Analysis and Optimization with Complex Computer Models

这个建议和我的研究计划的重点是开发工具来理解和量化复杂数学模型中的不确定性。科学的方法是建立一个计算效率高的代理模型，用来模拟复杂模型的输出。代理模型用于替代真实模型，用于预测、优化和敏感性分析等任务。将计算机代码视为未知，使用高斯过程先验来表示该未知函数建模中的不确定性。虽然高斯过程替代模型现在经常被研究人员使用，但仍然有许多与这些模型相关的基本问题需要更好地理解。本提案的主要目标是为贝叶斯顺序设计开发一个新的框架，该框架可以用于几乎任何标准，同时允许在实施顺序过程时对不确定性进行完整的表征。鉴于计算技术的快速发展，顺序程序可以同时用于收集多个试验，而不是传统的一次收集一个样本的方法，这是势在必行的。此外，程序的其他方面集中在与拟合高斯过程相关的基本问题上，这需要优化似然或运行马尔可夫链以从后验分布中获取样本。在任何一种情况下，模型的确切形式，包括参数化和所需运行的次数，都将对有效地拟合模型产生重大影响。对巨大空间场进行建模和开发拟合非平稳模型的新方法的其他方法学发展受到理解恒星形成问题的推动。接下来的计划中概述的工作有可能改变研究人员解决代理模型问题的方式，这将对从事代理模型工作的科学界产生巨大影响。此外，这种影响将转化为使用复杂模型的更广泛的建模者和决策者社区。政策制定者使用复杂的模型来研究和理解几乎所有科学和工程领域无法直接研究的行为。为了做出明智的决策，这些模型被用来研究各种各样的假设情景。可靠和有效的替代模型将允许决策者调查大量的情景，同时也提供一个框架，以更好地理解和量化各种问题中的不确定性。这将有助于促进使用定量数据作出明智的决定，从而使加拿大和世界受益。

This proposal and my research program is focussed on developing tools for understanding and quantifying uncertainty in complex mathematical models. The scientific approach is to build a computationally efficient surrogate model that is used to emulate the output of the complex model. The surrogate model is used in place of the true model for tasks such as prediction, optimization and sensitivity analysis. Treating the computer code as unknown, a Gaussian process prior is used to represent the uncertainty in modelling this unknown function. Although Gaussian process surrogate models are now routinely used by researchers, there are still many fundamental issues related to these models that need to be better understood. The main objective of this proposal is to develop a novel framework for Bayesian sequential design that can be used for virtually any criteria while allowing for a complete characterization of the uncertainty when implementing a sequential procedure. Given the rapid advancements in computational techniques it is imperative that sequential procedures can be used simultaneously collect multiple trials as opposed to the traditional approach of collecting samples one-at-a-time. Additionally, other aspects of the program are focused on fundamental issues relating to fitting the Gaussian process, which requires either optimizing the likelihood or running a Markov chain to obtain samples from the posterior distribution. In either case the exact form of the model, including parameterization and the number of required runs, will have a significant impact on efficiently fitting the model. Additional methodological developments of modelling huge spatial fields and developing new methods for fitting non-stationary models are motivated by the problem of understanding star formation. The work outlined in the program to follow has the potential to transform the way researchers tackle problems in surrogate modelling, which will have tremendous impact on the scientific community working on surrogate models. Additionally this impact will translate to the much wider community of modeller and decision makers using complex models. Policy makers use complex models to study and understand behaviour that cannot be directly studied across virtually every area of science and engineering. In order to make informed decisions these models are used to investigate various what-if scenarios. Reliable and efficient surrogate modelling will allow decision makers to investigate a significantly larger number of scenarios while also providing a framework to better understand and quantify uncertainty in various problems. This will provide a benefit to Canada and the world by helping facilitate the use of quantitative data to make informed decisions.