Sharif Digital Repository

The controllability problem may be formulated roughly as follows. Consider an evolution system (either described in terms of partial or ordinary differential equations) on which we are allowed to act by means of a suitable choice of the control (the right-hand side of the system, the boundary conditions, etc.). Given a time interval 0 < t < T, and initial and final states, the goal is to determine whether there exists a control driving the given initial data to the given final ones in time T. Now, consider the simplest parabolic equation, namely heat equation and suppose that one could act by appropraite controls on this system. The null controllability problem which is one of the very... 

In this thesis, issues related to the minimization of a class of energy functions defined over the space of vector valued functions, under a linear differential constraint are discussed. We collect a series of lower semi-continuity results with respect to the weak convergence obtained by using the tools of Young measures int the case of growth condition larger than one and using generalized Young measures in linear growth case.The general trend in these results is to obtain a good representation of the energy limit by a quantitative study of the oscillation and concentration effects in a weakly convergent sequence  

In this thesis, we first introduce the required biological preparations and popular models for modeling single neuron, synapse and cable. Then by introduction of limit cycle oscillators and the necessary prerequisites, investigations are limited to systems involving weakly coupled oscillators. As two examples of such models, famous Kuramoto and Wilson- Cowan models are described. In the following, we introduce some methods for reduction dimension of weakly coupled oscillators and finally we apply one of the expressed methods on the dynamics of cortical network  

There are some situations where we would like to solve a partial differential equation (PDE) in a domain whose boundary is not known a priori; such a problem is called free boundary problem and the boundary is called free boundary. For this kind of problems, aside from standard boundary conditions, an extra condition is imposed at the free boundary and the goal would be finding the free boundary in addition to finding a solution for PDE. One of the classical examples of free boundary problems is the Stefan problem (melting of ice) in which ice and water temperatures are determined from the heat equation and we would be interested in finding the boundary between ice and water. Another famous... 

The aim of this thesis is describes the method a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy... 

In this thesis, we study image restoration problems, which can be modeled as inverse problems. Our main focus is on inverse problems with Poisson noise; which are useful in many problems like positron emission tomography, fluorescence microscopy, and astronomy imaging. As a popular method in the literature, we use statistic modeling of inverse problem with Poisson noise, in the MAP-estimation framework. Then we introduce a semi-implicit minimization method, FB-EM-TV, that involves two alternate steps, including an EM step and a weighted ROF problem. Then we study well-posedness, existence and stability of the solution. This method can be interpreted as a forward-backward splitting strategy,... 

The In this thesis, we focus are attention on elliptic boundry value problems in domain with nonsmooth boundaries and problems with mixed boundry conditions. Indeed most of the available mathematical theories about elliptic boundry value problems deal with domains with very smooth boundaries;few of them deal with mixed boundry conditions.However, the majority of the elliptic boundry value problems which arise in practice are naturally posed in domains whose geometry is simlpe but not smooth. These domains are very often three-dimensional polyhedra. For the purpose of solving them numerically these problems are usually reduced to two-dimensional domains. Thus the domains are plane polygons... 

Through this thesis the Homogenization Theory for composite materials is studied assuming that the distribution of heterogeneities is periodic. In this theory two scales characterize the problem: microscopic and macroscopic scale. The first method that is used to solve the problem is the classical Asymptotic Expansion method where an error estimate is presented for the solution. The second method which was introduced by Luc Tartar for the first time is Oscillating Test Functions method. In the next chapter after introducing the concept of two-scale convergence, the Two-Scale Convergence method has been introduced. At last the unfolding
periodic method, which is based on the concept of... 

Level set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the Eulerian approach). Also, the level set method makes it very easy to follow shapes that change topology, for example when a shape splits in two, develops holes, or the reverse of these operations. All these make the level set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water  

Shape optimization can be viewed as a part of the important branch of computational mechanics called structural optimization. In structural optimization problems one tries to set up some data of the mathematical model that describe the behavior of a structure in order to find a situation in which the structure exhibits a priori given properties. Nowadays shape optimization represents a vast scientific discipline involving all problems in which the geometry (in a broad sense) is subject to optimization. The problem has to be well posed from the mechanical point of view, requiring a good understanding of the physical background. Then one has to find an appropriate mathematical model that can... 

