Advancements in Algorithms for Approximations of Rank Structured Matrices

Many problems in mathematical physics and engineering involve solving linear systems Ax = b which are highly structured. These structured matrices, which typically arise from discretizations of partial differential or integral equations, can be represented compactly through specific algebraic representations. Two such algebraic representations which fall into this category are the fast multipole method (FMM), originally introduced by Greengard and Rokhlin, and Hierarchically Semi-Separable (HSS) . These representations, which exploit the low-rank structure of off-diagonal blocks, have enabled fast solvers (linear time in certain scenarios) and are commonly used practice today. In this thesis, we provide new advancements which further enhance the performance of these algorithms. The contributions of this thesis are twofold: (i) we present a new fast memory efficient construction algorithm for Hierarchically Semi-Separable (HSS) representations, and (ii) we present a fast matrix-matrix multiply for the fast multipole method (FMM).The HSS representation takes advantage of the fact that off-diagonal blocks are known to have low rank in order to yield fast solvers. The memory consumption of the HSS representation itself is O(n) if the rank of the off-diagonal blocks is small. If the user is not required to store the matrix A, but instead only provides a functional interface in order to access the elements of the matrix, it is worthwhile to ask for the algorithm which computes the HSS representation to be memory efficient as well. Previous algorithms, have shown the HSS representation can be computed in O(n^2) flops. Randomized algorithms also exist. However, the memory requirements of these algorithms can be excessive, requiring as much as O(n^2) peak workspace memory. We deal with this issue and present an algorithm that requires O(n^{1.5}) peak workspace memory in the worst case, while still requiring only O(n^2) flops. The HSS Representation assumes off-diagonal blocks which have low rank, but in practice there are many cases for which this criteria is not satisfied, and in fact can be as much as O(√n). For this reason, there is much interest in FMM, as it relaxes this requirement, only demanding that off-diagonal blocks corresponding to well-separated clusters have low rank. However, the structure of the FMM inverse is not known. To better understand this problem, we consider the problem of computing a 1D FMM representation of the matrix-matrix product of two 1D FMM matrices. We show that the product of two standard (3 pt) 1D FMM matrices possesses a slightly modified 5 pt 1D FMM structure, and we provide a linear time (O(n) flop) algorithm for computing this product. Further, this work suggests that the inverse of an FMM matrix is not itself FMM

Nucleated dewetting in supported ultra-thin liquid films with hydrodynamic slip

This study reveals the influence of the surface energy and solid/liquid boundary condition on the breakup mechanism of dewetting ultra-thin polymer films. Using silane self-assembled monolayers, SiO$_2$ substrates are rendered hydrophobic and provide a strong slip rather than a no-slip solid/liquid boundary condition. On undergoing these changes, the thin-film breakup morphology changes dramatically -- from a spinodal mechanism to a breakup which is governed by nucleation and growth. The experiments reveal a dependence of the hole density on film thickness and temperature. The combination of lowered surface energy and hydrodynamic slip brings the studied system closer to the conditions encountered in bursting unsupported films. As for unsupported polymer films, a critical nucleus size is inferred from a free energy model. This critical nucleus size is supported by the film breakup observed in the experiments using high speed \emph{in situ} atomic force microscopy.Comment: 8 pages, 9 figures, including supplementary materia

Sampling inner experience in individuals with anxiety

The current study examines the inner experience of five individuals who report symptoms of anxiety and three control individuals. Descriptive Experience Sampling (DES) is the method that was used to examine inner (subjective) experience. Idiographic result chapters have been written for all eight of the participants involved in this study. These results include a description of an anxious participant who feels (emotionally) her way through life, another who is nearly always experiencing an acute awareness of sensory aspects of his environment or nothing at all, and another who solves problems through rapid fire reverse-order mental images. These results are followed by between participant nomothetic comparisons. Among the results it was found that anxious individuals are more likely than controls to be dominated by the experience of sensory awareness and experience a higher frequency of unsymbolized thought. The current study supports many of the findings of two previous DES studies on anxiety, both of which are discussed

An examination of inner experience: Anxiety

Descriptive Experience Sampling (DES) is used to examine the inner experience of seven individuals who have been diagnosed with at least one anxiety disorder and four control individuals. Idiographic results for each of the 11 participants are provided, including a description of frequent and rare/unique experiences of each participant. These results are followed by between participant nomothetic comparisons. Among the results, it was found that anxious participants experienced more indefinite figure-ground and concrete experiences when compared to controls. Anxious participants are also more likely than controls to engage in negative valence self-evaluations and rate moments as being anxious. There is also some evidence to support the notion that, overall, anxious and depressive symptoms decrease over the course of sampling regardless of group affiliation; Following the results, implications of findings from this study and future recommendations are discussed

Police Interaction Research Project: A Study of the Interactions that Occur Between the Police and Survivors Following a Sexual Assault

This study is intended to address problems that may exist for survivors and the NYPD. It will not only help the survivor struggling with the trauma of sexual assault, but may encourage more victims to utilize the resources of NYC's law enforcement community. If we want more rape victims to report these crimes to the authorities we need to understand more about what enhances or discourages this process. A better understanding of this process will enable us to develop a more collaborative approach to dealing with the issue of rape in New York City