2.3.2 Polynomial Approximation Algorithm

Given the experimental design matrix and the corresponding calculated model output values, the objective is to create a polynomial approximation with a minimum number of terms that minimizes the squared deviation error with respect to the TCAD model values. If the given order of dependency of all parameters is accurate, the objective is achieved by solving the linear least squares problem. In general, this is not usually the case. The complex interactions that exist between the parameters of a typical TCAD model prohibits the determination of the terms that are to be included in the final expression. Instead the given depedencies are only estimates that will only be confirmed in the final expression. A heuristic algorithm to select the appropriate terms to be included in the approximation was implemented. It is based on ideas similar to [95][65]. A formal description of the algorithm follows.

Given the order of dependencies specified, the set of all possible terms of the polynomial expansion is formed by multivariate combination or by including the terms from a specified design. The upper limit of the power of each parameter is given by the order of dependency of the output on this parameter. All terms that do not satisfy prespecified conditions are deleted. For example, terms of higher order than the specified degree of the polynomial are excluded. Let the set of remaining terms be denoted as and the set of selected terms as . Initially is the empty set. Terms are added to until the fitness criterion is met according to the following procedure:

• Step 1: Add each term in to the terms in , then perform a linear squares fit of the data and calculate the corresponding and fit figure of merit values:

  

  

  is a measure of the goodness of the fit and quantifies the capability of the model to explain the data variation around its average .

• Step 2: Select the term whose addition results in the highest combined figure of merit r:

  

  where and are user specified.

• Step 3: The selected term is deleted from and added to .
• Step 4: If the fit improvement is insignificant terminate, otherwise go to Step 1.

The solution of the linear least squares problem of Step 1 is now described. Let be the experimental run matrix, be the vector of l output values calculated using the TCAD model, and be the matrix whose elements are the values of the terms calculated using the parameter values in the run matrix:

The terms are in fact the basis functions of this general linear least squares problem. The corresponding normal equations can be formulated as:

and the solution is expressed as:

Due to possible ill-conditionning, the solution of the linear system above is implemented using singular value decomposition [84].