The aim of this thesis is to compare the efficiency of different algorithms on estimating parameters
that arise in partial differential equations: Kalman Filters (Ensemble Kalman Filter,
Stochastic Collocation Kalman Filter, Karhunen-Lo`eve Ensemble Kalman Filter, Karhunen-
Lo`eve Stochastic Collocation Kalman Filter), Markov-Chain Monte Carlo sampling schemes
and Adjoint variable-based method.
We also present the theoretical results for stochastic optimal control for problems constrained
by partial differential equations with random input data in a mixed finite element form. We
verify experimentally with numerical simulations using Adjoint variable-based method with
various identification objectives that either minimize the expectation of a tracking cost functional
or minimize the difference of desired statistical quantities in the appropriate Lp norm
