This paper explores some optimization methods such as the gradient descent method, Gauss-Newton method, and stochastic gradient method. Some examples of minimizing objective functions are given to validate the theories. Then we introduce a simple example of artificial neural networks, define its structure, and apply the optimization methods to it....
The gauge integral is a rather obscure object in mathematics. Like the Riemann integral, it is relatively simple to define, but unlike the Riemann integral, the gauge integral has the ability to integrate a very large class of functions. Here, we provide a development of the gauge integral at an...
Aggregation equations have been used to model phenomena such as insect swarming and chemotaxis. Previous work on aggregation equations in the area of analysis applied to PDE has proven well-posedness of certain classes of aggregation equations in Lebesgue spaces. We will prove local existence of solutions in H^1 to an...
In this work we consider a model for pattern-producing vegetation in semi-arid regions of the world proposed by Klausmeier. It is a coupled nonlinear diffusion-advection evolutionary PDE system describing the vegetation density and water amount. The model was studied extensively by J.Sherratt who took into consideration the field data for...
Charge transport in a semiconductor structure with heterojunction is described by a multiscale partial differential equation model. This model can be used, e.g., for the design of more efficient solar cells. Phenomena at the heterojunction must be resolved at the angstrom scale while the size of the device is that...
This dissertation presents some results from various areas of probability theory, the unifying theme being the use of functional analytic intuition and techniques. We first give a result regarding the existence of certain stochastic integral representations for Banach space valued Gaussian random variables. Next we give a spectral geometric construction...
This thesis is on the existence and uniqueness of weak solutions to the Navier-Stokes equations in R3 which govern the velocity of incompressible fluid with viscosity ν. The solution is obtained in the space of tempered distributions on R3 given an initial condition and forcing data which are dominated by...
An experimental study was performed in which an ammonia-water solution was desorbed within a branching fractal-like microchannel array. The solution entered in the center of a disk, and flowed out radially until discharging in to a gravity-driven separation chamber. Heat was added to the ammonia-water through a thin wall, above...
In this paper we develop an upscaling technique for non-Darcy flow in porous media.
Non-Darcy model of flow applies to flow in porous media when large velocities occur. The
well-posedness results for theory of quasilinear elliptic partial differential equations. To
discretize the model we used lowest order Raviart-Thomas mixed finite...
Water is one of the most biologically and economically important substances on Earth. A significant portion of Earth's water subsists in the subsurface. Our ability to monitor the flow and transport of water and other fluids through this unseen environment is crucial for a myriad of reasons.
One difficulty we...