Preconditioned SQP solvers for nonlinear optimization problems with partial differential equations
Final Report Abstract
Phenomena described by partial differential equations (PDEs) are ubiquitous in the natural and engineering sciences, but also in other disciplines. Numerical solution techniques help understand the underlying, often complex system behavior. This process is known as simulation. Beyond simulation one is often interested in an optimization of the said systems, for instance to drive the system under consideration towards a desired state. This leads to optimization problems with PDEs as side constraints. In the project we investigated certain aspects of fast solution methods for these very high dimensional and nonlinear but structured problems. In particular, we focused on a class of solution methods known as composite-step SQP algorithms. It turned out that the three sub-problems (normal, tangential, and multiplier update steps) which need to be solved in each iteration share a common algebraic structure. Moreover, this structure appears similarly in a class of simpler problems, known as linear-quadratic ones, whose fast solution has been the focus of intensive research in the past years. The method which was developed within the project paves the way of transferring these established techniques now to more challenging nonlinear problems. The design of our method takes into account the infinite dimensional nature of the problem at hand, and thus it exhibits a convergence behavior in practice which does not deteriorate as the discretization is refined. The practicality of the approach has been confirmed by means of a number of application driven problems, ranging from optimization problems involving semilinear elliptic model equations to problems featuring challenging nonlinear elasticity systems. We also addressed, for the first time, fast solution methods for optimization problems with a certain class of inequality constraints, known as state gradient constraints. In these problems one aims to keep the spatial variability of the system state at bay. This is of interest for example in cooling processes where high temperature gradients may cause material fatigue or damage and must be avoided.
Publications
- Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space, SIAM Journal on Numerical Analysis, Vol. 53. 2015, Issue 3, pp. 1304-1324.
Roland Herzog and Ekkehard Sachs
(See online at https://doi.org/10.1137/140973050) - Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems, SIAM Journal on Numerical Analysis, Vol. 54. 2016, Issue 2, pp.688-718.
Roland Herzog, Susann Mach
(See online at https://doi.org/10.1137/130948045)
