Gaussian processes are widely used for accurate emulation of unknown surfaces in sequential design of expensive simulation experiments. Integrated mean squared error (IMSE) is an effective acquisition function for sequential designs based on Gaussian processes. However, existing approaches struggle with its implementation because the required integrals often lack closed-form expressions for most kernel functions. We propose a novel and computationally efficient Hilbert space Gaussian process approximation for the IMSE acquisition function, where a truncated eigenbasis representation of the integral enables closed-form evaluation. We establish sharp global non-asymptotic bounds for both the approximation error of isotropic kernels and the resulting error in the acquisition function. In a series of numerical experiments with $γ$-stabilizing, the proposed method achieves substantially lower prediction error and reduced computation time compared to existing benchmarks. These results demonstrate that the proposed Hilbert space Gaussian process framework provides an accurate and computationally efficient approach for Gaussian process based sequential design.

翻译：高斯过程广泛用于昂贵仿真实验序贯设计中未知曲面的精确替代建模。综合均方误差（IMSE）是基于高斯过程的序贯设计的一种有效采集函数。然而，现有方法在实现该函数时面临挑战，因为所需积分对大多数核函数往往缺乏闭合表达式。我们提出一种新颖且计算高效的希尔伯特空间高斯过程近似方法用于IMSE采集函数，其中积分的截断特征基表示实现了闭合形式评估。我们建立了各向同性核近似误差及其在采集函数中误差的锐利全局非渐近界。在$γ$-稳定化的一系列数值实验中，与现有基准相比，所提方法实现了显著更低的预测误差和缩减的计算时间。这些结果表明，所提出的希尔伯特空间高斯过程框架为基于高斯过程的序贯设计提供了一种精确且计算高效的方法。