## Parameter Identification in One-Dimensional Partial Differential
Algebraic Equations

K. Schittkowski,
GAMM-Mitteilungen, Vol. 30, No. 2, 352-375
(2007)

**Abstract:**
In this paper we discuss a couple of situations, where algebraic equations are
to be attached to a system of one-dimensional partial differential equations.

Besides of models leading directly to algebraic equations because of the
underlying practical background, for example in case of steady-state equations,
there are many others where the specific mathematical structure requires a
certain reformulation leading to time-independent equations. To be able to apply
our approach to a large class of real-life problems, we have to take into
account flux formulations, constraints, switching points, different integration
areas with transition conditions, and coupled ordinary differential algebraic
equations (DAEs), for example. The system of partial differential algebraic
equations (PDAEs) is discretized by the method of lines leading to a large
system of differential algebraic equations which can be solved by any available
implicit integration method. Standard difference formulas are applied to
discretize first and second partial derivatives, and upwind formulae are used
for transport equations. Proceeding from given experimental data, i.e.,
observation times and measurements, the minimum least squares distance of
measured data from a fitting criterion is computed, which depends on the
solution of the system of PDAEs. Parameters to be identified can be part of the
differential equations, initial, transition, or boundary conditions, coupled
DAEs, constraints, fitting criterion, etc. Also the switching points can become
optimization variables. The resulting least squares problem is solved by an
adapted sequential quadratic programming (SQP) algorithm which retains typical
features of a classical Gauss-Newton method by retaining robustness and fast
convergence speed of SQP methods. The mathematical structure of the
identification problems is outlined in detail, and we present a number of case
studies to illustrate the different model classes which can be treated by our
approach.

To download a preprint, click here: pdaerev.pdf