BLIND SOURCE SEPARATION USING MAXIMUM ENTROPY PDF ESTIMATION BASED ON FRACTIONAL MOMENTS

Abstract. Recovering a set of independent sources which are linearly mixed is the main task of the blind source separation. Utilizing different methods such as infomax principle, mutual information and maximum likelihood leads to simple iterative procedures such as natural gradient algorithms. These algorithms depend on a nonlinear function (known as score or activation function) of source distributions. Since there is no prior knowledge of source distributions, the optimality of the algorithms is based on the choice of a suitable parametric density model. In this paper, we propose an adaptive optimal score function based on the fractional moments of the sources. In order to obtain a parametric model for the source distributions, we use a few sampled fractional moments to construct the maximum entropy probability density function (PDF) estimation . By applying an optimization method we can obtain the optimal fractional moments that best fit the source distributions. Using the fractional moments (FM) instead of the integer moments causes the maximum entropy estimated PDF to converge to the true PDF much faster . The simulation results show that unlike the most previous proposed models for the nonlinear score function, which are limited to some sorts of source families such as sub-gaussian and super-gaussian or some forms of source distribution models such as generalized gaussian distribution, our new model achieves better results for every source signal without any prior assumption for its randomness behavior

Passive scalar intermittency in random flows

This thesis concentrates on reconstructing the complete probability density function (PDF) for a passive scalar governed by a random advection-diffusion equation using a variety of mathematical tools, primarily from partial differential equations, perturbation theory, numerical analysis and statistics. First we present a one-dimensional model which is essentially a random translation of pure heat equation. For some deterministic initial data, the ensuing scalar PDF and its statistical moments can be explicitly calculated. We use this model as a testbed for validating a numerical reconstruction procedure for the PDF via orthogonal polynomial expansion. In this model, the Péclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF which affects the effectiveness of the series expansion, from only one algebraic singularity at unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet). Next, we study the more complicated, two-dimensional model in which the underlying flow is a random linear shear in one dimension. For planar, Gaussian random initial data, we identify the scalar PDF as an integral representing a conditional mixing of Gaussian probability measures averaged over all realizations of a single random variable, namely, the renormalized L2-norm of standard Wiener process. Rigorous asymptotic analyses and solid numerical simulation are performed to the integral formulation to study the evolution and the parametric dependence of the scalar PDF. During these analyses, we discover a transient, nonmonotonic "breathing" phenomena that is related to the multiple spatial scales in the initial random field. Lastly, some preliminary analytical and numerical results are presented to explore the potential of applying the reconstruction methodology to more general, physically relevent models, such as a rotating, viscous, wind-driven shallow water equation

Efficient Computational Methods for Structural Reliability and Global Sensitivity Analyses

Uncertainty analysis of a system response is an important part of engineering probabilistic analysis. Uncertainty analysis includes: (a) to evaluate moments of the response; (b) to evaluate reliability analysis of the system; (c) to assess the complete probability distribution of the response; (d) to conduct the parametric sensitivity analysis of the output. The actual model of system response is usually a high-dimensional function of input variables. Although Monte Carlo simulation is a quite general approach for this purpose, it may require an inordinate amount of resources to achieve an acceptable level of accuracy. Development of a computationally efficient method, hence, is of great importance. First of all, the study proposed a moment method for uncertainty quantification of structural systems. However, a key departure is the use of fractional moment of response function, as opposed to integer moment used so far in literature. The advantage of using fractional moment over integer moment was illustrated from the relation of one fractional moment with a couple of integer moments. With a small number of samples to compute the fractional moments, a system output distribution was estimated with the principle of maximum entropy (MaxEnt) in conjunction with the constraints specified in terms of fractional moments. Compared to the classical MaxEnt, a novel feature of the proposed method is that fractional exponent of the MaxEnt distribution is determined through the entropy maximization process, instead of assigned by an analyst in prior. To further minimize the computational cost of the simulation-based entropy method, a multiplicative dimensional reduction method (M-DRM) was proposed to compute the fractional (integer) moments of a generic function with multiple input variables. The M-DRM can accurately approximate a high-dimensional function as the product of a series low-dimensional functions. Together with the principle of maximum entropy, a novel computational approach was proposed to assess the complete probability distribution of a system output. Accuracy and efficiency of the proposed method for structural reliability analysis were verified by crude Monte Carlo simulation of several examples. Application of M-DRM was further extended to the variance-based global sensitivity analysis of a system. Compared to the local sensitivity analysis, the variance-based sensitivity index can provide significance information about an input random variable. Since each component variance is defined as a conditional expectation with respect to the system model function, the separable nature of the M-DRM approximation can simplify the high-dimension integrations in sensitivity analysis. Several examples were presented to illustrate the numerical accuracy and efficiency of the proposed method in comparison to the Monte Carlo simulation method. The last contribution of the proposed study is the development of a computationally efficient method for polynomial chaos expansion (PCE) of a system's response. This PCE model can be later used uncertainty analysis. However, evaluation of coefficients of a PCE meta-model is computational demanding task due to the involved high-dimensional integrations. With the proposed M-DRM, the involved computational cost can be remarkably reduced compared to the classical methods in literature (simulation method or tensor Gauss quadrature method). Accuracy and efficiency of the proposed method for polynomial chaos expansion were verified by considering several practical examples.1 yea

Hierarchical modelling of multiphase flows using fully resolved fixed mesh and PDF approaches

Fully–resolved simulations of multiphase flow phenomena and in particular particulate flow simulations are computationally expensive and are only feasible on massively parallel computer clusters. A 3D SIMPLE type pressure correction algorithm is implemented and extensively tested and parallelized to exploit the power of massively parallel computing clusters currently available. Domain decomposition and communication schemes applicable to a general unstructured or structured multi–block CFD codes are discussed and algorithms are proposed, implemented and tested. Several high–performance linear solvers and a multi–grid strategy for the current framework are implemented and the best types of solvers are identified. A 2D CFD code is developed by the author to test several possible fixed–mesh strategies. Variations of immersed boundary (IB) and fictitious domain (FD) methods are implemented and compared. FD methods are identified to have better properties especially if other transport phenomena are also considered. Therefore an FD method is adapted by the author for the SIMPLE type flow solvers and is extended to heat transfer problems. The method is extensively tested for the simulation of flow around stationary in addition to freely moving particles and forced motion where both natural and forced convection are considered. The method is used to study the flow and heat transfer around a stationary cylinder and a new high resolution correlation is devised for the estimation of the local Nusselt number curves. Free fall problem for a single circular cylinder is considered and the effects of internal heat generation and also long term behavior of single cold particle subject to natural convection are also studied in detail. A particle collision strategy is also adapted and tested for the particle–particle collision problems. The FD algorithm is extended to the 3D framework and the flow around single stationary sphere and also free fall of a single sphere are used to validate the FD algorithm in 3D. A unique polydispersed fluid-particle turbulent modelling process is reviewed and the closure problem for this framework is studied in detail. Two methods for the closure of the non–integer moments which results from the polydispersity of the particles are proposed namely PDF reconstruction using Laguerre polynomials and a unique direct method named Direct Fractional Method of Moments (DFMM). The latter is derived using the results of the fractional calculus by writing an equation for the fractional derivatives of the moment generating function. The proposed methods are tested on a number of problems consisting of analytical, experimental and DNS simulations to asses their validity and viability which shows that both methods provide accurate results with DFMM having more desirable properties.EThOS - Electronic Theses Online ServiceGBUnited Kingdo