The need for solving nonlinear optimization problems is pervasive in many fields. Particle swarm optimization, advantageous with the simple underlying implementation logic, and simultaneous perturbation stochastic approximation, which is famous for its saving in the computational power with the gradient-free attribute, are two solutions that deserve attention. Many researchers have exploited their merits in widely challenging applications. However, there is a known fact that both of them suffer from a severe drawback, non- effectively converging to the global best solution, because of the local “traps” spreading on the searching space. In this article, we propose two approaches to remedy this issue by combined their advantages. In the first algorithm, the gradient information helps optimize half of the particles at the initialization stage and then further updates the global best position. If the global best position is located in one of the local optima, the searching surface’s additional gradient estimation can help it jump out. The second algorithm expands the implementation of the gradient information to all the particles in the swarm to obtain the optimized personal best position. Both have to obey the rule created for updating the particle(s); that is, the solution found after employing the gradient information to the particle(s) has to perform more optimally. In this work, the experiments include five cases. The three previous methods with a similar theoretical basis and the two basic algorithms take participants in all five. The experimental results prove that the proposed two algorithms effectively improved the basic algorithms and even outperformed the previously designed three algorithms in some scenarios.

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Efficiency is one of the key issues in numerical simulation of large-scale problems with complex 3-D geometry. Traditional domain based methods, such as finite element methods, may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on boundary integral formulations (BIE), offers one possible solution to this issue by discretizing only the surface of the domain. However, to date, successful applications of the BEM are mostly limited to linear and continuum problems. The challenges in the extension of the BEM to nonlinear problems or problems with non-continuum boundary conditions (BC) include, but are not limited to, the lack of appropriate BIE and the difficulties in the treatment of the volume integrals that result from the nonlinear terms. In this thesis work, new approaches and techniques based on the BEM have been developed for 3-D nonlinear problems and Stokes problems with slip BC. For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the BIE. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh to evaluate volume integrals. The cubes enclosing part of the boundary are partitioned using surface panels. No volume discretization of the interior cubes is necessary. This grid is also used to accelerate volume integration. Based on this approach, accelerated BEM solvers for non-homogeneous and nonlinear problems are developed and tested. Good agreement is achieved between simulation results and analytical results. Qualitative comparison is made with current approaches. Stokes problems with slip BC are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based on the BEM combined with the precorrected-FFT technique has been proposed and various techniques have been developed to solve these problems. As the applications of the developed method, drag forces on oscillating objects immersed in an unbounded slip flow are calculated and validated with either analytic solutions or experimental results.

<p> The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, u_x]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.</p><p>The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.</p><p>The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.</p>

The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, ux]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0. The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense. The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:34:36Z No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:35:12Z (GMT) No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-06T10:35:12Z (GMT). No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-08-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study a singular partial differential problem in a bounded domain with smoth boundary. We have two main cases, one superlinear with weak singularity, and the other one sublinear with strong songularity. We use Variational Methods, such as the Ekeland Variational Principle and the Nehari Manifolds, to solve this problem, finding weak solutions and proving the multiplicity of solutions in one of the cases. / Neste trabalho estudaremos um problema diferencial parcial singular em um domínio limitado com bordo suave. Temos dois casos principais, um superlinear com singularidade fraca e um sublinear com singularidade forte. Usaremos Métodos Variacionais, como o Princípio Variacional de Ekeland e as Variedades de Nehari, para resolver este problema, encontrando soluções fracas e provando a multiplicidade das mesmas em um dos casos.