## 2 Proposed Order Reduction Methodology
### Multipoint Expansion  
The development of a feasible methodology for a wide-frequency band ROM requires compromising the accuracy of the constructed projection basis and computational cost of the procedure. Nevertheless, a basic MOR algorithm generally employs one expansion point in developing the projection basis. However, as aforementioned, the single-point moment-matching reduced model is a locally optimal one, and it is applicable to simulating the transient response only around the frequency in question. As the frequency band of interest increases, a large number of moments need to be included to guarantee the model's accuracy in the frequency band in question, increasing the computational and memory costs. Moreover, such a reduced model is inaccurate for simulating the transient performance of a frequency whose distance to the frequency expansion point is large. To address the locally supported approximation characteristics of a single-point moment-matching approach, the Reduced-Basis Method (RBM) [15] or Balanced Truncation Proper Orthogonal Decomposition (BT-POD) [16], based on solution snapshots [17; 18] from different frequency points, is introduced. However, it involves computing snapshots on a large number of points. Moreover, for each point, a factorization operation on the system matrix is required. Consequently, such an approach considerably increases computational cost. To take advantage of both a snapshot-based and a single-point moment-matching approach, a multipoint MOR method is developed [19; 20; 21; 22]. A multipoint MOR method obtains the projection basis from the moments of the transfer function at multiple frequency points, as explained in Figure 2.  
Figure 1: Taylor expansion around \(s=0\). Figure 2: Taylor expansion around several points. In [19], a fully adaptive scheme based on Krylov space was presented to reduce the size of a large-scale integrated circuit. This approach provides an interesting way of adaptively choosing expansion points by a posterior error estimation based on admittance or impedance. However, the computational process is rather complex and inefficient. Moreover, in 3-D transient field computations, the concern is more the precision or accuracy of the solution rather than the admittance or impedance parameters. Consequently, a posterior-error estimate, based on admittance or impedance, is not suitable. In [20], a greedy multipoint MOR technique chose expansion points from an arithmetic midpoint, using a posterior-error estimation based on scattering parameters, to reduce the size of a second-order electromagnetic system. In [21], an adaptive multipoint-MOR scheme, using eigen values as the stopping criterion and an error estimator, chose the new expansion point by computing error estimators at three points every time to reduce the computational cost of large-scale inductive partial element equivalent circuits (PEEC). In [22], a multipoint MOR scheme was used to reduce the size of the finite-element equation of second-order systems. The location of the expansion point and size of the projection basis were determined based on an error estimator. However, the scheme cannot be applied directly to the transient temperature problem. In this regard, we propose a novel block Arnoldi-based multipoint-MOR scheme for 3-D transient field problems. The proposed approach not only benefits from the work of [20; 21; 22], but also explores the application of Krylov subspaces. The originality of this study is a novel adaptive algorithm for determining expansion points, matching moments automatically, as well as a posterior-error estimator based on the temperature response, coupled with a network topological method (NTM). It should be noted that the optimal expansion point should be the one that gives the largest error between the current ROM and the high-fidelity model. However, finding such an ideal point is neither practical nor feasible [9].