Tensor decomposition sampling algorithms for Bayesian inverse problems

无论是在日常生活中还是在最紧迫的全球挑战中，了解我们所观察到的事件的原因都是一项至关重要的任务。由轻质复合材料制成的飞机机翼会断裂吗？或者核废料会从储存库泄漏到饮用水井吗？通常，一个简单的是/否的答案是不可能的。几乎所有的统计预测都是以事件发生的概率来进行的，这是预测中不确定性的定量特征。然而，如果只给出一些观测结果，并且对潜在的自然过程一无所知，那么这种预测可能非常糟糕。另一个极端是，如果没有数据来初始化它，即使是超级精确的模型也将毫无意义。如果我们今天没有收集任何测量结果，我们如何预测明天的天气？在实践中，我们通常既有模型又有数据，但质量有限：一个部分不准确的模型，一个部分不完整和嘈杂的观测数据。我们如何才能产生一个在某种意义上最好的预测，连同它的不确定性？几个世纪以来，这个问题的一个数学上严格的答案是众所周知的：贝叶斯定理。然而，由于所谓的维数灾难，在实践中使用它可能极具挑战性。贝叶斯定理以联合概率分布函数的形式描述答案，该函数取决于模型的所有可调参数。尽管感兴趣的预测可能只是一个数字，计算这个数字需要概率函数的数值积分。直接尝试这样做包括计算参数的所有可能组合的概率值。这使得计算量随着维数，即参数的数量，虽然一些只有一个参数的简单情况可能在毫秒内计算，但对于具有数十个参数的高维问题，即使宇宙的寿命也不足以直接解决它们。然而，在贝叶斯方法中出现的许多概率函数包含隐藏的结构，这些结构可能对计算方法有很大的帮助。本项目旨在揭示和利用这种结构，使贝叶斯统计预测在计算上易于处理。我将通过以下方法来实现这一点：结合联合收割机的优点，开发新的算法。该项目的核心是张量积分解。这是一个强大的数据压缩方法家族，起源于简单的分离变量。张量分解的效率依赖于模型参数在某种意义上弱相关的假设（例如，第一个参数对最后一个参数的影响很小）。另一个来自概率的经典方法，罗森布拉特变换，将开发一个自适应程序来计算坐标的变化，满足弱依赖性的假设转换参数。将使更好的预测驱动贝叶斯最优统计分析在复杂的逆问题，如那些出现在测试和认证的新复合材料的航空航天工业。此外，体现在开源软件与学术团体在统计和工程的合作，新的算法将铺平道路，甚至更广泛的吸收拟议的方法来处理不确定性。

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.