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定律的数学模型极其复杂，但尽管如此，计算机模拟对于准确的定量预测是不可或缺的。解释量子效应或不确定信息的创新模型是高维的，而且非常昂贵——宇宙的生命周期不足以用经典方法解决这些问题。然而，许多这样的问题显示出结构，如果加以利用，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，并将其开放到计算机模拟的常规应用中。这将通过开发新方法来实现，这些新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。我将极大地扩展这些方法的功能，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呢？想象一下湖里有一个细菌。在任何时间点，它都向任意方向移动。我们不能肯定地说，细菌会到达湖中的某个区域，感染生长在那里的植物，但我们可以计算出这种情况发生的概率。如果我们想考虑所有的植物一起生长在一个湖中，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，我们将对计算机内存的消耗进行平方，因为与第一个细菌的位置无关，第二个细菌仍然可以自由地去任何地方。随着细菌数量的增加，数据量呈指数级增长，并迅速耗尽任何合理的内存。所以“尺寸”一词指的是细菌的数量，自然是非常高的。然而，如果细菌独立移动，那么仅存储其中一个的概率值就足够了：湖中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，并受到湖泊环境流量的影响。然而，湖一边的细菌不太可能受到另一边细菌的影响。所以有效地近似整个系统的数据量将比总的自由度要小很多个数量级。在这个例子中，高维的概念实际上无处不在。一架机翼承受波动载荷的飞机，被磁共振光谱仪揭开的量子效应，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置新的预测精度水平，特别是在不确定介质中污染物的地下流动或随机病毒动力学等重要问题上，并扩展可以解决的问题类别。我将要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，但实际上它只不过是连续函数的变量分离概念的直接推广，可以通过奇异值分解精确计算。张量积框架中的主力是交替优化算法。采用经典的克雷洛夫迭代法求解矩阵方程，可以大大提高其收敛性。我将进一步扩展张量积方法，并将它们体现在具有透明用户界面的公开软件中，以流行的科学软件包，以鼓励其他研究人员在实际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.