A direct method for solving matrix Wiener-Hopf equations

应用数学中的许多问题都可以用维纳-霍普夫技术来解决。这些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；许多人无法接受任何已知的替代方法。特别是，Wiener-Hopf方程可以用来模拟计算机模拟很难捕捉到的突变，因为这些变化通常是在物理量可以使用平滑函数精确近似的假设下工作的。事实证明，这对于发展无损评估技术的理论至关重要，无损评估技术是通过将超声波发射到结构（如飞机机翼、潜艇船体或核电站部件）并研究散射响应来检测和确定裂纹和其他缺陷的大小。为了使维纳-霍普夫方程的应用可视化，考虑一个声波穿过圆柱形管道，朝向开口的一端传播。如果观察者正对着管道的一端站着，并移动到一边，声音将逐渐变得听不见；它不会突然消失，因为观察者移动与管道的直线。声场的连续特性与波从管道传播到室外空气时发生的衍射效应有着复杂的联系，这可以通过求解适当的Wiener-Hopf方程来精确地建模[1，第3.4节]。本课题研究同时求解耦合Wiener-Hopf方程的一种新方法。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，它们在现代应用数学中经常出现。在一定的约束条件下，已知存在解决方案。然而，这个证明是非建设性的，这意味着它没有提供任何关于如何实际获得解的指示。即使是近似解的构造也是困难的，因为维纳-霍普夫方程非常微妙，系数的微小变化会极大地影响解，导致它违反物理定律，并使其无法使用。这个项目的动机是一个非常复杂的矩阵Wiener-Hopf方程的解，它出现在最近的一篇论文[2]中。这似乎不适用于早先基于在方程本身中进行简化近似的方法。相反，它是由一个“隐式正交方案”来解决的，该方案通过使用柯西积分公式来表示未知项。粗略地说，这表明具有某些属性的函数完全由其沿单一路径的值决定。沿此路径分布一组节点，节点处的函数值使用矩阵Wiener-Hopf方程构造。最后，使用其在节点处的值来近似完整解。增加节点数量可以提高解的准确性。在Wiener-Hopf方程本身中没有近似应用，因此没有产生无效解的风险。我们的主要目标是进一步探索隐式正交方案，扩大其可以应用的问题范围并优化其性能。我们还将把它与其他早期可用的方法进行比较。然后，我们将开始开发一个数字库，以有效地实现隐式正交方案。这将使物理学家、工程师和其他数学家能够迅速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bbb . Noble“基于Wiener-Hopf技术的方法”。切尔西,1988。[2]I. Thompson，《由明德林理论模拟的板中刚性条的波衍射》。英国皇家学会学报A 476(2243), 2020。

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.