  The Heston model is a well-known two-dimensional financial model. Because the
Heston model contains implicit parameters that cannot be determined directly
from real market data, calibrating the parameters to real market data is
challenging. In addition, some of the parameters in the model are non-linear,
which makes it difficult to find the global minimum of the optimization problem
within the calibration. In this paper, we present a first step towards a novel
space mapping approach for parameter calibration of the Heston model. Since the
space mapping approach requires an optimization algorithm, we focus on deriving
a gradient descent algorithm. To this end, we determine the formal adjoint of
the Heston PDE, which is then used to update the Heston parameters. Since the
methods are similar, we consider a variation of constant and time-dependent
parameter sets. Numerical results show that our calibration of the Heston PDE
works well for the various challenges in the calibration process and meets the
requirements for later incorporation into the space mapping approach. Since the
model and the algorithm are well known, this work is formulated as a proof of
concept.
