Tensor product numerical methods for high-dimensional problems in probability and quantum calculations

当代技术过程的巨大复杂性导致描述自然物理规律的数学模型具有极高的复杂性，但计算机模拟对于准确的定量预测是必不可少的。考虑到量子效应或不确定信息的创新模型是高维的，而且成本高昂--宇宙的寿命不足以用经典方法解决这些问题。然而，许多这样的问题表现出这样的结构，如果利用这些结构，可以显著减少计算工作量。该项目旨在打破高维模型的复杂性，将其开放给计算机模拟的常规使用。这将通过开发新的方法来实现，这些方法使用经典的数学方法来揭示隐藏的低维结构，例如分离变量和奇异值分解。我将大大扩展这些方法的能力，并将它们应用于各种重要的物理问题。我们知道我们的世界是三维的，那么高维模型是如何产生的？想象一下湖里有一种细菌。在任何时间点，它都会朝着任意的方向移动。我们不能肯定地说细菌会到达湖中的某个区域，感染那里生长的植物，但我们可以计算出发生这种情况的可能性。如果我们想要考虑一起生长在一个湖里的所有植物，我们将不得不在计算机内存中存储所有植物的概率值。如果我们同时描述两个细菌的行为，我们将平方计算机内存的消耗，因为第二个细菌仍然有自由去任何地方的自由，而不是第一个细菌的位置。随着细菌数量的增加，数据量呈指数级爆炸，并迅速耗尽任何合理的内存。所以“维度”这个术语指的是细菌的数量，自然是很高的。但是，如果细菌独立运动，那么只存储其中一个的概率值就足够了：那么湖泊整体情况的联合概率就是边际概率的乘积。在现实中，细菌会与附近的细菌发生一定程度的相互作用，并会受到湖中周围水流的影响。然而，湖的一边的细菌不太可能受到另一边的细菌的影响。因此，有效接近整个系统的数据量将比总自由度小许多个数量级。这个例子中的高维概念实际上是无处不在的。一架飞机，经历机翼上的波动载荷，磁共振光谱仪揭示的量子效应，昼夜节律或病毒复制-所有这些现象都可以用高维模型来描述。我将设计方法，在这些现有模型中设定新的预测精度水平，特别是诸如污染物在不确定介质中的地下流动或随机病毒动力学等关键问题，并扩大可以解决的问题的类别。我将开发的新方法结合了一系列数学技术。张量积概念是一种强大的数据压缩方法，但它实际上只不过是连续函数变量分离概念的直接推广，可以通过奇异值分解准确地计算出来。张量积框架中的主力是交替的优化算法。通过采用矩阵方程的经典Krylov迭代方法的思想，可以大大增强它们的收敛。我将进一步扩展张量乘积方法，并将它们体现在公开可用的软件中，该软件具有透明的用户界面，适用于流行的科学包，以鼓励其他研究人员在现实生活问题中尝试新的方法。

The tremendous complexity of contemporary technological processes induces a extremely high complexity in mathematical models that describe the physical laws of nature, but nonetheless computer simulations are indispensable for accurate quantitative predictions. Innovative models accounting for quantum effects or uncertain information are high-dimensional and hugely expensive - the lifetime of the Universe would not be enough to solve them by classical means. However, many such problems exhibit structure that if exploited can significantly reduce the computational effort. This project aims to break the complexity of high-dimensional models, and open them for routine use in computer simulation. This will be accomplished by the development of new methods that reveal hidden low-dimensional structures using classic mathematical methods, such as separation of variables and singular value decomposition. I will substantially extend the power of these methods, and apply them across a variety of important physical problems.We know that our world is three-dimensional, so how do high-dimensional models arise? Imagine a bacterium in a lake. At any point in time it makes a move in an arbitrary direction. We cannot say definitely that the bacterium will reach a certain region in the lake and infect a plant growing there, but we can calculate the probability that this will happen. If we would like to consider all plants growing in a lake together we will have to store in computer memory the probability values for all plants. If we describe the behaviour of two bacteria at the same time, we will square the consumption of computer memory, since independent of the position of the first bacterium the second one still has the freedom to go anywhere. With an increasing number of bacteria, the amount of data explodes exponentially and quickly exhausts any reasonable memory. So the term "dimension" refers to the number of bacteria, and is naturally very high.However, if bacteria move independently it is enough to store probability values just for one of them: the joint probability of the overall situation in the lake is then simply the product of the marginal probabilities. In reality, the bacteria will interact to some extent with near-by bacteria and will be influenced by the ambient flow in the lake. However, it is highly unlikely that a bacterium on one side of the lake will be affected by bacteria at the other. So the volume of data that effectively approximates the whole system will be many orders of magnitude smaller than the total number of degrees of freedom. The concept of high dimension in this example is in fact ubiquitous. An airplane, experiencing a fluctuating load on its wings, quantum effects unravelled by a magnetic resonance spectrometer, circadian rhythms or virus replication - all these phenomena are described by high-dimensional models. I will design methods that set new levels of prediction accuracy in these existing models, in particular such vital problems as subsurface flows of pollutants in an uncertain medium or stochastic virus dynamics, and extend the class of problems that can be tackled.The new methods I am going to develop combine a range of mathematical techniques. The tensor product concept is a powerful data compression method, but it is in fact nothing more than an immediate generalisation of the separation of variable concept for continuous functions that can be accurately computed via the singular value decomposition. The workhorses in the tensor product framework are alternating optimisation algorithms. Their convergence can be substantially enhanced by employing ideas from classical Krylov iterative methods for matrix equations. I will extend tensor product methods further, and embody them in publicly available software with transparent user interfaces to popular scientific packages in order to encourage other researchers to try the new methodology in real-life problems.