Numerical Methods and Uniquness for the Canham-Helfrich Model of Biomembranes

The classical Canham-Helfrich models of biomembranes consist of a family of geometric constrained variational problems. Their physical importance and mathematical challenge attract the attention of both biophysicists and geometric analysts. In this PhD thesis, we develop a numerical method for these models. Our method uses a high-order approximation of surfaces with arbitrary topology based on subdivision methods. We also develop multiscale and parallel versions of our method which substantially speed up computations. An implementation based on Matlab and CUDA is provided along with this thesis. We use our solver to explore a phenomenon known as conformal diusion in the biophysical literature, which is also connected to the open uniqueness question for the Canham and Helfrich variation problems. We establish the uniqueness of solution of the Canham problem in a special case related to the Willmore conjecture (now the Marques-Neves theorem).Ph.D., Mathematics -- Drexel University, 201

Modeling Route Choice Behavior From Smartphone GPS data

Abstract GPS capable smart phones are emerging survey tools in transportation research field, especially in modeling individuals' mobility patterns. In route choice modeling, path observations need to be generated explicitly for the estimation. It is a challenge because the recorded data is not as dense or accurate as those from dedicated GPS devices. In this paper, we develop a methodology for generating probabilistic path observations from sparse and inaccurate location data, for state-of-the-art discrete route choice models. The difference of the proposed algorithm and the map matching algorithms is that instead of giving a unique matching result, the new algorithm generates a set of potential true paths, along with probabilities for each one to have been the true path. More importantly, the algorithm uses not only the topological measurement, but also temporal information (speed and time) in the GPS data to calculate the probability for observing the data while traveling on the proposed path. We emphasis traveling as a dynamic movement on a path, and model it as such in the algorithm. A short trip and two longer trips are used to analyze the performance of the algorithm on real data. Then, 19 trips recorded from a single user's cell phone are used in a preliminary study that estimates route choice behaviors using state-of-the-art discrete route choice modeling methodologies with the proposed probabilistic path observation generation algorithm. Abstract Keyword: route choice modeling, path observation generation, smart-phone data, GPS data, map matchin

GENERATING PROBABILISTIC PATH OBSERVATION FROM GPS DATA FOR ROUTE CHOICE MODELING

Abstract: Map matching algorithms are the conventional way to generate path observations from GPS data for route choice models. The deterministic matching may introduce extra biases to parameters of route choice models if the matching is wrong. In this paper, a new methodology is proposed to probabilistically generate path representation from GPS location data and the underlying network. This methodology takes advantage of both spatial and temporal relationships existing in the location data and the network. The generated result includes a set of potential true paths, along with a probability of each proposed path to have been the actual path. An algorithm is designed and applied to a simulated trip. Keywords: path probability, spatial temporal, route choice modelling, map matching