Uncertainty Quantification for Numerical Models with two Regions of Solution

如今，复杂的数字模型在科学和政策领域的应用无处不在。对于实际应用，此类模型的预测需要伴随着对不确定性的估计。提供这些估计是一个困难的问题。高斯过程仿真器已被证明是一种非常有效的解决方案。高斯过程仿真器是一种运行速度很快的数字代码的统计模型，因此可以用于计算不确定度估计，例如，通过蒙特卡罗方法，这在整个模型的计算中是不可能的。仿真器已被广泛应用于从气候和空气污染到工程和地质学到系统生物学的各种应用。然而，在一些数值模型中，有两个或更多的解被分叉或临界点分开。气候科学的一个简单例子是Stommel模型，它对何时开启或关闭颠覆的环流有不同的解决方案。我们还有来自气候、油藏建模和生物学的其他例子。本博士将着眼于这个问题，开发方法来确定每个解决方案所在的模型输入空间区域，以及如何为每个解决方案构建单独但链接的仿真器。我们还将考虑反问题，即我们对一些模型输出进行真实世界的观测，这些输出被用来对模型输入做出推断，实际上是向后运行模型。此外，学生还将研究实验设计的问题，其中最好是在有限的计算机预算下运行模型，开发新的顺序方法。虽然学生将开发一种通用的方法，但它将在整个博士学位中应用到许多现实世界中。

The use of complex numerical models in science and policy is now ubiquitous. For practical application predictions from such models need to be accompanied by estimates of uncertainty. Supplying these estimates is a difficult problem. Gaussian process emulation has proved to be a very effective solution A Gaussian process emulator is a statistical model of the numerical code that is fast to run so can be used to calculate uncertainty estimates, for example by Monte Carlo methods that are computationally impossible with the full model. Emulators have been used across a wide variety of applications from climate and air pollution through engineering and geology to systems biology. However there are numerical models where there are two or more solutions separated by bifurcations or tipping points. A simple example from climate science is the Stommel model which has a different solution for when the overturning circulation is turned on or off. We have other examples from climate, oil reservoir modelling and biology. This PhD will look at this problem developing methods to identify the areas of model input space where each solution holds and how to build separate, but linked, emulators for each solutions. The inverse problem where we have real world observations of some of the model outputs and these are used to make inferences about the model inputs, in effect running the model 'backwards', will also be considered. In addition the student will look at the problem of experimental design, where best to run the model with a limited computer budget, developing new sequential methods. Although the student will develop a general methodology it will be applied to a number of real world applications throughout the PhD.