The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate were considered. These equations suggest certain successive approximations iterative procedures for obtaining maximum likelihood estimates. The procedures, which are generalized steepest ascent (deflected gradient) procedures, contain those of Hosmer as a special case

On Quasi‐Newton methods in fast Fourier transform‐based micromechanics

This work is devoted to investigating the computational power of Quasi‐Newton methods in the context of fast Fourier transform (FFT)‐based computational micromechanics. We revisit FFT‐based Newton‐Krylov solvers as well as modern Quasi‐Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) method, one of the most powerful Quasi‐Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi‐Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT‐based context, we promote a Dong‐type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi‐)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast

Some numerical methods for inverse problems.

Tsang, Ka Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves [121]-123).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 2 --- Inverse problems and formulations --- p.6Chapter 3 --- Review of some existing methods --- p.8Chapter 4 --- Trust Region Method --- p.16Chapter 4.1 --- Some Auxiliary Tools --- p.18Chapter 4.2 --- Trust Region Algorithm --- p.23Chapter 4.3 --- Convergence of trust region method --- p.28Chapter 4.3.1 --- Notations and Assumptions --- p.28Chapter 4.3.2 --- Convergence for exact data --- p.29Chapter 4.3.3 --- Regularity For Inexact Data --- p.36Chapter 4.4 --- Experiment On Trust Region Method --- p.39Chapter 4.4.1 --- Problem Setting --- p.39Chapter 4.4.2 --- Algorithm --- p.40Chapter 4.4.3 --- Experiment Results --- p.42Chapter 4.5 --- Trust Region Conjugate Gradient Method --- p.46Chapter 4.5.1 --- Notations and Assumptions --- p.49Chapter 4.5.2 --- Convergence Properties for Exact Data --- p.52Chapter 4.5.3 --- Regularity for Inexact Data --- p.57Chapter 5 --- Parameter Identification Problems --- p.60Chapter 5.1 --- Introduction --- p.60Chapter 5.1.1 --- Computation of VJ(x) --- p.67Chapter 5.2 --- Algorithm for Parameter Identification Problems --- p.72Chapter 5.2.1 --- "Finite Element Method in Two Dimensions:Ω =[0,1] x [0,1]" --- p.75Chapter 5.3 --- Experiments on Trust Region-CG Method for Parameter Identification Problems --- p.82Chapter 5.3.1 --- One Dimension Problem --- p.82Chapter 5.3.2 --- Two Dimensions Problem --- p.95Chapter 5.4 --- Conclusion --- p.119Bibliography --- p.12

Modeling a fish population with diffusive and advective movement in a spatial environment

This dissertation has developed an individual-based, physiologically structured model for a fish population with diffusive and advective movement in a spatial environment. It incorporates spatio-temporal processes and individual processes simultaneously into the population dynamic model of a McKendrick-von Foerster type partial differential equation. Anindividualfish is physiologically structured according to age, lipid and structure (protein and carbohydrates). Fish are assumed to be immobile in their embryonic stage and the fish begin to feed and might move after the embryonic stage. Advective processes are induced by environmental heterogeneity, in which fish move toward neighboring areas with different levels of, for instance, resource density or/and chemical toxicant concentration. The population dynamic model is complicated, in that it is a mixed type partial differential equation that combines a quasi-linear hyperbolic equation in the embryonic stage and degenerate parabolic equation in the older life stage. Some mathematical aspects of the model of primary interest have been discussed. The existence of a local weak solution has been shown. By the constructive analysis used to demonstrate the existence of a local solution, a computational scheme for the mathematical model has been developed. For the individual growth model, we simply use the implicit Runge-Kutta method. For the population dynamic model of partial differential problem, we use a characteristic finite difference method in the age-time domain and a finite element method with numerical integration and upwind modification in the spatial domain. Furthermore, the numerical scheme has been proved to yield numerical approximations with optimal error estimates and produce biologically reasonable approximate solutions as well. The mathematical and computational models have been used to study a specific model of a population of rainbow trout, Oncorhynchus mykiss, in a spatial environment. We Have investigated numerically the dynamics of spatio-temporal population distribution variations as they are viewed through the fish population density, total fish biomass, total fish age, total fish lipid, total fish structure (protein) and total fish protected protein. Furthermore, the model has also been used to study the effects of a spatially distributed nonpolar narcotic chemical on a rainbow trout population. The combined effects of lethal and sublethal toxicant effects have been considered. The methodologies and conclusions in this dissertation can be extended immediately into other populations and even some community settings, such as the fish-Daphnia predator-prey model if Daphnia are assumed to be immobile

