Investigating uncertainties in computer simulations can be prohibitive in terms of computational costs, since the simulator needs to be run over a large number of input values. Building an emulator, i.e., a statistical surrogate model of the simulator constructed using a design of experiments made of a comparatively small number of evaluations of the forward solver, greatly alleviates the computational burden to carry out such investigations. Nevertheless, this can still be above the computational budget for many studies. Two major approaches have been used to reduce the budget needed to build the emulator: efficient design of experiments, such as sequential designs, and combining training data of different degrees of sophistication in a so-called multifidelity method, or multilevel in case these fidelities are ordered typically for increasing resolutions. We present here a novel method that combines both approaches, the multilevel adaptive sequential design of computer experiments in the framework of Gaussian process (GP) emulators. We make use of reproducing kernel Hilbert spaces as a tool for our GP approximations of the differences between two consecutive levels. This dual strategy allows us to allocate efficiently limited computational resources over simulations of different levels of fidelity and build the GP emulator. The allocation of computational resources is shown to be the solution of a simple optimization problem in a special case where we theoretically prove the validity of our approach. Our proposed method is compared to other existing models of multifidelity Gaussian process emulation. Gains in orders of magnitudes in accuracy or computing budgets are demonstrated in some numerical examples for some settings.

中文翻译：

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多级计算机实验顺序设计的自适应策略

就计算成本而言，调查计算机模拟中的不确定性可能会让人望而却步，因为模拟器需要在大量输入值上运行。构建模拟器，即使用由相对少量的正向求解器评估构成的实验设计构建的模拟器的统计替代模型，大大减轻了进行此类研究的计算负担。尽管如此，这仍然可能超出许多研究的计算预算。已使用两种主要方法来减少构建仿真器所需的预算：高效的实验设计，例如顺序设计，以及在所谓的多保真方法中组合不同复杂程度的训练数据，或多级，以防这些保真度通常是为了增加分辨率而订购的。我们在这里提出了一种结合这两种方法的新方法，即在高斯过程 (GP) 仿真器框架下进行计算机实验的多级自适应顺序设计。我们利用再生核 Hilbert 空间作为我们对两个连续级别之间的差异进行 GP 近似的工具。这种双重策略使我们能够在不同保真度级别的模拟上有效地分配有限的计算资源，并构建 GP 模拟器。在我们从理论上证明我们方法的有效性的特殊情况下，计算资源的分配被证明是一个简单优化问题的解决方案。将我们提出的方法与其他现有的多保真高斯过程仿真模型进行了比较。