In¬¬ this thesis we review adomian decomposition method in solving differential equations.First described the technique and then we will introduce two ways to obtain adomian polynomials.In the next section to ompare two methods of adomian decomposition and Taylor series with two examples, we will solve an example and compared decomposition method with the Picard method and finally using Cauchy-Kowalewski Theorm Convergence of answer Series shown and it will present a rate of convergence  

In this Thesis, we investigate the Modeling of Oscillator Neural Networks. Let Oscillators are coupled to each other Weakly. agood way to use Phase Model to describe each Oscillator. Then provide specific Examples to see the nesseceryConditions forExsistence and Stability of Synchrony and desyncrony  

We study synaptically coupled neuronal networks to identify the role of coupling delays in network synchronized behavior. Identifying the effect of delay is important for understanding information processing functions in the brain and Many studies have been done on the effects of delays on networks where the synapses are exclusively excitatory or inhibitory, but few address networks with both. Thus we focus on this case. In this thesis, we consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory are coupled with time-delayed synapses. Using singular perturbation analysis, we will study the dynamic behavior of the network... 

In this thesis, unconstrained convex quadratic optimization problems subject to parameter perturbations are considered. A robustification approach is proposed and analyzed which reduces the sensitivity of the optimal function value with respect to the parameter. Since reducing the sensitivity and maintaining a small objective value are competing goals, strategies for balancing these two objectives are discussed. Numerical examples illustrate the approach  

In recent decades, the study of Nano and Micro Electromechanical Systems (NEMS and MEMS) as in micro sensors and actuators or in Atomic Force Microscopes (AFM) attracted considerable attention. Since an indispensable part of such systems is a nano or micro flexible beam, the inevitable undesirable vibrations of the flexible beam necessitate the implementation of vibration control strategies. Recently, some cutting edge experiments on statics and dynamics behavior of micro beams have been conducted that reveal some deviations from the prediction of classical theories, and hence this leads to development of strain gradient elasticity theory. In addition, the often-adopted finite dimensional... 

As a matter of fact, power systems are always faced with a variety of uncertainties. These uncertainties change system parameters in an unwanted manner which may deteriorate the system security and reliability. For instance, a transmission line which is designed for a certain capacity may be suffered as a result of load increment and consequently may influence the normal operation of other network elements. In recent years, as a result of environmental considerations and especially after the oil shock at 1973 which caused the the energy prices to increase unprecedentedly, the human has taken more attention to renewable energies exploitation. Theses energies have a fluctuating nature and are... 

The tree structure of neuron morphologies has excited neuroscientists since their discovery in the 19-th century. Many theories assign computational meaning to morphologies, but it is still hard to generate realistic looking morphologies. There are a few growth models for generating neuron morphologies that correctly reproduce some features (e.g. branching angles) of morphologies, but they tend to fall short on other features. Here we present an approach that builds a generative model by extracting a set of human-chosen features from a database of neurons by using the naïve Bayes approach. Then by starting from a neuron with a soma we use statistical sampling techniques to generate... 

A major medical revolution that is helping us overcome some of the worst diseases, including cancer, is angiogenesis, which is based on the processes our bodies use to develop blood vessels. Most human cancers have acquired six essential capabilities: self-sufficiency in growth signals, insensitivity to growth-inhibitory signals, programmed cell death escape, unlimited replication potential, persistent angiogenesis, and tissue invasion that can induce metastasis. In other words, the defense mechanism that prevents any of these acquired capabilities must be neutralized before the cells become malignant and invasive tumors. In fact, tumors in the non-vascular growth stage can only grow up to... 

Our main focus in this thesis is on optimal control methods for the analysis of deep neural networks, such as the supervised learning problem. A neural network with a large number of layers can be modeled with an ordinary differential equation, whose control parameters play the role of intermediary functions in the neural network. This model gives us a more powerful tool to analyze the asymptotic behavior of the neural network. Also, in this thesis, the problems of numerical implementation of these methods will be discussed