In this paper, we propose a new methodology for solving stochastic inversion problems through computer experiments, the stochasticity being driven by a functional random variables. This study is motivated by an automotive application. In this context, the simulator code takes a double set of simulation inputs: deterministic control variables and functional uncertain variables. This framework is characterized by two features. The first one is the high computational cost of simulations. The second is that the probability distribution of the functional input is only known through a finite set of realizations. In our context, the inversion problem is formulated by considering the expectation over the functional random variable. We aim at solving this problem by evaluating the model on a design, whose adaptive construction combines the so-called stepwise uncertainty reduction methodology with a strategy for an efficient expectation estimation. Two greedy strategies are introduced to sequentially estimate the expectation over the functional uncertain variable by adaptively selecting curves from the initial set of realizations. Both of these strategies consider functional principal component analysis as a dimensionality reduction technique assuming that the realizations of the functional input are independent realizations of the same continuous stochastic process. The first strategy is based on a greedy approach for functional data-driven quantization, while the second one is linked to the notion of space-filling design. Functional PCA is used as an intermediate step. For each point of the design built in the reduced space, we select the corresponding curve from the sample of available curves, thus guaranteeing the robustness of the procedure to dimension reduction. The whole methodology is illustrated and calibrated on an analytical example. It is then applied on the automotive industrial test case where we aim at identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle.

中文翻译：

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通过功能量化进行数据驱动的随机反演

在本文中，我们提出了一种通过计算机实验解决随机反演问题的新方法，该随机性是由函数随机变量驱动的。这项研究是受汽车应用推动的。在这种情况下，模拟器代码采用了双重模拟输入集：确定性控制变量和功能不确定性变量。该框架具有两个功能。第一个是模拟的高计算成本。第二个是功能输入的概率分布只有通过有限的一组实现才能知道。在我们的上下文中，通过考虑对函数随机变量的期望来公式化反演问题。我们旨在通过评估设计模型来解决此问题，其自适应构造将所谓的逐步不确定性降低方法与有效的预期估计策略结合在一起。引入了两种贪婪策略，以通过从初始实现集中自适应选择曲线来依次估计功能不确定变量的期望值。这两种策略都将功能主成分分析视为降维技术，假设功能输入的实现是同一连续随机过程的独立实现。第一种策略基于贪婪的方法，用于功能数据驱动的量化，而第二种策略则与空间填充设计的概念相关联。功能性PCA用作中间步骤。对于缩小空间中的设计的每个点，我们从可用曲线的样本中选择相应的曲线，从而保证了该程序对降维的鲁棒性。整个方法在一个分析示例上进行了说明和校准。然后将其应用到汽车工业测试用例中，我们的目标是确定一组能够满足车辆污染物排放标准的控制参数。