## Parameter Estimation

Mathematical modeling of dynamic processes in science, engineering and finance frequently leads to large systems of ordinary (ODE), partial (PDE), delay (DDE), stochastic (SDE), differential-algebraic (DAE), and partial differential-algebraic equations (PDAE), which are used in process simulation. Typically, these mathematical models involve unknown model parameters and initial conditions that have to be fitted against experimental data to ensure a good match between mathematical model and reality.

For this parameter estimation task, also known as model calibration or model fitting, boundary value problem (BVP) methods with multiple shooting or collocation discretization have proven to be very successful.

The underlying idea is to treat the discretized DAE model as a nonlinear constraint of the optimization problem. This problem is then solved by algorithms that allow infeasible points, such as SQP generalized Gauss-Newton methods, the latter ensuring that computed solutions are not attracted to statistically instable local solutions.

The group has developed and implemented methods and algorithms for efficient and reliable parameter estimation tasks for usage with all of the above model classes (ODE, DAE, PDE, PDAE, DDE, SDE).

Software Packages:

PARFIT l1, l2, Huber estimation in ODE models

ParamEDE parameter estimation in delay differential equation (DDE) models

PAREMERA parameter estimation in ODE models

:sfit parameter estimation and simulation of stochastic differential equation (SDE) models

VPLAN optimum experimental design, parameter estimation and simulation in ODE and DAE models