In the TCAD area, splines and tensor product spline (TPS) usage has been mainly restricted to interpolation applications. Such a role is illustrated in [9], where spline interpolation forms the basis of an automatic table-lookup technology independent device model generation technique. In [52] their use to model impurity concentration variation in inverse modeling techniques to determine doping profiles has been proposed. The details of this technique are presented in Chapter 4. A complete study of the mathematical properties of spline approximation and modeling and all potential applications for TCAD purposes are beyond the scope of this thesis. The description in this section is limited to the definition and characterization of polynomial basis splines or B-splines and their properties as a general formulation of continuous functions in regression analysis.
Polynomial piecewise approximation using splines in one-dimension, and the extension to higher dimensions by the TPS constructs is a well established technique [14] which is widely used for interpolation and approximation of discrete data. Piecewise spline functions do not suffer from the essential limitation of single polynomial approximation, namely the Runge phenomenon [14]. Recently, their use as non-parameteric models for experimental data has been the subject of active research effort within the statistical community [105][27].
In general, the spline representation is very flexible and can easily adapt to data variation by adjusting the order of the spline and the set of breakpoints. One can also enforce constraints on the spline coefficients to conserve monotonicity or convexity of the original data set. Furthermore, they have a reasonable speed of evaluation.