Optimized Schwarz methods for time-harmonic wave problems in resonating cavities

时谐波问题的数值模拟在医学成像等关键技术的发展中具有重要意义（例如，在快速中风检测中的应用），粒子加速器（例如，在癌症治疗中的应用），或光子学（例如在高速电信中的应用），大型结构的数值处理（相对于所考虑的波长）越来越被需要，特别是在粒子加速器领域。为了给所考虑的粒子提供所需的加速，许多腔通常被组织成长链，从而产生如此大的结构。除了腔的加速模式之外，必须小心地研究高阶寄生模式，因为它们可能危及粒子束的稳定性，从而促使使用数值计算。然而，当考虑大型结构时，求解时谐波问题所需的计算工作量变得非常大，因此需要使用超级计算设施以及利用计算资源的数值技术。不幸的是，由于其数学性质，只有少数技术可用于有效地求解大规模时谐波问题，而大规模时间谐波问题的处理仍然是一项困难的（如果不是不可能的话）任务。显然，所有这些技术都有优点和缺点。就优化的施瓦茨（OS）方法而言，该方法已成功应用于模拟天线阵列、光子波导或重建医学图像，可以表明，当处理空腔结构时，即不能忽略反向传播波的影响的结构时，它们的性能下降。如已经提到的，在粒子加速器中发现了这样的空腔结构，而且在其他关键技术，如激光器或量子电动力学器件中也是如此。本研究计划旨在发展一种精确可靠的数值方法来求解腔结构中的大尺度时间谐波问题，通过减轻OS方法在处理反向传播波时表现出的性能下降。

Numerical simulations of time-harmonic wave problems are nowadays of paramount importance in the development of many key technologies, such as medical imaging (e.g. with applications in rapid stroke detection), particle accelerators (e.g. with applications in cancer treatment), or photonics (e.g. with applications in high-speed telecommunication), just to cite a few.The numerical treatment of large structures (with respect to the considered wavelength) is more and more required, especially in the field of particle accelerators.Indeed, in this case, many cavities are typically organized in long chains, in order to provide the required acceleration to the considered particles, creating thus large structures.In addition to the accelerating mode of the cavities, higher-order parasitic modes must be studied with care, as they might jeopardize the stability of the particle beam, motivating thus the use of numerical computations.However, when considering large structures, the computational effort needed for solving time-harmonic wave problems becomes very large, requiring thus the use of super-computing facilities together with numerical techniques taking advantage of the computational resources.Unfortunately, due to their mathematical nature, only a few techniques are available for efficiently solving large-scale time-harmonic wave problems, and the treatment of large-scale time-harmonic wave problems remains a difficult, if not impossible, task.Obviously, all these techniques have advantages and drawbacks.In the case of optimized Schwarz (OS) methods, which were successfully applied for simulating antenna arrays, photonic waveguides or for reconstructing medical images, it can be shown that their performance drops when treating cavity structures, i.e. structures where the effect of back-propagating waves cannot be neglected.Such cavity structures are found, as already mentioned, in particle accelerators, but also in other key technologies such as lasers or quantum electrodynamic devices.This research proposal aims at developing an accurate and reliable numerical method for solving large-scale time-harmonic wave problems in cavity structures, by alleviating the performance drop exhibited by OS methods when treating back-propagating waves.