  The accuracy of the reduced-order model (ROM) mainly depends on the selected
basis. Therefore, it is essential to compute an appropriate basis with an
efficient numerical procedure when applying ROM to nonlinear problems. In this
paper, we propose an online adaptive basis technique to increase the quality of
ROM while decreasing the computational costs in nonlinear problems. In the
proposed method, the adaptive basis is defined by the low-rank update
formulation, and two auxiliary vectors are set to implement this low-rank
condition. To simultaneously tackle the issues of accuracy and the
computational cost of the ROM basis, the auxiliary vectors are algebraically
derived by optimizing a local residual operator. As a result, the reliability
of ROM is significantly improved with a low computational cost because the
error information can be contained without inverse operations of the full model
dimension required in conventional approaches. The other feature of the
proposed iterative algorithm is that the number of the initial incremental ROM
basis could be varied, unlike in the typical online adaptive basis approaches.
It may provide a fast and effective spanning process of the high-quality ROM
subspace in the iteration step. A detailed derivation process of the proposed
method is presented, and its performance is evaluated in various nonlinear
numerical examples.
