We consider the constrained convex minimization problem min x2C f(x) we will present gradient projection method which generates a sequence fxkg according to the formula xk+1 = PC(xk

The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.

We consider the constrained convex minimization problem: min_x∈C f(x) we will present gradient projection method which generates a sequence x^k according to the formula x^(k+1) = P_c(x^k − £\_k∇f(x^k)), k= 0, 1, ¡P ¡P ¡P , our ideal is rewritten the formula as a xed point algorithm: x^(k+1) = T_(£\k)x^k, k = 0, 1, ¡P ¡P ¡P is used to solve the minimization problem. In this paper, we present the gradient projection method(GPM) and different choices of the stepsize to discuss the convergence of gradient projection method which converge to a solution of the concerned problem.

Orientador: Nancy Lopes Garcia / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T16:45:43Z (GMT). No. of bitstreams: 1 Goto_DanielaBentoFonsechi_M.pdf: 3513260 bytes, checksum: ff6f9e35005ad9015007d1f51ee722c1 (MD5) Previous issue date: 2008 / Resumo: O objetivo desta dissertação é estudar estimação de máxima verossimilhança para processos de nascimento puro espacial para dois diferentes tipos de amostragem: a) quando há observação permanente em um intervalo [0, T]; b) quando o processo é observado após um tempo T fixo. No caso b) não se conhece o tempo de nascimento dos pontos, somente sua localização (dados faltantes). A função de verossimilhança pode ser escrita para o processo de nascimento puro não homogêneo em um conjunto compacto através do método da projeção descrito por Garcia and Kurtz (2008), como projeção da função de verossimilhança. A verossimilhança projetada pode ser interpretada como uma esperança e métodos de Monte Carlo podem ser utilizados para estimar os parâmetros. Resultados sobre convergência quase-certa e em distribuição são obtidos para a aproximação do estimador de máxima verossimilhança. Estudos de simulação mostram que as aproximações são adequadas. / Abstract: The goal of this work is to study the maximum likelihood estimation of a spatial pure birth process under two different sampling schemes: a) permanent observation in a fixed time interval [0, T]; b) observation of the process only after a fixed time T. Under scheme b) we don't know the birth times, we have a problem of missing variables. We can write the likelihood function for the nonhomogeneous pure birth process on a compact set through the method of projection described by Garcia and Kurtz (2008), as the projection of the likelihood function. The fact that the projected likelihood can be interpreted as an expectation suggests that Monte Carlo methods can be used to compute estimators. Results of convergence almost surely and in distribution are obtained for the aproximants to the maximum likelihood estimator. Simulation studies show that the approximants are appropriate. / Mestrado / Inferencia em Processos Estocasticos / Mestre em Estatística

The aim of this project is to study the viscous Burgers' equation for the case where the viscosity is constant, but also when it contains a jump in viscosity. In the first case where the viscosity is constant, Burgers' is simply solved on a singular domain. For the case with jump in viscosity, Burgers' is solved on multiple domains with different viscosity. The different domains are then connected by applying inner boundary conditions at an interface in order to produce a singular solution. The inner boundary conditions are imposed using three different methods; simultaneous approximation term (SAT), projection and hybrid method, where the hybrid method is a combination of both the SAT and projection method. These methods are used in combination with a stable and high-order accurate summation by parts (SBP) finite difference approximation in MATLAB. The three methods are then compared to each other with respect to the least square error and the corresponding convergence rate to determine which method is the most preferable to use. The methods resulting in the highest convergence rates are the projection and the hybrid methods. These methods manage to live up to the expected convergence rates for all operators with different orders of accuracy and are therefore both good methods to use. However, the best method to use is the projection method since it is much simpler to implement than the two other methods but still achieves just as good convergence rates as the hybrid method.

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We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L2 inner product. We examine this L2 projection and an H1 projection induced by the H1 semi-inner product. The H1 projection has an advantage over the L2 projection in that it retains tangential Dirichlet boundary conditions for the flow. Both the H1 and L2 projections are solutions to saddle point problems that are efficiently solved by a preconditioned Uzawa algorithm. / Ph. D.

In this study we investigate the propagation of uncertainties in the input forces through a mechanical system. The system of interest was a wheel loader, but the methodology developed can be applied to any multibody systems. The modeling technique implemented focused on efficiently modeling stochastic systems for which the equations of motion are not available. The analysis targeted the reaction forces in joints of interest. The modeling approach developed in this thesis builds a foundation for determining the uncertainties in a Caterpillar 980G II wheel loader. The study begins with constructing a simple multibody deterministic system. This simple mechanism is modeled using differential algebraic equations in Matlab. Next, the model is compared with the CAD model constructed in ProMechanica. The stochastic model of the simple mechanism is then developed using a Monte Carlo approach and a Linear/Quadratic transformation method. The Collocation Method was developed for the simple case study for both Matlab and ProMechanica models. Thus, after the Collocation Method was validated on the simple case study, the method was applied to the full 980G II wheel loader in the CAD model in ProMechanica. This study developed and implemented an efficient computational method to propagate computational method to propagate uncertainties through "black-box" models of mechanical systems. The method was also proved to be reliable and easier to implement than traditional methods. / Master of Science

This thesis presents a GPU accelerated implementation of a high order splitting scheme with a spectral element discretization for the incompressible Navier Stokes (INS) equations. While others have implemented this scheme on clusters of processors using the Nek5000 code, to my knowledge this thesis is the first to explore its performance on the GPU. This work implements several of the Nek5000 algorithms using OpenCL kernels that efficiently utilize the GPU memory architecture, and achieve massively parallel on chip computations. These rapid computations have the potential to significantly enhance computational fluid dynamics (CFD) simulations that arise in areas such as weather modeling or aircraft design procedures. I present convergence results for several test cases including channel, shear, Kovasznay, and lid-driven cavity flow problems, which achieve the proven convergence results.

Micromagnetism is the study of magnetic materials on the microscopic length scale (of nano to micrometers), this scale does not take quantum mechanical effects into account, but is small enough to neglect certain macroscopic effects of magnetism in a material. The Landau-Lifshitz-Gilbert (LLG) equation is used within micromagnetism to determine the time evolution of the magnetisation vector field in a ferromagnetic solid. It is a partial differential equation with high non linearity, which makes it very difficult so solve analytically. Thus numerical methods have been developed for approximating the solution using computers. In this report we compare the performance of three different numerical methods for the LLG equation, the implicit midpoint method (IMP), the midpoint with extrapolation method (MPE), and the Gauss-Seidel Projection method (GSPM). It was found that all methods have convergence rates as expected; second order for IMP and MPE, and first order for GSPM. Energy conserving properties of the schemes were analysed and neither MPE or GSPM conserve energy. The computational time required for each method was determined to be very large for the IMP method in comparison to the other two. Suggestions for different areas of use for each method are provided.