Ph.D. Theses (Mathematics)
http://hdl.handle.net/1957/15739
2016-02-13T13:15:12ZSome Results in Single-Scattering Tomography
http://hdl.handle.net/1957/58015
Some Results in Single-Scattering Tomography
Sherson, Brian
Single-scattering tomography describes a model of photon transfer through a object in which photons are assumed to scatter at most once. The Broken Ray transform arises from this model, and was first investigated by Lucia Florescu, Vadim A. Markel, and John C. Schotland, [2], in 2010, followed by an inversion, [1], in the case of fixed initial and terminal directions. Later, in 2013, Katsevich and Krylov, [6], investigated settings where terminal rays were permitted to vary, either heading towards or away from a focal point, providing inversion formulas in two- and three-detector settings.
In this thesis, we will explore these transforms, give them distributional meaning, and analyze how these transforms propagate singularities. This requires analysis of the relationships between a function and its antiderivative obtained from integration over a ray.
This thesis also introduces the Polar Broken Ray transform, in which the source position is fixed, initial direction varies, and the scattering angle is held constant. We will discover that the Polar Broken Ray transform is injective on spaces of functions supported in an annulus that is bounded away from the origin, and also derive distributional meaning to the Polar Broken Ray transform, and derive a relationship between a wavefront set of a function and that of its Polar Broken Ray transform.
Provided in the appendix are implementations of numerical inversions of the Broken Ray transform with fixed initial and terminal directions, as well as the Polar Broken Ray transform.
Graduation date: 2016
2015-12-10T00:00:00ZRuin Problems with Risky Investments
http://hdl.handle.net/1957/57263
Ruin Problems with Risky Investments
Loke, Sooie-Hoe
In this dissertation, we study two risk models. First, we consider the dual risk process which models the surplus of a company that incurs expenses at a constant rate and earns random positive gains at random times. When the surplus is invested in a risky asset following a geometric Brownian motion, we show that the ruin probability decays algebraically for small volatility and that ruin is certain for large volatility. We use numerical methods to approximate the ruin probability when the surplus is invested in a risk-free asset. When there are no investments, we recover the exact expression for the ruin probability via Wiener-Hopf factorization. Second, we are concerned with incurred but not reported (IBNR) claims, modeled by delaying the settlement of each claim by a random time. When the investments follow a geometric Brownian motion, we derive a parabolic integro-partial-differential equation (IPDE) for the ultimate ruin probability with final value condition given by the ruin probability under risky investments with no delay. Assuming that the delay times are bounded by a constant, we obtain an existence theorem of the final value IPDE in the space of bounded functions, and a uniqueness theorem in the space of square integrable functions. When the delay times are deterministic, we show that delaying the settlement of claims does not reduce the probability of ruin when the volatility is large.
Graduation date: 2016
2015-09-10T00:00:00ZThe Variable Speed Wave Equation and Perfectly Matched Layers
http://hdl.handle.net/1957/56308
The Variable Speed Wave Equation and Perfectly Matched Layers
Kim, Dojin
A perfectly matched layer (PML) is widely used to model many different types of wave propagation in different media. It has been found that a PML is often very effective and also easy to set, but still many questions remain.
We introduce a new formulation from regularizing the classical Un-Split PML of the acoustic wave equation and show the well-posedness and numerical efficiency. A PML is designed to absorb incident waves traveling perpendicular to the PML, but there is no effective absorption of waves traveling with large incident angles. We suggest one method to deal with this problem and show well-posedness of the system, and some numerical experiments. For the 1-d wave equation with a constant speed equipped a PML, stability and the exponential decay rate of energy has been proved, but the question for variable sound speed equation remained open. We show that the energy decays exponentially in the 1-d PML wave equation with variable sound speed.
Most PML wave equations appear as a first-order hyperbolic system with as a zero-order perturbation. We introduce a general formulation and show well-posedness and stability of the system. Furthermore we develop a discontinuous Galerkin method and analyze both the semi-discrete and fully discretized system and provide a priori error estimations.
Graduation date: 2016
2015-06-04T00:00:00Z3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curve
http://hdl.handle.net/1957/51754
3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curve
Wongsason, Patcharee
3D vector tomography has been explored and results have been achieved in the last few decades. Among these was a reconstruction formula for the solenoidal part of a vector field from its Doppler transform with sources on a curve. The Doppler transform of a vector field is the line integral of the component parallel to the line. In this work, we shall study the transverse ray transform of a vector field, which instead integrates over lines the component of the vector field perpendicular to the line. We provide a reconstruction procedure for the transverse ray transform of a vector field with sources on a curve fulfilling Tuy’s condition of order 3. We shall recover both the potential and solenoidal parts. We present two steps for the reconstruction. The first one is to reconstruct the solenoidal part and the techniques we use are inspired by work of Katsevich and Schuster. A procedure for recovering the potential part will be the second step. The main ingredient is the difference between the measured data and the reprojection of the solenoidal part. We also provide a variation of the Radon inversion formula for the vector part of a quaternionic-valued function (or vector field) and an inversion formula in cone-beam setting with sources on the sphere.
Graduation date: 2015
2014-07-23T00:00:00Z