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The accuracy of the reduced-order model (ROM) mainly depends on the selectedbasis. Therefore, it is essential to compute an appropriate basis with anefficient numerical procedure when applying ROM to nonlinear problems. In thispaper, we propose an online adaptive basis technique to increase the quality ofROM while decreasing the computational costs in nonlinear problems. In theproposed method, the adaptive basis is defined by the low-rank updateformulation, and two auxiliary vectors are set to implement this low-rankcondition. To simultaneously tackle the issues of accuracy and thecomputational cost of the ROM basis, the auxiliary vectors are algebraicallyderived by optimizing a local residual operator. As a result, the reliabilityof ROM is significantly improved with a low computational cost because theerror information can be contained without inverse operations of the full modeldimension required in conventional approaches. The other feature of theproposed iterative algorithm is that the number of the initial incremental ROMbasis could be varied, unlike in the typical online adaptive basis approaches.It may provide a fast and effective spanning process of the high-quality ROMsubspace in the iteration step. A detailed derivation process of the proposedmethod is presented, and its performance is evaluated in various nonlinearnumerical examples.

Online adaptive basis construction for nonlinear model reduction through local error optimization