Incremental modeling of a new high-order polynomial surrogate model

• The proposal of a new high-order polynomial surrogate model, termed as HOPSM.
• Zeros of Chebyshev polynomials with the highest allowable order used as sampling candidates.
• An incremental sampling scheme using the maximin principle to sequentially collect samples.
• An order incremental method to adaptively improve the order of the polynomial.

This study will develop a new high-order polynomial surrogate model (HOPSM) to overcome routines of expensive computer simulations in engineering. The proposed HOPSM is expected to keep advantages of the traditional low-order polynomial models in efficiency, transparency and simplicity, while avoid their disadvantage in accuracy. The zeros of Chebyshev polynomials having the highest allowable order will be utilized as the sampling candidates to improve stability and accuracy of the approximation. In the numerical process, a space-filling scheme is used to generate the initial set of samples, and then an incremental method based on the maximin principle is established to select more samples from all candidates. At the same time, the order of HOPSM is sequentially updated by using an order incremental scheme, to adaptively increase the polynomial order along with the increase of the sample size. After the order increment, the polynomial with the largest adjusted R-square is determined as the final HOPSM. Several typical test functions and two engineering applications are used to demonstrate the effectiveness of the proposed surrogate modeling method.