Tensor decomposition sampling algorithms for Bayesian inverse problems

在日常生活和最紧迫的全球挑战中，理解我们观察到的事件的原因是一项至关重要的任务。多雨的夏天真的是由于极地冰川融化吗？由轻质复合材料制成的飞机机翼会破裂吗？或者核废料会从仓库泄漏并到达饮用水水井吗？通常，简单的是/否的答案是不可能的。实际上，所有的统计预测都是根据发生事件的概率进行的，这是预测中不确定性的量化特征。然而，如果只给出一些观察结果，而对潜在的自然过程一无所知，这样的预测可能会非常糟糕。在另一个极端，如果没有数据来初始化，即使是一个超精确的模型也是没有意义的。如果我们今天没有收集到任何测量数据，我们如何预测明天的天气呢？实际上，我们通常有模型和数据，但质量有限：一个部分不准确的模型，以及部分不完整和有噪声的观测数据。我们如何才能产生在某种意义上最好的预测，以及它的不确定性？对于这个问题，一个数学上严格的答案是几个世纪以来众所周知的：贝叶斯定理。然而，贝叶斯定理以依赖于模型的所有可调参数的联合概率分布函数的形式描述了答案。尽管感兴趣的预测可以只是单个数字，但计算该数字需要对概率函数进行数值积分。这样做的直接尝试涉及计算参数的所有可能组合的概率值。这使得在问题中计算量随维度指数增长，即参数的数量。虽然只有一个参数的一些简单情况可以在毫秒内计算，对于具有几十个参数的高维问题，即使是宇宙的寿命也不足以直接解决它们。然而，在贝叶斯方法中产生的许多概率函数包含着隐藏的结构，这可能会对计算方法有很大的帮助。本项目的目的是揭示和利用这种结构来使贝叶斯统计预测在计算上是可追踪的。我将通过开发结合几种经典数学方法的优点的新算法来实现这一点。该项目的核心是张量积分解。这是一系列强大的数据压缩方法，源于简单的变量分离。张量分解的效率依赖于假设模型参数在某种意义上是弱依赖的(例如，第一个参数对最后一个参数的影响很小)。另一个经典的概率方法，Rosenblatt变换，将被利用来开发一个自适应的程序来计算坐标的变化，满足变换后的参数的弱相关性的假设。新的方法将使更好的预测由贝叶斯-最优统计分析驱动的复杂逆问题，如航天工业新复合材料的测试和认证。此外，将新算法与统计学和工程学的学术团体合作，将新算法应用于开源软件，将为更广泛地采用所提出的处理不确定性的方法铺平道路。

Understanding the cause of events we observe is a vital task in both everyday life and most pressing global challenges.Was the rainy summer really due to the polar ice meltdown?Will an aircraft wing made from lightweight composite materials break?Or will a nuclear waste leak from the storage and reach a drinking water well?Often, a simple yes/no answer is impossible.Virtually all statistical forecasts operate with the probability of an event to happen,that is a quantitative characteristics of the uncertainty in the prediction.However, such predictions can be very poor if only some observations are given, and nothing is known about the underlying natural processes.On the other extreme, even a super-accurate model would be meaningless if there is no data to initialise it.How can we predict weather for tomorrow if we have collected no measurements today?In practice we usually have both model and data, but of limited quality:a partially inaccurate model, and a partially incomplete and noisy observation data.How can we produce a forecast that is best in some sense, together with its uncertainty?A mathematically rigorous answer to this question is known for centuries: the Bayes theorem.However, it might be extremely challenging to employ it in practice due to the so-called curse of dimensionality.The Bayes theorem describes the answer in the form of a joint probability distribution function that depends on all tunable parameters of the model.Although a forecast of interest can be just a single number,computing this number requires numerical integration of the probability function.Straightforward attempt to do so involves computing probability values for all possible combinations of the parameters.This renders the amount of computations growing exponentially with the dimensionality, that is the number of parameters, in the problem.While some simple case with only one parameter might be calculable in milliseconds,for high-dimensional problems with tens of parameters even the lifetime of the Universe could be not enough to solve them straightforwardly.However, many probability functions arising in the Bayesian approach contain hidden structure that may aid computational methods significantly.This project aims to reveal and exploit this structure to make Bayesian statistical predictions computationally tractable.I will approach this by developing new algorithms that combine advantages of several classical mathematical methods.The core of the project is the tensor product decompositions.This is a powerful family of methods for data compression that originate from the simple separation of variables.The efficiency of tensor decompositions relies on assumption that the model parameters are weakly dependent in a certain sense (for example, the first parameter has little influence on the last one).Another classical method from probability, the Rosenblatt transformation, will be exploitedto develop an adaptive procedure to compute a change of coordinates that fulfils the assumption of weak dependence for the transformed parameters.The new methods will enable better predictions driven by Bayes-optimal statistical analysis in complicated inverse problems such as those arising in testing and certification of new composite materials for aerospace industry.Moreover, embodying the new algorithms in open-source software in collaboration with academic groups in statistics and engineering will pave the way to even wider uptake of the proposed methodology for treatment of uncertainty.