Functional data are ubiquitous in scientific modeling. For instance, quantities of interest are modeled as functions of time, space, energy, density, etc. Uncertainty quantification methods for computer models with functional response have resulted in tools for emulation, sensitivity analysis, and calibration that are widely used. However, many of these tools do not perform well when the model's parameters control both the amplitude variation of the functional output and its alignment (or phase variation). This paper introduces a framework for Bayesian model calibration when the model responses are misaligned functional data. The approach generates two types of data out of the misaligned functional responses: one that isolates the amplitude variation and one that isolates the phase variation. These two types of data are created for the computer simulation data (both of which may be emulated) and the experimental data. The calibration approach uses both types so that it seeks to match both the amplitude and phase of the experimental data. The framework is careful to respect constraints that arise especially when modeling phase variation, but also in a way that it can be done with readily available calibration software. We demonstrate the techniques on a simulated data example and on two dynamic material science problems: a strength model calibration using flyer plate experiments and an equation of state model calibration using experiments performed on the Sandia National Laboratories' Z-machine.

翻译：函数型数据在科学建模中无处不在。例如，研究关注的量常被建模为时间、空间、能量、密度等的函数。针对具有函数响应的计算机模型的不确定性量化方法，已产生了广泛使用的替代、敏感性分析和校准工具。然而，当模型参数同时控制函数输出的幅度变化及其对齐（或相位变化）时，许多此类工具表现欠佳。本文提出了一种针对模型响应为未对齐函数数据的贝叶斯模型校准框架。该方法从未对齐的函数响应中生成两类数据：一类分离幅度变化，另一类分离相位变化。这两类数据同时从计算机模拟数据（两者均可通过替代模型近似）和实验数据中生成。校准方法利用这两类数据，旨在同时匹配实验数据的幅度和相位。该框架谨慎地考虑了在建模相位变化时产生的约束，同时确保能够借助现有的校准软件实现。我们通过模拟数据示例以及两个动态材料科学问题展示了该技术：基于飞片板实验的强度模型校准，以及基于桑迪亚国家实验室Z机实验的状态方程模型校准。