# Interdisciplinary Applied Mathematics

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solver, and the electrostatic force is computed using the capacitance solver based on the boundary element method. The functions that represent the relationship between the variables, in this case fs and fe, are generated from tables of numerical data obtained from device simulations. During the simulations, the tabl-based models are evaluated by interpolating the data; typically, a cubic spline interpolation is used. As an example, suppose y depends on    x,    and    for    some    value    of x between    Xi    and    xi+1, y can be

computed by

y

aiyi + a2yi+1 + a3

д2Уг

dx2

+ a4

d2yi+1 dx2

ai

аз

•&i-1-1 %i—1

7(a? — ai)(xi+i — ^i)2,

6

a2 —

a4

0(а2 a2){xiJrl

xi )2.

Since second derivatives are required, typically, a second table comprising second derivatives is also constructed.

The merits of table-based reduced-order modeling are: (i) The procedure eliminates the difficulties and errors associated with analytical model extraction. (ii) The method can be easily automated once the order and state variables of the ODE are decided. The problems with table-based macromodeling are: (i) The method does not preserve the physical meaning of the system. (ii) It is not easy for the designer to propose modifications and expect the system to behave as desired. (iii) At least some expensive full-scale physical simulations are required to generate the table(s) of data.

###### 17.3.2 Linear Dynamic Models

The reduced-order modeling methods that fall in this category are the Krylov subspace technique and the moment matching technique, which are discussed in this section.

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