# Interdisciplinary Applied Mathematics

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Further improvements to the method have been reported in (Rewien-ski and White, 2002), where a richer aggregated reduced basis is obtained by applying the Arnoldi method at each linearization point instead of only once. This results in improved accuracy and consequently reduces the order of the reduced model further. In the original implementation, the projection matrix V was constructed using a Krylov subspace based on a linearization about the initial state x0. In the new implementation, the above approach has been replaced by a three-step procedure. First, at each of the linearization points Xj, a reduced-order basis is generated in a suitable Krylov space corresponding to a linearized model generated at x*. Second, the union of all the bases is formed, and third, the set is reduced using singular value decomposition. The method for basis generation was replaced from the Arnoldi-based Krylov subspace method to the TBR (Truncated Balanced Realization) algorithm in (Vasilyev et al., 2003), and a hybrid method using both TBR and Krylov subspace has been implemented. It was found that the TBR-based methods gave better accuracy than the method in which the Krylov subspace was used solely.

##### 17.4 Galerkin Methods

Galerkin methods are popular techniques for reduced-order modeling. In this section, we summarize both linear and nonlinear Galerkin methods for reduced-order modeling.

###### 17.4-1 Linear Galerkin Methods

The objective is to create a set of coupled ordinary differential equations that give an accurate representation of the dynamical behavior of the device. The approach is to formulate the dynamical behavior in terms of a finite set of orthonormal spatial basis functions, each with a time-dependent coefficient. Though this method is not typically analytical, it still forms a very important tool in the reduced-order modeling of microsystems that cannot be represented as lumped elements. Because of the relatively small number of state variables, the models can be quickly evaluated, and integrated over time. Such models can be readily inserted into circuit simulators for behavioral representation at the system level, including feedback effects around nonlinear devices. For completely numerical sets of ODEs, automatic model-order reduction can be implemented, at least for linear problems, and for nonlinear problems by using a combination of methods like the Krylov subspace techniques. The two most popular methods that fall under this category are the linear modes of vibration and Karhunen-Loeve decomposition.

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