Enabling Quantification of Uncertainty for Large-Scale Inverse Problems (EQUIP)

物理实验的数学模型是一组把输入和输出联系起来的方程。输入代表可以在实验发生前调整的物理变量；输出表示可以作为实验结果测量的量。前向问题是指利用数学模型从给定的输入预测实验的输出。反问题是指利用数学模型对数学模型的输入进行推断，从而得到给定的测量输出。一个例子涉及油藏模拟的数学模型。该模型的一个重要输入是地下岩石的渗透率。自然产量是指从生产井中流出的油和/或水。由于地下不能直接观测到，因此从生产井的测量中推断其性质的问题尤为重要。准确的推断有助于对钻井的经济可行性和井位进行决策。在许多反问题中，测量数据受到噪声的影响，数学模型可能不完善。因此，量化作为反问题解决方案的一部分所做的任何推论中固有的不确定性是非常重要的。这项工作汇集了一组数学科学家，他们在应用数学、计算机科学和统计学以及工程应用方面具有专业知识，以开发解决反问题的新方法，包括不确定性的量化。这项工作将以确定地下性质的应用为驱动力，但也将应用于生物、物理和社会科学的一系列问题。

A mathematical model for a physical experiment is a set of equations which relate inputs to outputs. Inputs represent physical variables which can be adjusted before the experiment takes place; outputs represent quantities which can be measured as a result of the experiment.The forward problem refers to using the mathematical model to predict the output of an experiment from a given input. The inverse problem refers to using the mathematical model to make inferences about input(s) to the mathematical model which would result in a given measured output.An example concerns a mathematical model for oil reservoir simulation. An important input to the model is the permeability of the subsurface rock. A natural output would be measurements of oil and/or water flow out of production wells. Since the subsurface is not directly observable, the problem of inferring its properties from measurements at production wells is particularly important. Accurate inference enables decisions to be made about the economic viability of drilling a well, and about well-placement. In many inverse problems the measured data is subject to noise, and the mathematical model may be imperfect. It is then very important to quantify the uncertainty inherent in any inferences made as part of the solution to the inverse problem. The work brings together a team of mathematical scientists, with expertise in applied mathematics, computer science and statistics, together with engineering applications, to develop new methods for solving inverse problems, including the quantification of uncertainty. The work will be driven by applications in the determination of subsurface properties, but will have application to a range of problems in the biological, physical and social sciences.