FAST TRANSFORMS BASED ON STRUCTURED MATRICES WITH APPLICATIONS TO THE FAST MULTIPOLE METHOD

The solution of many problems in engineering and science is enabled by the availability of a fast algorithm, a significant example being the fast Fourier transform, which computes the matrix-vector product for a $N \times N$ Fourier matrix in $O(N\log(N))$ time. Related fast algorithms have been devised since to evaluate matrix-vector products for other structured matrices such as matrices with Toeplitz, Hankel, Vandermonde, etc. structure. A recent fast algorithm that was developed is the fast multipole method (FMM). The original FMM evaluates all pair-wise interactions in large ensembles of $N$ particles in $O(p^2N)$ time, where $p$ is the number of terms in the truncated multipole/local expansions it uses. Analytical properties of translation operators that shift the center of a multipole or local expansion to another location and convert a multipole expansion into a local expansion are used. The original translation operators achieve the translation in $O(p^2)$ operations for a $p$ term expansion. Translation operations are among the most important and expensive steps in an FMM algorithm. The main focus of this dissertation is on developing fast accurate algorithms for the translation operators in the FMM for Coulombic potentials in two or three dimensions. We show that the matrices involved in the translation operators of the FMM for Coulombic potentials can be expressed as products of structured matrices. Some of these matrices have fast transform algorithms available, while for others we show how they can be constructed. A particular algorithm we develop is for fast computation of matrix vector products of the form $Px$, $P'x$, and $PP'x$, where $P$ is a Pascal matrix. When considering fast translation algorithms for the 3D FMM we decompose translations into an axial translation and a pair of rotations. We show how a fast axial translation can be performed. The bottleneck for achieving fast translation is presented by the lack of a fast rotation transform. A fast rotation algorithm is also important for many other applications, including quantum mechanics, geoscience, computer vision, etc, and fast rotation algorithms are being developed based on the properties of spherical harmonics. We follow an alternate path by showing that the rotation matrix $R$ can be factored in two different ways into the product of structured matrices. Both factorizations allow a fast matrix-vector product. Our algorithm efficiently computes the coefficients of spherical harmonic expansions on rotation. Numerical experiments confirm that the new $O(p\log p)$ translation operators for both the 2D and 3D FMM have the same accuracy as the original ones, achieve their asymptotic complexity for modest $p$, and significantly speed up the FMM algorithms in 2D and 3D. We hope that this thesis will also lead to promising future research in establishing fast translation for the FMM for other potentials, as well as applying the transforms in other applications such as in harmonic analysis on the sphere

Safe storage guidelines for soybeans at different temperatures and moisture contents: Poster

Poor storage capacity of soybean makes it prone to fungal spoilage and heating during storage, resulting in lower quality. Early prediction of the fungal spoilage in stored soybeans is very difficult because fungi are often too small to be seen with the naked eye. Here a new method for fungus to early detection is adopted: it is called counting fungal spores. Soybeans with moisture contents of 11.4, 12.1, 13.0, 13.9, 14.3 and 14.7%, were held at 6 temperatures 10, 15, 20, 25, 30 and 35? for180d. Samples were taken at regular intervals and the fungal spores counted. The safe storage conditions (temperature, moisture content, duration) were estimated by means of a curve fitted using the power function fitting. It can predict of soybean spoilage by fungus before there is visible damage.Poor storage capacity of soybean makes it prone to fungal spoilage and heating during storage, resulting in lower quality. Early prediction of the fungal spoilage in stored soybeans is very difficult because fungi are often too small to be seen with the naked eye. Here a new method for fungus to early detection is adopted: it is called counting fungal spores. Soybeans with moisture contents of 11.4, 12.1, 13.0, 13.9, 14.3 and 14.7%, were held at 6 temperatures 10, 15, 20, 25, 30 and 35? for180d. Samples were taken at regular intervals and the fungal spores counted. The safe storage conditions (temperature, moisture content, duration) were estimated by means of a curve fitted using the power function fitting. It can predict of soybean spoilage by fungus before there is visible damage

A marginal structural model for normal tissue complication probability

The goal of radiation therapy for cancer is to deliver prescribed radiation dose to the tumor while minimizing dose to the surrounding healthy tissues. To evaluate treatment plans, the dose distribution to healthy organs is commonly summarized as dose-volume histograms (DVHs). Normal tissue complication probability (NTCP) modelling has centered around making patient-level risk predictions with features extracted from the DVHs, but few have considered adapting a causal framework to evaluate the comparative effectiveness of alternative treatment plans. We propose causal estimands for NTCP based on deterministic and stochastic interventions, as well as propose estimators based on marginal structural models that parametrize the biologically necessary bivariable monotonicity between dose, volume, and toxicity risk. The properties of these estimators are studied through simulations, along with an illustration of their use in the context of anal canal cancer patients treated with radiotherapy

The Global Geometry of Centralized and Distributed Low-rank Matrix Recovery without Regularization

Low-rank matrix recovery is a fundamental problem in signal processing and machine learning. A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UV^T, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry---with no spurious local minima and satisfying the so-called strict saddle property---ensuring convergence to a global minimum for many local-search algorithms. Such results hold whenever the original objective function is restricted strongly convex and smooth. However, most of these results actually consider a modified cost function that includes a balancing regularizer. While useful for deriving theory, this balancing regularizer does not appear to be necessary in practice. In this work, we close this theory-practice gap by proving that the unaltered factorized non-convex problem, without the balancing regularizer, also has similar benign global geometry. Moreover, we also extend our theoretical results to the field of distributed optimization