  Parameters in mathematical models for glioblastoma multiforme (GBM) tumour
growth are highly patient specific. Here we aim to estimate parameters in a
Cahn-Hilliard type diffuse interface model in an optimised way using model
order reduction (MOR) based on proper orthogonal decomposition (POD). Based on
snapshots derived from finite element simulations for the full order model
(FOM) we use POD for dimension reduction and solve the parameter estimation for
the reduced order model (ROM). Neuroimaging data are used to define the highly
inhomogeneous diffusion tensors as well as to define a target functional in a
patient specific manner. The reduced order model heavily relies on the discrete
empirical interpolation method (DEIM) which has to be appropriately adapted in
order to deal with the highly nonlinear and degenerate parabolic PDEs. A
feature of the approach is that we iterate between full order solves with new
parameters to compute a POD basis function and sensitivity based parameter
estimation for the ROM problems. The algorithm is applied using neuroimaging
data for two clinical test cases and we can demonstrate that the reduced order
approach drastically decreases the computational effort.
