<p>We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.</p><p>Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.</p><p>Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.</p> / Dissertation

In today’s computer age, the numerical simulations are replacing the traditional laboratory experiments. Researchers around the world are using advanced computer software and multiprocessor computer technology to perform experiments, and analyse these simulation results to advance in their respective endeavours. With a wide variety of tools and technologies available, it could be a tedious and time taking task for a non-computer science researcher to choose appropriate methodologies for developing simulation software The research of this thesis addresses the use of Message Passing Interface (MPI) using object-oriented programming techniques and discusses the methodologies suitable to scientific computing, also, propose a customized software engineering development model.

We elaborate upon the Herty-Illner macroscopic traffic models which include special non-local forces. The first chapter presents these in relation to the traffic models of Aw-Rascle and Zhang, arguing that non-local forces are necessary for a realistic description of traffic. The second chapter considers travelling wave solutions for the Herty-Illner macroscopic models. The travelling wave ansatz for the braking scenario reveals a curiously implicit nonlinear functional differential equation, the jam equation, whose unknown is, at least to conventional tools, inextricably self-argumentative! Observing that analytic solution methods fail for the jam equation yet succeed for equations with similar coefficients raises a challenging problem of pure and applied mathematical interest. An unjam equation analogous to the jam equation explored by Illner and McGregor is derived. The third chapter outlines refinements for the Herty-Illner models. Numerics allow exploration of the refined model dynamics in a variety of realistic traffic situations, leading to a discussion of the broadened applicability conferred by the refinements: ultimately the prediction of stop-and-go waves. The conclusion asserts that all of the above contribute knowledge pertinent to traffic control for reduced congestion and ameliorated vehicular flow. / Graduate

There has been much work on electron emission. It has lead to the concept of the photon and new electron sources for imaging such as electron microscopes and the rst formulation of holographic reconstructions [1-6]. Analytical derivations are important to gain physical insight into the problem of developing better electron sources. However, to date, such formulations have su ered by a number of approximations that have masked important physics. In this thesis, a new approach is provided that solves the Schrodinger wave equation for photoemission from a single atom tungsten tip barrier or more generally, for photoemission from a Schottky triangular barrier potential, with or without image potential e ects. We describe the system, then introduce the mathematical derivation. We conclude with the applications of the theory.

There has been much work on electron emission. It has lead to the concept of the photon and new electron sources for imaging such as electron microscopes and the rst formulation of holographic reconstructions [1-6]. Analytical derivations are important to gain physical insight into the problem of developing better electron sources. However, to date, such formulations have su ered by a number of approximations that have masked important physics. In this thesis, a new approach is provided that solves the Schrodinger wave equation for photoemission from a single atom tungsten tip barrier or more generally, for photoemission from a Schottky triangular barrier potential, with or without image potential e ects. We describe the system, then introduce the mathematical derivation. We conclude with the applications of the theory.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

A unique mathematical tool is developed to deal with global optimization of a set of engineering problems. These include image processing, mechanical topology optimization, and optimal path planning in a variational framework, as well as some benchmark problems in parameter optimization. The optimization tool in these applications is based on the level set theory by which an evolving contour converges toward the optimum solution. Depending upon the application, the objective function is defined, and then the level set theory is used for optimization. Level set theory, as a member of active contour methods, is an extension of the steepest descent method in conventional parameter optimization to the variational framework. It intrinsically suffers from trapping in local solutions, a common drawback of gradient based optimization methods. In this thesis, methods are developed to deal with this drawbacks of the level set approach. By investigating the current global optimization methods, one can conclude that these methods usually cannot be extended to the variational framework; or if they can, the computational costs become drastically expensive. To cope with this complexity, a global optimization algorithm is first developed in parameter space and compared with the existing methods. This method is called "Spiral Bacterial Foraging Optimization" (SBFO) method because it is inspired by the aggregation process of a particular bacterium called, Dictyostelium Discoideum. Regardless of the real phenomenon behind the SBFO, it leads to new ideas in developing global optimization methods. According to these ideas, an effective global optimization method should have i) a stochastic operator, and/or ii) a multi-agent structure. These two properties are very common in the existing global optimization methods. To improve the computational time and costs, the algorithm may include gradient-based approaches to increase the convergence speed. This property is particularly available in SBFO and it is the basis on which SBFO can be extended to variational framework. To mitigate the computational costs of the algorithm, use of the gradient based approaches can be helpful. Therefore, SBFO as a multi-agent stochastic gradient based structure can be extended to multi-agent stochastic level set method. In three steps, the variational set up is formulated: i) A single stochastic level set method, called "Active Contours with Stochastic Fronts" (ACSF), ii) Multi-agent stochastic level set method (MSLSM), and iii) Stochastic level set method without gradient such as E-ARC algorithm. For image processing applications, the first two steps have been implemented and show significant improvement in the results. As expected, a multi agent structure is more accurate in terms of ability to find the global solution but it is much more computationally expensive. According to the results, if one uses an initial level set with enough holes in its topology, a single stochastic level set method can achieve almost the same level of accuracy as a multi-agent structure can obtain. Therefore, for a topology optimization problem for which a high level of calculations (at each iteration a finite element model should be solved) is required, only ACSF with initial guess with multiple holes is implemented. In some applications, such as optimal path planning, objective functions are usually very complicated; finding a closed-form equation for the objective function and its gradient is therefore impossible or sometimes very computationally expensive. In these situations, the level set theory and its extensions cannot be directly employed. As a result, the Evolving Arc algorithm that is inspired by "Electric Arc" in nature, is proposed. The results show that it can be a good solution for either unconstrained or constrained problems. Finally, a rigorous convergence analysis for SBFO and ACSF is presented that is new amongst global optimization methods in both parameter and variational framework.

We study a class of differential inclusions involving maximal monotone relations, which cover a huge class of problems in mathematical physics. For this purpose we introduce the time derivative as a continuously invertible operator in a suitable Hilbert space. It turns out that this realization is a strictly monotone operator and thus, the question on existence and uniqueness can be answered by well-known results in the theory of maximal monotone relations. Furthermore, we show that the resulting solution operator is Lipschitz-continuous and causal, which is a natural property of evolutionary processes. Finally, the results are applied to a system of partial differential equations and inclusions, which describes the diffusion of a compressible fluid through a saturated, porous, plastically deforming media, where certain hysteresis phenomena are modeled by maximal montone relations.

Many problems in continuum mechanics, especially in the theory of elastic materials, lead to nonlinear partial differential equations. The nonconvexity of their underlying energy potential is a challenge for mathematical analysis, since convexity plays an important role in the classical theories of existence and regularity. In the last years one main point of interest was to develop techniques to circumvent these difficulties. One approach was to use different notions of convexity like quasi-- or polyconvexity, but most of the work was done only for static (time independent) equations. In this thesis we want to make some contributions concerning existence, regularity and numerical approximation of nonconvex dynamical problems.