Model order reduction for stochastic dynamical systems with continuous symmetries

Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not well captured by current order reduction techniques and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices, a symmetry reduction tool, with (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the Dynamically Orthogonal (DO) equations in the second step, we obtain a novel nonlinear Symmetry-reduced Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes equations.Comment: Minor revision

Characteristic eddy decomposition of turbulence in a channel

The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions, a generaliztion of the shot-noise expansion is used to determine the characteristic eddies. In this expansion, the Fourier coefficients of the characteristic eddy cannot be obtained from the second-order statistics. Three different techniques are used to determine the phases of these coefficients. They are based on: (1) the bispectrum, (2) a spatial compactness requirement, and (3) a functional continuity argument. Results from these three techniques are found to be similar in most respects. The implications of these techniques and the shot-noise expansion are discussed. The dominant eddy is found to contribute as much as 76 percent to the turbulent kinetic energy. In both 2D and 3D, the characteristic eddies consist of an ejection region straddled by streamwise vortices that leave the wall in the very short streamwise distance of about 100 wall units

Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

2011 Summer.Includes bibliographical references.In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Lévy measure ν as the master equation for a pure-jump Lévy process. In the case ν ∈ L1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case ν ∉ L1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Lévy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Lévy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Lévy processes and fractional diffusion on bounded, non-simple geometries with volume constraints. In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain Ω(θ) ⊂ R2 with a randomly perturbed boundary, where (θ) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of Ω(θ) to a reference domain Ω in order to avoid the complications associated with solving the problems on Ω(θ). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample

Demarcation of coding and non-coding regions of DNA using linear transforms

Deoxyribonucleic Acid (DNA) strand carries genetic information in the cell. A strand of DNA consists of nitrogenous molecules called nucleotides. Nucleotides triplets, or the codons, code for amino acids. There are two distinct regions in DNA, the gene and the intergenic DNA, or the junk DNA. Two regions can be distinguished in the gene- the exons, or the regions that code for amino acid, and the introns, or the regions that do not code for amino acid. The main aim of the thesis is to study signal processing techniques that help distinguish between the regions of the exons and the introns. Previous research has shown the fact that the exons can be considered as a sequence of signal and noise, whereas introns are noise-like sequences. Fourier Transform of an exonic sequence exhibits a peak at frequency sample value k N/3 where N is the length of the FFT transform. This property is referred to as the period -3 property. Unlike exons, introns have a noise-like spectrum. The factor that determines the performance efficiency of a transform is the figure of merit, defined as the ratio of the peak value to the arithmetic mean of all the values. A comparative study was conducted for the application of the Discrete Fourier Transform and the Karhunen Loeve Transform. Though both DFT and KLT of an exon sequence produce a higher figure of merit than that for an intron sequence, it is interesting to note that the difference in the figure of merits of exons and introns was higher when the KLT was applied to the sequence than when the DFT was applied. The two transforms were also applied on entire sequences in a sliding window fashion. Finally, the two transforms were applied on a large number of sequences from a variety of organisms. A Neyman Pearson based detector was used to obtain receiver operating curves, i.e., probability of detection versus probability of false alarm. When a transform is applied as a sliding window, the values for exons and introns are taken separately. The exons and the introns served as the two hypotheses of the detector. The Neyman Pearson detector helped indicate the fact the KLT worked better on a variety of organisms than the DFT

Transform processing and coding of images Final report

Transform processing and image codin

A survey of face detection, extraction and recognition

The goal of this paper is to present a critical survey of existing literatures on human face recognition over the last 4-5 years. Interest and research activities in face recognition have increased significantly over the past few years, especially after the American airliner tragedy on September 11 in 2001. While this growth largely is driven by growing application demands, such as static matching of controlled photographs as in mug shots matching, credit card verification to surveillance video images, identification for law enforcement and authentication for banking and security system access, advances in signal analysis techniques, such as wavelets and neural networks, are also important catalysts. As the number of proposed techniques increases, survey and evaluation becomes important

Neural field models with threshold noise

The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models. Moreover, for a prescribed spatial covariance function we explore the differences in behaviour that can emerge when the underlying stationary distribution is changed from Gaussian to non-Gaussian. For travelling front solutions, in a system with exponentially decaying spatial interactions, we make use of an interface approach to calculate the instantaneous wave speed analytically as a series expansion in the noise strength. From this we find that, for weak noise, the spatially averaged speed depends only on the choice of covariance function and not on the shape of the stationary distribution. For a system with a Mexican-hat spatial connectivity we further find that noise can induce localised bump solutions, and using an interface stability argument show that there can be multiple stable solution branches