|Authors||S. W. Funke and M. Nordaas|
|Title||PDE-Constrained Optimisation in Hilbert Spaces|
|Afilliation||Scientific Computing, , Scientific Computing|
|Project(s)||Center for Biomedical Computing (SFF)|
|Publication Type||Talks, contributed|
|Year of Publication||2014|
|Location of Talk||FEniCS'14 workshop, Paris|
The solution of optimisation problems constrained by partial differential equations becomes increasingly more feasible due to the increase of computational resources and the development of solution algorithms. The computational cost and the large number of degrees of freedoms in PDE applications raise a particular challenge for partical algorithms. A key property for an efficient optimisation algorithm is that the required number of iterations is independent on the local mesh refinement and its element size. However, a straight-forward application on standard optimisation algorithms on the reduced optimisation problem results in mesh-size dependent iteration numbers and a poor performance on non-homogoenous meshes. In this talk we present on how good convergence is obtained by rethinking these optimisation algorithms to respect the underlying inner products and induced norms of the involved function spaces. Based on this idea we develop a new optimisation framework specifically designed for PDE-constrained optimisation which honors the inner product of the underlying Hilbert spaces. We show that this strategy is equivalent to a variance of preconditioned conjugate gradient methods or for the BFGS a user defined initial estimate of the Hessian matrix. Parallel , currently supports unconstrained optimisation algorithms and integrates seemlesly with FEniCS and dolfin-adjoint. Numerical results demonstrate the effectiveness of the framework.