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：在许多需要求解类似反问题的任务中，评估代价高昂的仿真计算是不切实际的。因此，用能够快速评估的代理模型 $y_s$ 替换原模型 $y$ 可以显著提升计算速度。代理模型的近似质量很大程度上取决于样本点的数量、位置和精度。在有限的计算预算下，这将导致（计算机）实验设计问题。与样本点选择不同，精度与计算开销之间的权衡鲜有系统研究。为此，我们提出一种自适应算法，以在位置和精度两方面寻求最优设计。通过逐步追加计算预算进行序贯设计，可转化为一个凸约束优化问题。我们构建高斯过程回归模型作为代理，以全局近似误差对辨识参数精度的影响为度量，并假设 $y_s$ 由有限元计算获得，从而设定均匀绝对容差。通过先验误差估计和计算开销的粗略评估，将代理模型误差的预期改进与计算开销相关联，进而获得样本点与评估容差的最优组合方案。我们还允许通过延续先前截断的有限元求解过程，提升已有样本点的精度。