Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity

In this work we propose and analyze a numerical method for electrical impedance tomography of recovering a piecewise constant conductivity from boundary voltage measurements. It is based on standard Tikhonov regularization with a Modica-Mortola penalty functional and adaptive mesh refinement using suitable a posteriori error estimators of residual type that involve the state, adjoint and variational inequality in the necessary optimality condition and a separate marking strategy. We prove the convergence of the adaptive algorithm in the following sense: the sequence of discrete solutions contains a subsequence convergent to a solution of the continuous necessary optimality system. Several numerical examples are presented to illustrate the convergence behavior of the algorithm.Comment: 26 pages, 12 figure

Landscape and flux for quantifying global stability and dynamics of game theory

Game theory has been widely applied to many areas including economics, biology and social sciences. However, it is still challenging to quantify the global stability and global dynamics of the game theory. We developed a landscape and flux framework to quantify the global stability and global dynamics of the game theory. As an example, we investigated the models of three-strategy games: a special replicator-mutator game, the repeated prison dilemma model. In this model, one stable state, two stable states and limit cycle can emerge under different parameters. The repeated Prisoner's Dilemma system has Hopf bifurcation transitions from one stable state to limit cycle state, and then to another one stable state or two stable states, or vice versa. We explored the global stability of the repeated Prisoner's Dilemma system and the kinetic paths between the basins of attractor. The paths are irreversible due to the non-zero flux. One can explain the game for $Peace$ and $War$.Comment: 25 pages, 15 figure

Fabrication and function of microfluidic devices for monitoring of in-vitro fertilization processes

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2007.Includes bibliographical references (leaf 36).The process of assistive reproduction is often a headache and heartache for those who choose to go through it. The field currently relies heavily on morphological characteristics to determine embryo health and development success, a highly unreliable method. While they appear healthy at implantation, many embryos, in reality, have poor development potential and fail to survive within the womb. Therefore, to offset the high chances of miscarriage, multiple eggs are implanted in the uterus. This has occasionally lead to multi-fetal pregnancies, which have a higher maternal mortality risk, and, in general, is more physically demanding. This thesis researches a microfluidic device that aids in the crucial stages of in vitro- fertilization. The device allows for a fertilized egg to be cultured within, and provides the ability to carefully monitor its health through a series of metabolic assays, a better indication of embryo health. This microfluidic embryo health monitoring device is comprised of two layers of channel networks. It works through passing fluids along flow channels that are driven by control channels. The control layer, when pressurized with gas, operates as valves and peristaltic pumps along the flow layer to pump and transport fluids through the flow channels. As embryonic fluids are passed through the channels, the status of the fertilized egg can be monitored with metabolic assays taken of the embryo at various detection sites.by Jin Xu.S.B