Collaborative Research: SI2-SSI: Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion

科学家经常使用数学模型来预测自然和工程系统的行为。因此，这些模型是科学和工程进步的基础，因此与NSF的科学使命有关。现实物理系统的大多数模型使用涉及许多变量的复杂公式（如偏微分方程）。当使用这种模型来预测系统的未来行为时，科学家必须为所有变量提供初始值。 这可能是困难的，因为输入值可能无法直接测量。因此，科学家经常必须使用“逆”计算来计算基于真实的世界的外部观测的系统模型的变量的初始输入值。换句话说，科学家试图从输出的真实的观测数据中推断出物理过程计算机模型的输入。逆计算的例子有很多，从计算机断层扫描计算器官的重要尺寸，通过测量不同地方的音量和频率重建声源，从重力场的测量计算地球的密度，或计算大气的初始条件（温度，压力等）。从卫星和气象站的观测中得到的数据。逆问题在所有科学和工程（以及其他领域）中无处不在。逆问题存在许多解，即将数据拟合到观测值的解。然而，所确定的解决办法各不相同。也就是说，反问题的解是不确定的。贝叶斯推理提供了一个系统的数学框架来描述这种不确定性。然而，大规模复杂模型的逆问题的贝叶斯解需要巨大的计算能力。直到最近才开始出现计算上易处理的算法。然而，这些算法仍然超出了解决逆问题的主流科学家的范围，因为它们的复杂性和对来自正演模型的更深层次信息的需求。该项目旨在开发，分发和支持开源软件，该软件编码用于解决大规模复杂贝叶斯逆问题的最先进算法，并且具有鲁棒性，可扩展性，灵活性，模块化，广泛访问和易于使用。该项目主要基于该团队一直在开发的两个互补开源软件库：MIT的MUQ和UT-Austin/UC-Merced的hIPPYlib。MUQ提供了一系列功能强大的贝叶斯反演模型和算法，但预计前向模型将配备梯度/Hessian，以允许大规模解决方案。hIPPYlib在一个可以自动生成所需导数的环境中实现了强大的大规模梯度/基于Hessian的逆求解器，但它缺乏完整的贝叶斯功能。通过整合这两个互补的库，该项目将产生一个强大的，可扩展的，高效的软件框架，实现每一个的好处，以解决复杂的大规模贝叶斯逆问题，跨越广泛的科学和工程学科。由此产生的软件将在开放源码许可证下分发，将提供一个环境，用于快速开发配备有梯度/Hessian信息的逆向模型;用于评估和比较算法的基准问题;以及用于培训和测试目的的教程问题。

Scientists often use mathematical models to predict the behavior of natural and engineered systems. These models are therefore fundamental to scientific and engineering progress and hence relevant to NSF's science mission. Most models of realistic physical systems use complex formulae (such as, partial differential equations) involving many variables. When using such a model for predicting the future behavior of a system, a scientist has to provide initial values for all the variables. This can be difficult because input values may not be directly measureable. Thus, scientists often must use "inverse" computations to calculate the initial input values of the variables of a system model based on external observations of the real world. In other words, scientists seek to infer inputs to a computer model of a physical process from real observational data of the outputs. There are many examples of inverse computations, ranging from computing the important dimensions of an organ from its CAT scan, reconstructing the source of a sound by measuring its volume and frequency at various places, calculating the density of the Earth from measurements of its gravity field, or calculating the initial condition of the atmosphere (temperature, pressure, etc.) from satellite and weather station observations over a time interval. Inverse problems are ubiquitous across all of science and engineering (and beyond). Many solutions exist for inverse problems, i.e. solutions that fit the data to the observations. However, there are variations in the solutions identified. That is, the solutions of an inverse problem are subject to uncertainty. Bayesian inferencing provides a systematic mathematical framework for characterizing this uncertainty. However, the Bayesian solution of inverse problems for large-scale complex models require enormous computational power. Only recently have algorithms begun to emerge that are computationally tractable. However, these algorithms have remained out of the reach of the mainstream of scientists who solve inverse problems, due to their complexity and the need for deeper information from the forward model. This project aims to develop, distribute, and support open-source software that encodes state-of-the-art algorithms for the solution of large-scale complex Bayesian inverse problems and is robust, scalable, flexible, modular, widely accessible, and easy to use.The project builds heavily on two complementary open-source software libraries the team has been developing: MUQ at MIT, and hIPPYlib at UT-Austin/UC-Merced. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients/Hessians to permit large-scale solution. hIPPYlib implements powerful large-scale gradient/Hessian-based inverse solvers in an environment that can automatically generate needed derivatives, but it lacks full Bayesian capabilities. By integrating these two complementary libraries, the project will result in a robust, scalable, and efficient software framework that realizes the benefits of each to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. The resulting software, that will be distributed under an open-source license, will provide an environment for rapid development of inverse models equipped with gradient/Hessian information; benchmark problems for evaluation and comparison of algorithms; and tutorial problems for training and testing purposes.