Browsing by Subject "Wave Equation"

In this thesis we consider the inverse problem for the one-dimensional wave equation. That is, we would like to recover the velocity function, the wave speed, from the equation given Neumann and Dirichlet boundary conditions, when the solution to the equation is known. It has been shown that an operator Λ corresponding to the boundary conditions determines the volumes of the domain of influence, which is the set where the travel time for the wave is limited. These volumes then in turn determine the velocity function. We present some theorems and propositions about determining the wave speed and present proofs for a few of them. Artificial neural networks are a form of machine learning widely used in various applications. It has been previously proven that a one-layer feedforward neural network with a non-polynomial activation function with some additional constraints on the activation function can approximate any continuous real valued functions. In this thesis we present proof of this result for a continuous non-polynomial activation function. Furthermore, in this thesis we apply two neural network architectures to the volume inversion problem, which means that we train the networks to approximate a single volume when the operator Λ is given. The neural networks in question are the feedforward neural network and the operator recurrent neural network. Before the volume inversion problem, we consider a simpler problem of finding an inverse matrix of a small invertible matrix. Finally, we compare the performances of these two neural networks for both the volume and matrix inversion problems.