevel strategies for non-convex minimization


We present and discuss globally convergent domain decomposition
and multilevel strategies for the solution of non-convex - and
possible constrained - minimization problems. Our approach is
inherently nonlinear in the sense that we decompose the original
nonlinear problem into many small, but also nonlinear,
problems. In this way, strongly local nonlinearities or even
heterogeneous problems can be handled easily and
consistently. Starting from ideas from Trust-Region methods, we
show how global convergence can be obtained for the case of a
nonlinear domain decomposition as well as for the case of a
nonlinear multilevel method - or combinations thereof. These
ideas also allow us for deriving a globally convergent variant of
the ASPIN method (G-ASPIN). We will illustrate our findings along
examples from computational mechanics in 3D.


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