Parameter Identification Problems

In the preceding chapters we started from a mathematical description of a mechanical system and used it to predict its behavior. 

This chapter is devoted to the inverse task We use now knowledge about the system s behavior in order to determine unknown parameters in the mathematical description. 

A typical example for parameter identification (PI) in vehicle dynamics is the identification of unknown parameters in a tire model One starts by setting up a mathematical model with a couple of unknown parameters. Then the tire itself or in combination with other components of a vehicle is investigated under various conditions. Measurements are taken in order to determine the unknown parameters. 

In most cases the unknown parameters cannot be measured directly, they might even have no direct physical interpretation. The selection of the quantities to be measured and the way how the measurements are set up are important engineering questions with a strong impact on the quality of the parameter identification. 

As the measurements are usually randomly perturbed by measurement noise, the unknown parameters are determined in such a way that they describe the system with the largest likelihood maximum likelihood approach). 

---

Our approach to determine the properties of heterogeneous media utilizes mathematical models of the measurement process and, as appropriate, the flow process itself. To determine the desired properties, we solve an associated system and parameter identification problem (also termed an inverse problem) to estimate the properties from the measured data. 

The parameter identification problem associated with the conventional permeability experiments is within the first class (with m= 1). By contrast, the problems we consider here are within the second and third classes these areJunctional estimation problems. Ultimately, however, these are solved with finite-dimensional representations, although an essential aspect of the solution of these infinite-dimensional (function) estimation problems is the selection of the appropriate representations. 

We have developed a method to spatially resolve permeability distributions. We use MRI to determine spatially resolved velocity distributions, and solve an associated system and parameter identification problem to determine the permeability distribution. Not only is such information essential for investigating complex processes within permeable media, it can provide the means for determining improved correlations for predicting permeability from other measurements, such as porosity and NMR relaxation [17-19]. 

We presented a novel method to determine spatially resolved permeability distributions. We used MRI to measure spatially resolved flow velocities, and estimated the permeability from the solution of an associated system and parameter identification problem. 

A special case occurs when some material or transport parameters are still unknown at the starting point and yet, at the same time, we have a lot of experimental data for the model validation. In this situation, we consider both data and model by formulating a parameter identification problem. The validation test for... 

There are two kinds of inverse problems in the solid mechanics. One is the source reconstruction problem and the other is the parameter identification problem (Bezerra Saigal, 1995). For our application only source reconstruction technique is used. In the case of source reconstruction, usually unknown boundary conditions are determined using observed field quantities inside the domain of interest or overprescribed boundary conditions. If on some portion of the boundaries both displacements and tractions are unknown, it cannot be solved directly in a forward boundary value problem senses because the number of unknowns is larger than number of equations. However, if some quantities, say stresses for our problem at hand, are known at some points inside the domain, the number of the equations can be increased to obtain a solution. Usually the solution of an inverse problem does not always satisfy stability and uniqueness... 

Each algorithm has its own influential parameters that affect its performance in terms of solution quality and computational time. In order to increase the performance of the FA and GA, it is necessary to provide the adjustments of the parameters depending on the problem domain. With the appropriate choice of the algorithm settings the accuracy of the decisions and the execution time can be optimized. Parameters of the FA and GA are tuned on the basis of a large number of pre-tests according to the parameter identification problem, considered here. 

This is an confirmation of the better performance of the FA compared to GA for considered model parameter identification problem. 

When more than two kinds of liquids are used, a useful technique introduced by Erbil and Meric [22] can be used. The method, based on a parameter identification problem, uses only the total surface free energy of liquids, which can be obtained precisely by experiment. For this method, an equation that shows the interaction between the solid and liquid surfaces is needed. 

This questions includes parameter optimization, optimization of the structure, optimal control problems and parameter identification problems. The latter problem arises if some of the system parameters are unknown and measurements are taken to determine them. 

We will state in this chapter the mathematical task of parameter identification and discuss the corresponding numerical methods. Techniques from various branches of numerical mathematics are required, e.g. numerical solution of differential equations, numerically solving nonlinear problems especially large-scale constrained nonlinear least squares problem. Thus, some of the methods discussed in the previous chapters will reappear here. We will see how parameter identification problems can be treated efficiently by boundary value problem (BVP) methods and extend the discussion of solution techniques for initial value problems (IVPs) to those for BVPs. 

In the first part of this chapter we will give a mathematical formulation of parameter identification problems for unconstrained differential equations. In the second part we discuss essential aspects of the numerical treatment of these systems. In the last... 

Summarizing, the parameter identification problem can now be formulated as the following nonlinear constrained optimization problem... 

---