Search ETDs:
Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology
Chen, Weitao

2013, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis consists of three parts.

In the first part, we develop a numerical solver for steady states of hyperbolic conservation problems with high order of accuracy and the capability to resolve shocks. Fast sweeping methods are efficient iterative numerical schemes originally designed for solving stationary Hamilton-Jacobi equations. Their efficiency relies on Gauss-Seidel type nonlinear iterations, and a finite number of sweeping directions. We generalize the fast sweeping methods to hyperbolic conservation laws with source terms. The algorithm is obtained through finite difference discretization, with the numerical fluxes evaluated in WENO (Weighted Essentially Non-oscillatory) fashion, coupled with Gauss-Seidel iterations. In particular, we consider mainly the Lax-Friedrich numerical fluxes. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods.

In the second part, an inverse problem, arising from the design of some optical materials to localize waves at a specific wavelength, is solved by the steepest descent method. The method of steepest descent is a classical approach to find the minimum/maximum of an objective function or functional based on a first order approximation. The method works in spaces of any number of dimensions, even in infinite-dimensional spaces. This method can converge more efficiently than methods which do not require derivative information; however, in certain circumstances the "cost function space" may become discontinuous and as a result, the derivatives may be difficult or impossible to determine. Here, we discuss eigenfunction optimization for representing the topography of a dielectric environment and efficient techniques to solve different material design problems. Numerous results are shown to demonstrate the robustness of the gradient-based approach.

In the last part, we model and simulate an important biological process called cell polarization. Yeast cell mating is a well-studied biological system which involves external chemical stimuli, reactions and surface diffusion of multiple proteins on the cell membrane. In this part, we use the level set method to do the simulation and couple it with the extended diffusion equations. The level set method efficiently captures the motion of a closed curve by embedding the curve of interest as the zero contour of a level set function defined on a Cartesian grid. It allows us to update the level set function on a fixed grid mesh instead of moving grid points on the curve. This strategy not only works well for the complex system of a single cell, but also shows great advantage in multi-cell environment since for different cells different level set functions with the same kind of systems of equations will be assigned. In this part, we apply the level set method to reproduce the results obtained from using Lagrangian framework for a single cell with artificial pheromone gradient in extracellular space, and use it to establish a more refined model with extracellular diffusion based on previous work. Moreover, it allows us to investigate simulations involving multiple cells and reveal more biological mechanisms behind that.
Ching-Shan Chou (Advisor)
Ed Overman (Committee Member)
Chuan Xue (Committee Member)
124 p.

Recommended Citations

Hide/Show APA Citation

Chen, W. (2013). Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

Hide/Show MLA Citation

Chen, Weitao. "Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology." Electronic Thesis or Dissertation. Ohio State University, 2013. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

Hide/Show Chicago Citation

Chen, Weitao "Fast Sweeping Methods for Steady State Hyperbolic Conservation Problems and Numerical Applications for Shape Optimization and Computational Cell Biology." Electronic Thesis or Dissertation. Ohio State University, 2013. https://etd.ohiolink.edu/

Files

WeitaoChen_thesis.pdf (1.49 MB) View|Download