Infinite-Dimensional Eigenvalue Problems

Lee0209931 研究者和他的同事们开发了无限维空间中本征值问题的数值方法。这项工作的核心是最近开发的混合对角化/蒙特卡罗方法，他们认为这种方法在一般低能本征值问题的计算处理方面取得了重大进展。 他们研究了与无穷维空间中低能本征值和本征向量的近似方法有关的许多未解决的数学问题。 其中，合作者严格分析了对角化/蒙特卡罗方法的准确性，研究了特征值分布和特征向量结构之间的关系，将该方法推广到非Hermitian矩阵，并改进了聚类特征值的计算处理。 除了研究基本的数学，研究人员也开始生产的公共领域图书馆的无穷维特征值问题。 特征值问题在21世纪科学和工程的许多不同分支中起着至关重要的作用。 这包括例如纳米结构和新材料的电子特性;质子和中子在重核中的相互作用;以及生物信息学中的基因相似性数据库研究。 具体来说，一个是寻求列向量，v，使得M乘以v再次与v成比例。比例常数被称为M的特征值，v是相应的特征向量。 在这个项目中，计算机科学、应用数学和物理学的计算专家致力于研究高维矩阵特征值问题。 合作者研究了与此问题相关的基本问题，并开发了一个可以应用于无限维矩阵的公共领域库。无限维的扩展对于许多应用至关重要，特别是量子物理和化学中的应用，但这个库的存在将影响许多其他领域，这些领域的特征在于高维本征值问题，并且目前难以用现有方法解决。

Lee0209931 The investigator and his colleagues develop numericalmethods for eigenvalue problems in infinite-dimensional spaces.At the heart of the effort is the recently developed hybriddiagonalization/Monte Carlo approach, which they feel has led tosignificant advances in the computational treatment of thegeneral low energy eigenvalue problem. They study the manyunresolved mathematical issues related to approximate methods forlow energy eigenvalues and eigenvectors in infinite-dimensionalspaces. Among other things, the collaborators rigorously analyzethe accuracy of the diagonalization/Monte Carlo approach, studythe relation between eigenvalue distribution and eigenvectorstructure, generalize the approach to non-Hermitian matrices, andimprove the computational treatment of clustered eigenvalues. Inaddition to studying the underlying mathematics, theinvestigators also begin production on a public domain libraryfor infinite-dimensional eigenvalue problems. Eigenvalue problems play a crucial role in many diversebranches of 21st-century science and engineering. This includesfor example the electronic properties of nanoscale structures andnew materials; the interactions of protons and neutrons in heavynuclei; and gene similarity database studies in bioinformatics.The eigenvalue problem consists of finding certain attributes ofa given square matrix, M. Specifically one is seeking columnvectors, v, such that M times v is again proportional to v. Theproportionality constant is called an eigenvalue of M and v isthe corresponding eigenvector. In this project computationalexperts in computer science, applied mathematics, and physicscollaborate to investigate large-dimensional matrix eigenvalueproblems. The collaborators study the underlying mathematicalissues related to this problem and develop a public domainlibrary that can be applied to matrices of infinite dimension.The extension to infinite dimensions is crucial for manyapplications, particularly those in quantum physics andchemistry, but the existence of this library will impact manyother areas that feature large-dimensional eigenvalue problemsand are currently intractable with existing methods.