Hongkai Zhao

A fast sweeping method for Eikonal equations

In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

Fast surface reconstruction using the level set method

We describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from a scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surface-like model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.

Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations

We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes

Super-resolution in time-reversal acoustics

H(p,q)=\sqrt{ap^{2}+bq^{2}-2cpq},

An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface

c^{2}<ab.

High Order Fast Sweeping Methods for Static Hamilton---Jacobi Equations

We combine our Godunov numerical fluxes with simple Gauss--Seidel-type iterations for solving the corresponding Hamilton--Jacobi (HJ) equations. The resulting algorithm is fast since it does not require a sorting strategy as found, e.g., in the fast marching method. In addition, it providesa way to compute solutions to a class of HJ equations for which the conventional fast marching method is not applicable. Our experiments indicate convergence after a few iterations, even in rather difficult cases.

A level-set method for interfacial flows with surfactant

The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.

Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

In this paper we consider a fundamental visualization problem: shape reconstruction from an unorganized data set. A new minimal-surface-like model and its variational and partial differential equation (PDE) formulation are introduced. In our formulation only distance to the data set is used as our input. Moreover, the distance is computed with optimal speed using a new numerical PDE algorithm. The data set can include points, curves, and surface patches. Our model has a natural scaling in the nonlinear regularization that allows flexibility close to the data set while it also minimizes oscillations between data points. To find the final shape, we continuously deform an initial surface following the gradient flow of our energy functional. An offset (an exterior contour) of the distance function to the data set is used as our initial surface. We have developed a new and efficient algorithm to find this initial surface. We use the level set method in our numerical computation in order to capture the deformation of the initial surface and to find an implicit representation (using the signed distance function) of the final shape on a fixed rectangular grid. Our variational/PDE approach using the level set method allows us to handle complicated topologies and noisy or highly nonuniform data sets quite easily. The constructed shape is smoother than any piecewise linear reconstruction. Moreover, our approach is easily scalable for different resolutions and works in any number of space dimensions.

A Fast Sweeping Method for Static Convex Hamilton-Jacobi Equations

In this paper we study an Eulerian formulation for solving partial differential equations (PDE) on a moving interface. A level set function is used to represent and capture the moving interface. A dual function orthogonal to the level set function defined in a neighborhood of the interface is used to represent some associated quantity on the interface and evolves according to a PDE on the moving interface. In particular we use a convection diffusion equation for surfactant concentration on an interface passively convected in an incompressible flow as a model problem. We develop a stable and efficient semi-implicit scheme to remove the stiffness caused by surface diffusion.