A direct method for solving matrix Wiener-Hopf equations

应用数学中的许多问题都可以用Wiener-Hopf技术来解决。这样的问题出现在许多应用领域，包括电磁学、固体力学、水波理论、声学、金融数学、理论物理和概率论；许多问题不适用于任何已知的替代方法。特别是，Wiener-Hopf方程可以用来对使用计算机模拟很难捕获的突变进行建模，因为这些方程通常是在假设物理量可以使用光滑函数精确逼近的情况下工作的。事实证明，这在发展非破坏性评估技术的理论方面是至关重要的，在非破坏性评估技术中，通过向结构(如飞机机翼、潜艇船体或核电站部件)发射超声波并研究散射响应来检测裂纹和其他缺陷并确定其大小。为了直观地描述Wiener-Hopf方程的应用，考虑声波通过圆柱形管道，朝向开口端的传播。如果观察者直接面向管道末端站着，并移动到一侧，声音将逐渐变得听不见；当观察者移动到管道外时，声音不会突然消失。声场的连续性质与波从管道传播到露天时产生的绕射效应有着复杂的联系，这可以通过求解适当的Wiener-Hopf方程来精确地模拟[1，3.4节]。本项目涉及一种同时求解耦合Wiener-Hopf方程的新方法。这些所谓的矩阵问题是最困难的问题之一，但由于需要对复杂的结构和材料进行建模，它们经常出现在现代应用数学中。在一定的约束条件下，解决方案是已知存在的。然而，这一点的证明是非建设性的，这意味着它没有提供任何关于如何实际获得解决方案的指示。即使是近似解的构造也很困难，因为Wiener-Hopf方程非常精细，系数的微小变化可能会严重影响解，导致它违反物理定律，并使其不可用。该项目的动机是最近一篇论文[2]中出现的一个非常复杂的矩阵Wiener-Hopf方程的解。这似乎不适用于早期基于方程本身进行简化近似的方法。取而代之的是一种“隐式求积格式”，它的工作原理是用柯西积分公式表示未知项。粗略地说，这表明一个具有某些属性的函数完全由它沿一条路径的值决定。沿着这条路径分布一组节点，并使用矩阵Wiener-Hopf方程构造节点处的函数值。最后，使用其在节点处的值来近似完整解。增加节点数可以提高解的精度。Wiener-Hopf方程本身没有应用任何近似，因此不存在产生无效解的风险。我们的主要目标是进一步探索隐式求积方案，扩大它可以应用的问题范围，并优化它的性能。我们还会将它与其他早期的方法进行比较，在这些方法中可以使用这些方法。然后，我们将开始开发一个数值库，以有效地实施隐式求积格式。这将使物理学家、工程师和其他数学家能够快速地将该方法应用于使用现有方法难以解决的重要实际问题。参考文献[1]B.著名的基于Wiener-Hopf技术的方法。切尔西，1988.[2]I.Thompson《用Mindlin理论模拟的板中刚性条带的波的绕射》。《皇家学会论文集》A 476(2243)，2020年。

Many problems in applied mathematics can be solved using the Wiener-Hopf technique. Such problems arise in numerous application areas, including electromagnetism, solid mechanics, water wave theory, acoustics, financial mathematics, theoretical physics and probability theory; many are not amenable to any known alternative approaches. In particular, Wiener-Hopf equations can be used to model abrupt changes that are very difficult to capture using computer simulations, since these often work under the assumption that physical quantities can be accurately approximated using smooth functions. This has proved essential in developing theory that underpins non-destructive evaluation techniques, in which cracks and other defects are detected and sized by transmitting ultrasonic waves into a structure (such as an aeroplane wing, submarine hull or nuclear power plant component) and studying the scattered response. To visualise an application of a Wiener-Hopf equation, consider a sound wave travelling through a cylindrical pipe, towards an open end. If an observer stands directly facing the end of the pipe and moves to one side, the sound will gradually become inaudible; it will not suddenly disappear as the observer moves out of line with the pipe. The continuous nature of the acoustic field is intricately linked to the diffraction effect that occurs as the wave propagates from the pipe into the open air, and this can be accurately modelled by solving an appropriate Wiener-Hopf equation [1, Section 3.4].This project is concerned with a new approach to simultaneously solving coupled Wiener-Hopf equations. These so-called matrix problems are amongst the most difficult, but they arise frequently in modern applied mathematics, due to the need to model complicated structures and materials. Within certain constraints, solutions are known to exist. However, the proof of this is nonconstructive, meaning it does not provide any indication as to how the solutions can actually be obtained. Even the construction of approximate solutions is difficult, because Wiener-Hopf equations are extremely delicate, and a small change to the coefficients can drastically affect the solution, causing it to violate physical laws, and rendering it unusable.The project is motivated by the solution to a very complicated matrix Wiener-Hopf equation which appeared in a recent paper [2]. This does not appear to be amenable to earlier approaches based on making simplifying approximations in the equation itself. Instead, it was solved by an 'Implicit Quadrature Scheme' which works by representing the unknown terms using Cauchy's integral formula. Roughly, this shows that a function which possesses certain attributes is wholly determined by its values along a single path. A set of nodes is distributed along this path, and function values at the nodes are constructed using the matrix Wiener-Hopf equation. Finally, the full solution is approximated using its values at the nodes. Increasing the number of nodes improves the accuracy of the solution. No approximations are applied in the Wiener-Hopf equation itself, so there is no risk of generating an invalid solution. Our principal objective is to further explore the Implicit Quadrature Scheme, widening the range of problems to which it can be applied and optimising its performance. We will also compare it with other, earlier methods where these are available. We will then begin development of a numerical library to efficiently implement the Implicit Quadrature Scheme. This will enable physicists, engineers and other mathematicians to quickly apply the method to important practical problems which are intractable using existing approaches.References[1] B. Noble "Methods Based on the Wiener-Hopf Technique". Chelsea, 1988.[2] I. Thompson "Wave diffraction by a rigid strip in a plate modelled by Mindlin theory". Proceedings of the Royal Society A 476(2243), 2020.