Density ridge manifold traversal

The density ridge framework for estimating principal curves and surfaces has in a number of recent works been shown to capture manifold structure in data in an intuitive and effective manner. However, to date there exists no efficient way to traverse these manifolds as defined by density ridges. This is unfortunate, as manifold traversal is an important problem for example for shape estimation in medical imaging, or in general for being able to characterize and understand state transitions or local variability over the data manifold. In this paper, we remedy this situation by introducing a novel manifold traversal algorithm based on geodesics within the density ridge approach. The traversal is executed in a subspace capturing the intrinsic dimensionality of the data using dimensionality reduction techniques such as principal component analysis or kernel entropy component analysis. A mapping back to the ambient space is obtained by training a neural network. We compare against maximum mean discrepancy traversal, a recent approach, and obtain promising results

Schrodinger wave-mechanics and large scale structure

In recent years various authors have developed a new numerical approach to cosmological simulations that formulates the equations describing large scale structure (LSS) formation within a quantum mechanical framework. This method couples the Schrodinger and Poisson equations. Previously, work has evolved mainly along two different strands of thought: (1) solving the full system of equations as Widrow & Kaiser attempted, (2) as an approximation to the full set of equations (the Free Particle Approximation developed by Coles, Spencer and Short). It has been suggested that this approach can be considered in two ways: (1) as a purely classical system that includes more physics than just gravity, or (2) as the representation of a dark matter field, perhaps an Axion field, where the de Broglie wavelength of the particles is large. In the quasi-linear regime, the Free Particle Approximation (FPA) is amenable to exact solution via standard techniques from the quantum mechanics literature. However, this method breaks down in the fully non-linear regime when shell crossing occurs (confer the Zel'dovich approximation). The first eighteen months of my PhD involved investigating the performance of illustrative 1-D and 3-D ``toy" models, as well as a test against the 3-D code Hydra. Much of this work is a reproduction of the work of Short, and I was able to verify and confirm his results. As an extension to his work I introduced a way of calculating the velocity via the probability current rather than using a phase unwrapping technique. Using the probability current deals directly with the wavefunction and provides a faster method of calculation in three dimensions. After working on the FPA I went on to develop a cosmological code that did not approximate the Schrodinger-Poisson system. The final code considered the full Schrodinger equation with the inclusion of a self-consistent gravitational potential via the Poisson equation. This method follows on from Widrow & Kaiser but extends their method from 2D to 3D, it includes periodic boundary conditions, and cosmological expansion. Widrow & Kaiser provided expansion via a change of variables in their Schrodinger equation; however, this was specific only to the Einstein-de Sitter model. In this thesis I provide a generalization of that approach which works for any flat universe that obeys the Robertson-Walker metric. In this thesis I aim to provide a comprehensive review of the FPA and of the Widrow-Kaiser method. I hope this work serves as an easy first point of contact to the wave-mechanical approach to LSS and that this work also serves as a solid reference point for all future research in this new field

Fast Solvers and Simulation Data Compression Algorithms for Granular Media and Complex Fluid Flows

Granular and particulate flows are common forms of materials used in various physical and industrial applications. For instance, we model the soil as a collection of rigid particles with frictional contact in soil-vehicle simulations, and we simulate bacterial colonies as active rigid particles immersed in a viscous fluid. Due to the complex interactions in-between the particles and/or the particles and the fluid, numerical simulations are often the only way to study these systems apart from typically expensive physical experiments. A standard method for simulating these systems is to apply simple physical laws to each of the particles using the discrete element method (DEM) and evolve the resulting multibody system in time. However, due to the sheer number of particles in even a moderate-scale real-world system, it quickly becomes expensive to timestep these systems unless we exploit fast algorithms and high-performance computing techniques. For instance, a big challenge in granular media simulations is resolving contact between the constituent particles. We use a cone-complementarity formulation of frictional contact to resolve collisions; this approach leads to a quadratic optimization problem whose solution gives us the contact forces between particles at each timestep. In this thesis, we introduce strategies for solving these optimization problems on distributed memory machines. In particular, by imposing a locality-preserving partitioning of the rigid bodies among the computing nodes, we minimize the communication cost and construct a scalable framework for collision detecting and resolution that can be easily scaled to handle hundreds of millions of particles. For robust and efficient simulation of axisymmetric particles in viscous fluids, we introduce a fast method for solving Stokes boundary integral equations (BIEs) on surfaces of revolution. By first transforming the Stokes integral kernels into a rotationally invariant form and then decoupling the transformed kernels using the Fourier series, we reduce the dimensionality of the problem. This approach improves the time complexity of the BIE solvers by an order of magnitude; additionally we can use high-order one-dimensional singular quadrature schemes to construct highly accurate solvers. Finally, coupling our solver framework with the fast multipole method, we construct a fast solver for simulating Stokes flow past a system of axisymmetric bodies. Combining this with our complementarity collision resolution framework, we have the potential to simulate dense particulate suspensions. Physics-based simulations similar to those described above generate large amounts of output data, often in the hundreds of gigabytes range. We introduce data compression techniques based on the tensor-train decomposition for DEM simulation outputs and demonstrate the high compressibility of these large datasets. This allows us to keep a reduced representation of simulated data for post-processing or use in learning tasks. Finally, due to the high cost of physics-based models and limited computational budget, we can typically run only a limited number of simulations when exploring a high-dimensional parameter space. Formally, this can be posed as a matrix/tensor completion problem, and Bayesian inference coupled with a linear factorization model is often used in this setup. We use Markov chain Monte Carlo (MCMC) methods to sample from the unnormalized posteriors in these inference problems. In this thesis, we explore the properties of the posterior in a simple low-rank matrix factorization setup and develop strategies to break its symmetries. This leads to better quality MCMC samples and lowers the reconstruction errors with various synthetic and real-world datasets.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169614/1/saibalde_1.pd