Further studies of the hybrid methods proposed in this work are suggested in two aspects. One is extending the range of applications, such as systems with not only non-magnetic materials, systems with lossy dielectrics, or high-Q materials in systems with open boundaries. The other is to enhancing the performance, for
example, faster convergence or less memory requirement in the model order reduction process.
Extending the range of applications can be accomplished by introducing the Krylov subspace methods for non-symmetric systems in the model order reduction process, in the mean time complex eigenpairs will also be generated for the approximation of both the attenuation constants and wave numbers of resonant modes, symmetric systems, systems with lossy media, or high-Q materials in systems with open boundaries will then be applicable. In order to integrate the Krylov subspace methods for non-symmetric systems into the hybrid method, however, modification of existing time-domain simulation code will be necessary because both the original and its adjoint problems are required. Convergence problems will also arise with Krylov subspace methods for non-symmetric systems [15]. Therefore minimizing the modification of existing to assure code reuse and avoiding the convergence issues with asymmetric Krylov subspace methods will be the primary challenges in extending the application range of the hybrid method.
The performance of the hybrid method may be enhanced by applying the model order reduction process to smaller vectors containing a smaller subset of field values
in the solution space of the systems. The Krylov subspace can then be constructed faster and vectors that are necessary for the construction of Krylov subspace can also be stored with less memory requirement. However, less space information may result in longer converging time since more time information is needed to compensate the missing space information. Tradeoffs between space and time information for the eigenmodes of systems may become a major problem for enhancing the performance of the hybrid methods.
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