Sharif Digital Repository

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... 

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  

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... 

Partial Differential Equations (PDEs) play a crucial role in modeling various physical, biological, and engineering phenomena. One of the most important and complex of these equations is the Monge-Ampère equation, which appears in various fields including differential geometry and optimization theory. In this thesis, after defining and examining the preliminary properties of the Monge-Ampère measure and defining the Alexandrov weak solution, the existence and uniqueness of this weak solution for the Dirichlet problem are addressed, and finally, the regularity problem is studied  

The role of a mathematical model is to explain a set of experiments, and to make predictions. In setting up a mathematical model of a biological process, by a set of differential equations, it is very important to determine the numerical value of the parameters. For biological processes are typically valid only within a limited range of parameters. In the last decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. Some of these models use only ordinary differential equations (ODEs), ignoring the spatial effects of tumor... 

A typical problem in applied mathematics and science is to estimate the future state of a dynamical system given its current state. One approach aimed at understanding one or more aspects determining the behavior of the system is mathematical modeling. This method frequently entails formulation of a set of equations, usually a system of partial or ordinary differential equations. Model parameters are then measured from experimental data or estimated from computer simulation or other methods. Solutions to the model are then studied through mathematical analysis and numerical simulation usually for qualitative fit to the dynamical system of interest and any relative time-series data that is... 

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, first, the modeling method of neural fields is precisely presented. Then, Existence and Stability of different solutions of one dimensional neural fields like Standing Pulses, Traveling Waves and ... are investigated in three different models of neural fields. In order for proving Existence and Stability of the solutions the mathematical tools like Fourier transform and Evans function are applied. All the models which analysed in this thesis have the following Integro-Differential Equation form:
τ
∂u(x, t)
∂t
= −u(x, t) +
∫ +∞
−∞
w(x, y)f[u(y, t)]dy + I(x, t) + s(x, t)
and also in some models the parameters might be changed  

The aim of homogenization theory is to establish the macroscopic behaviour of a system which is ‘microscopically’ heterogeneous, in order to describe some characteristics of the heterogeneous medium (for instance, its thermal or electrical conductivity). This means that the heterogeneous material is replaced by a homogeneous fictitious one (the ‘homogenized’ material), whose global (or overall) characteristics are a good approximation of the initial ones. From the mathematical point of view, this signifies mainly that the solutions of a boundary value problem, depending on a small parameter, converge to the solution of a limit boundary value problem which is explicitly described. In... 

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 1989 DiPerna and Lions proved the stability of Boltzmann’s evolution equation using their theory of renormalized solutions. Their result , for the first time, proved the existence of solutions without extra assumptions on the initial condition and on arbitrary time intervals. While the wellknown Grad’s cut-off assumption is present in the original theory, there have been successful generalizations to account for the singular case. Also the renormalization theory has shed light on limiting regimes of the Boltzmann equation. Here we discuss the new techniques that are essential for such generalizations  

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  

The aim of this thesis is to investigate recent methods for Image Processing(Any signal process which it’s input is an image and it’s ouput is an image or a set of Image parameters) using Calculus of variation tools. Methods which are to be investigated has been divided into two well known parts of Image Processing : Image Restoration and Image Segmentation.Image Processing Chapter includes two sections: one calculus of variations methods(energy method), other methods based on PDEs(heat equation and Malik-Perona equation). In studing each of this methods, It has been tried to include experimental results and negative and positive points of them.In Image Segmentation Chapter, first... 

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  

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... 

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... 

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 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,... 

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...