Computational Tools for Exploring Eigenvector Localization

复杂系统中的振动最好理解为振幅随时间变化的驻波的叠加。这些驻波是相关的与时间无关的微分算子的本征向量（振动模式），并且它们的对应本征值表示与这些振动模式相关的能量。更好地理解这样的特征向量可能局部化的地方，以及这种局部化发生的特征值，在具有期望的声学或电磁特性的结构的设计中具有实际意义：声音减轻户外屏障和下一代有机LED和太阳能电池是这种设计原理的例子。用于探索特征向量局部化现象的计算工具是指导设计决策的模拟的重要组成部分，但是这样的工具是非常新的并且非常少。这项资助支持的工作将提供新的技术，用于比目前可行的更广泛地探索局部化现象，允许更深入地探索更广泛类别的算子/模型，并具有基于可测量量接受/拒绝局部化声明的内置机制。该项目侧重于以下基本任务：给定一个子域，一个小的公差和一个有限的区间，找到所有的本征值/本征向量对（本征对）的运营商，其本征值是在区间内，其本征向量是本地化的子域内给定的公差。构造原始算子的复杂且紧凑的扰动，使得满足上述条件的原始算子的任何本征对将具有新算子的“回声”，即，将存在新算子的本征对，其本征值在由局部化条件确定的复平面的明确定义的“目标区域”中，并且大多数（！）不满足局部化条件的原始算子的本征对将不具有这样的回波。然后使用基于轮廓积分的特征值求解器来有效地识别特征值在目标区域内的新算子的特征对。这构成了该方法的第一阶段，并且经验证据已经强烈地表明，在该阶段中最不可能的候选者被过滤掉，并且所有可能的候选者被保留。剩下的候选者然后（廉价地）后处理，以确定它们是回声的原始算子的本征对。对这些特征对的最后一个简单检查决定了哪些特征对最终被接受。这种方法的几个方面是并行的，这一事实将被利用来进行比以前更彻底的本地化调查。这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为是值得支持的。

Vibrations in complex systems are best understood in terms of superpositions of stationary waves whose amplitudes vary in time. These stationary waves are eigenvectors (modes of vibration) of an associated time-independent differential operator, and their corresponding eigenvalues represent energies associated with these modes of vibration. A better understanding of where such eigenvectors are likely to localize, and for which eigenvalues this localization occurs, is of practical interest in the design of structures having desired acoustic or electro-magnetic properties: sound-mitigating outdoor barriers and next generation organic LEDs and solar cells are examples of this design principle in action. Computational tools for exploring eigenvector localization phenomena are an essential component of simulations guiding design decisions, but such tools are very new and very few. The work supported by this grant will provide new techniques for a much broader exploration of localization phenomena than is currently feasible, allowing for exploration much deeper into the spectrum for a broader class of operators/models, with built-in mechanisms for accepting/rejecting localization claims based on measurable quantities.This project focuses on the following fundamental task: Given a subdomain, a small tolerance and a finite interval, find all eigenvalue/eigenvector pairs (eigenpairs) of the operator whose eigenvalue is in the interval and whose eigenvector is localized in the subdomain to within the given tolerance. A complex and compact perturbation of the original operator is constructed such that any eigenpair of the original operator satisfying the conditions above will have an "echo" for the new operator, i.e. there will be an eigenpair of the new operator whose eigenvalue is in a well-defined "target region" of the complex plane determined by the localization conditions, and most(!) eigenpairs of the original operator not satisfying the localization condition will not have such an echo. A contour-integral based eigenvalue solver is then used to efficiently identify eigenpairs of the new operator whose eigenvalues are within the target region. This constitutes the first phase of the method, and empirical evidence already strongly indicates that most unlikely candidates are filtered out in this phase, and all likely candidates are maintained. The remaining candidates are then (cheaply) post-processed to determine the eigenpairs of the original operator of which they were echos. A final simple check of these eigenpairs determines which are ultimately accepted. Several aspects of this approach are embarrassingly parallel, and that fact will be exploited to conduct more thorough investigations of localization than have been previously attempted. The project involves the development of software that will be made publicly available.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.