Analysis of Linkage-Friendly Genetic Algorithms

Evolutionary algorithms (EAs) are stochastic population-based algorithms inspired by the natural processes of selection, mutation, and recombination. EAs are often employed as optimum seeking techniques. A formal framework for EAs is proposed, in which evolutionary operators are viewed as mappings from parameter spaces to spaces of random functions. Formal definitions within this framework capture the distinguishing characteristics of the classes of recombination, mutation, and selection operators. EAs which use strictly invariant selection operators and order invariant representation schemes comprise the class of linkage-friendly genetic algorithms (lfGAs). Fast messy genetic algorithms (fmGAs) are lfGAs which use binary tournament selection (BTS) with thresholding, periodic filtering of a fixed number of randomly selected genes from each individual, and generalized single-point crossover. Probabilistic variants of thresholding and filtering are proposed. EAs using the probabilistic operators are generalized fmGAs (gfmGAs). A dynamical systems model of lfGAs is developed which permits prediction of expected effectiveness. BTS with probabilistic thresholding is modeled at various levels of abstraction as a Markov chain. Transitions at the most detailed level involve decisions between classes of individuals. The probability of correct decision making is related to appropriate maximal order statistics, the distributions of which are obtained. Existing filtering models are extended to include probabilistic individual lengths. Sensitivity of lfGA effectiveness to exogenous parameters limits practical applications. The lfGA parameter selection problem is formally posed as a constrained optimization problem in which the cost functional is related to expected effectiveness. Kuhn-Tucker conditions for the optimality of gfmGA parameters are derived

Glosarium Matematika

273 p.; 24 cm

High Performance Computing Based Methods for Simulation and Optimisation of Flow Problems

The thesis is concerned with the study of methods in high-performance computing for simulation and optimisation of flow problems that occur in the framework of microflows. We consider the adequate use of techniques in parallel computing by means of finite element based solvers for partial differential equations and by means of sensitivity- and adjoint-based optimisation methods. The main focus is on three-dimensional, low Reynolds number flows described by the instationary Navier-Stokes equations

Methods mathematical modeling and identification of complex processes and systems on the basis of high-performance calculations (neuro- and nanoporous feedback cyber systems, models with sparse structure data, parallel computations)

PREFACE ...9 INTRODUCTION ...15 Chapter1. High-performance methods of diagnostics and identification of the abnormal neurological state parameters caused by cognitive feedback influences of the cerebral cortex ...21 1.1 Problems of human neurological conditions ...21 1.2 Comprehensive methodology and analysis tools for the diagnosis of neurological conditions of T-objects based on the hybrid ANM model. Problems of human neurological conditions ...23 1.3 Hybrid mathematical model for the analysis of the ANM of the T-object based on feedback-connections and the effects of the neural nodes of the CC...26 1.4 Identification of AMM amplitude components. Inverse heterogeneous boundary value problem taking into account the cognitive feedback influences of the neuronodes of the CC ...32 1.5 Initial-boundary value problems accompanying algorithms for identifying parameters in the ANM ...35 1.6 Statement and methodology for the ANM conjugate boundary value problem solving ...36 1.7 Statement and methodology for solving conjugate initial-boundary value problems of functional identification of the ANM ...37 1.8 Expressions for gradient components and regularization expressions ...39 1.9 Modeling and identification of parameters of complex multicomponent non-biofeedback systems on multicore computers ...42 Chapter 2. High-performance methods of modeling and identification of feedback influences of competitive adsorption of gaseous air pollutants at micro- and macro-levels in nanoporous systems ...50 2.1. Analysis of research state ...50 2.2 Experimental setap ...52 2.3 Experimental results: Gaseous benzene and hexane competitive adsorption curves ...52 2.4 A mathematical model of competitive adsorption and competitive diffusion in microporous solids ...54 2.5 Numerical simulation and analysis: Competitive diffusion coefficients. Concentration profiles in inter- and intracrystallite spaces ...62 2.6 Iterative gradient method of the identification of competitive diffusion coefficients ...65 2.7 The linearization schema of the nonlinear competitive adsorption model. System of linearized problems and construction of solutions ...69 Chapter 3. High computational methods and simulation technology nanoporous systems with feedback adsorption for gas purification ...76 3.1 Nonlinear mathematical model of nonisothermal adsorption and desorption based on the generalized Langmuir adsorption equilibrium equation ...77 3.2 The methodology for constructing analytical solution systems to heterogeneous adsorption / desorption problems ...81 3.3 Computer simulation. Analysis of the distributions of the adsorbent concentration in the gas phase and nanopores of zeolite and temperatures ...86 Chapter 4. High-performance algorithms for solving systems of nonlinear equations on supercomputers with parallel organization of computations ...92 4.1 Layered parallel computing model ...93 4.2 Parallel algorithms for solving SNE with a sparse data structure ...97 4.3 Parallel algorithms for solving systems of linear equations with a sparse matrix ...99 4.4 Hybrid algorithms for solving linear systems with sparse matrices of irregular structure based on LLT-decomposition of block-diagonal matrices with framing .. 125 4.5 Experimental study of parallel algorithms ...131 Chapter 5. The methods of integral transformations for creation of hybrid ANM-models ...137 5.1. Finite integral Fourier transformation with spectral parameter for homogeneous media ...137 5.2 Finite hybrid integral Fourier transformation for bounded heterogeneous ncomponent media ...147 5.3 Integral Fourier transformation for semi-bounded heterogeneous n – component media ...169 Conclusions ...187 References ...18