University of Macau / Faculty of Science and Technology. / Department of Mathematics

University of Macau / Faculty of Science and Technology. / Department of Mathematics

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to free-boundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013). / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / October 30, 2017. / backward stochastic differential equations, Hamilton-Jacobi-Bellman equation, Monte Carlo approximation, portfolio optimization, stochastic control, transaction costs / Includes bibliographical references. / Arash Fahim, Professor Directing Dissertation; Jen Atkins, University Representative; Giray Ökten, Committee Member; Lingjiong Zhu, Committee Member.

Biofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like self-produced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque removing process. Differential equations are used for modelling the microfiltration process and the optimal control method is applied to analyze the efficiency of the filtration. The multiphase fluid system is introduced to describe the dental plaque removing process and results are obtained by numerical schemes. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / October 27, 2017. / Includes bibliographical references. / Nick Cogan, Professor Directing Dissertation; Eric Chicken, University Representative; Kyle A. Gallivan, Committee Member; Monica K. Hurdal, Committee Member.

To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$-solver (Section~\ref{3F2 type solution}) and $F_1$-solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,|\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,|\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm~\ref{hom}) of two $D$-modules, where $D$ is a multivariate differential ring, and compute projective homomorphisms (Algorithm~\ref{algo ProjHom}) using the tensor product module and Algorithm~\ref{hom}. As an application, all irreducible order $2$ subsystems from reducible order $3$ systems turn out to come from Gauss hypergeometric function $_2F_1(a,b;c\,|\,x)$ (Chapter~\ref{chapter applications}). / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / November 16, 2017. / Includes bibliographical references. / Mark van Hoeij, Professor Directing Dissertation; Laura Reina, University Representative; Amod Agashe, Committee Member; Ettore Aldrovandi, Committee Member; Paolo Aluffi, Committee Member.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaves 33-34). / by Hi-joon Chae. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 133-142). / Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work such as multimaterial fiber drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem. The resulting mathematical model, based on linear-stability analysis, can quickly predict the breakup lengthscale and timescale of concentric cylindrical fluids, and provides useful guidance for material selections and design parameters in fiber-drawing experiments. A three-fluid system with competing breakup processes at very different length scales is demonstrated with a full Stokes flow simulation. In the second half of this thesis, we study large-scale PDE-constrained microcavity topology optimization. Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. While most microcavities are designed mostly by hand using some understanding of the physical principles of the confinement, we let the computer discover its own structures. We formulate and solve a full 3d optimization scheme over all possible 2d-lithography patterns in a thin dielectric film. The key to our formulation is a frequency-averaged local density of states (LDOS), where the frequency averaging corresponds to the desired bandwidth, evaluated by a novel technique: solving a single electromagnetic wave scattering problem at a complex frequency. / by Xiangdong Liang. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000. / Includes bibliographical references (p. 100-104). / by Jing-Rebecca Li. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 245-254). / Granular materials such as sand or gravel surround us everyday and yet remain poorly understood. In this thesis, two models are developed for dense granular flow, each capable of predicting flows with accuracy in multiple environments. The models are based on differing perspectives of grain-level dynamics, with one deriving flow from a stochastic mechanism and the other from a deterministic deformation law. The Stochastic Flow Rule (SFR): This work models granular flow as a sequence of localized collective grain displacements. As in the Spot Model for drainage (Bazant 2001), grain clusters move as dictated by "spots" which travel through the material as biased random-walkers. The SFR derives spot motion directly from the material stresses, thus generalizing and extending the Spot Model beyond drainage to any quasi-2D geometry with a computable stress field. Limit-State Mohr-Coulomb Plasticity is used to approximate the stress profile in a slow flowing granular assembly. The SFR then describes quantitatively how to convert the slip-line field and stresses into the necessary parameters to fully define a spot's random trajectory through the material and generate a steady flow profile. Results are compared to known flow data. Nonlinear Granular Elasto-Plasticity: This work models granular deformation at the meso-scale as a deterministic consequence of the local stresses and state parameters. Recently proposed models for granular elasticity (Jiang and Liu 2003) and plastic flow (Jop et al. 2006) are combined into one universal granular continuum law, capable of predicting both flowing regions and stagnant zones simultaneously in any arbitrary 3D flow geometry. / (cont.) The unification is performed by first motivating physically, and then implementing a Kroner-Lee elasto-plastic decomposition. The model is then numerically solved in multiple geometries and results are compared to experiments and discrete simulations. / by Kenneth Norman Kamrin. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993. / Includes bibliographical references (leaves 138-142). / by Jöns Richard Gustaf Ehrenborg. / Ph.D.