In a task where many similar inverse problems must be solved, evaluating costly simulations is impractical. Therefore, replacing the model $y$ with a surrogate model $y_s$ that can be evaluated quickly leads to a significant speedup. The approximation quality of the surrogate model depends strongly on the number, position, and accuracy of the sample points. With an additional finite computational budget, this leads to a problem of (computer) experimental design. In contrast to the selection of sample points, the trade-off between accuracy and effort has hardly been studied systematically. We therefore propose an adaptive algorithm to find an optimal design in terms of position and accuracy. Pursuing a sequential design by incrementally appending the computational budget leads to a convex and constrained optimization problem. As a surrogate, we construct a Gaussian process regression model. We measure the global approximation error in terms of its impact on the accuracy of the identified parameter and aim for a uniform absolute tolerance, assuming that $y_s$ is computed by finite element calculations. A priori error estimates and a coarse estimate of computational effort relate the expected improvement of the surrogate model error to computational effort, resulting in the most efficient combination of sample point and evaluation tolerance. We also allow for improving the accuracy of already existing sample points by continuing previously truncated finite element solution procedures.

翻译：在需要求解大量相似反问题的任务中，评估成本高昂的仿真计算是不切实际的。因此，用可快速评估的替代模型 $y_s$ 替代原始模型 $y$ 可显著加速计算。替代模型的近似质量强烈依赖于样本点的数量、位置和精度。在有限的计算预算下，这引出了（计算机）实验设计问题。与样本点选择相比，精度与计算投入之间的权衡尚未得到系统研究。为此，我们提出一种自适应算法，旨在从位置和精度两方面寻找最优实验设计。通过逐步追加计算预算的序贯设计方法，可归结为一个带约束的凸优化问题。我们构建高斯过程回归模型作为替代模型，从参数识别精度受全局近似误差影响的角度衡量该误差，并基于有限元计算得到的 $y_s$ 设定统一绝对容差。利用先验误差估计与粗略计算开销估计，将替代模型误差的预期改进与计算投入相关联，从而得到样本点及其评估容差的最优组合方案。此外，我们还允许通过延续先前截断的有限元求解过程来提升已有样本点的精度。