Li-Tien Cheng

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

A level set method for dislocation dynamics

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

A level set-based Eulerian approach for anisotropic wave propagation

c^{2}<ab.

Dewetting-controlled binding of ligands to hydrophobic pockets.

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.

Redistancing by flow of time dependent eikonal equation

We propose a three-dimensional level set method for dislocation dynamics in which the dislocation lines are represented in three dimensions by the intersection of the zero levels of two level set functions. Since the level set method does not discretize nor directly track individual dislocation line segments, it easily handles topological changes occurring in the microstructure. The dislocation dynamics are not limited to glide along a slip plane, but also account for threedimensional aspects of their motion: cross-slip occurs naturally and climb is included by fixing the relative climb and glide mobility. The level set dislocation dynamics method was implemented using an accurate finite difference scheme on a uniform grid. To demonstrate the versatility, utility and simplicity of this new model, we present examples including the motion of dislocation loops under applied and self-stresses (including glide, cross-slip and climb), intersections of dislocation lines, operation of Frank–Read sources and dislocations bypassing particles.

Estimation of 3D Surface Shape and Smooth Radiance from 2D Images: A Level Set Approach

A level-set method is developed for numerically capturing the equilibrium solute-solvent interface that is defined by the recently proposed variational implicit solvent model [Dzubiella, Swanson, and McCammon, Phys. Rev. Lett. 104, 527 (2006); J. Chem. Phys. 124, 084905 (2006)]. In the level-set method, a possible solute-solvent interface is represented by the zero level set (i.e., the zero level surface) of a level-set function and is eventually evolved into the equilibrium solute-solvent interface. The evolution law is determined by minimization of a solvation free energy functional that couples both the interfacial energy and the van der Waals type solute-solvent interaction energy. The surface evolution is thus an energy minimizing process, and the equilibrium solute-solvent interface is an output of this process. The method is implemented and applied to the solvation of nonpolar molecules such as two xenon atoms, two parallel paraffin plates, helical alkane chains, and a single fullerence C(60). The level-set solutions show good agreement for the solvation energies when compared to available molecular dynamics simulations. In particular, the method captures solvent dewetting (nanobubble formation) and quantitatively describes the interaction in the strongly hydrophobic plate system.

Interfaces and hydrophobic interactions in receptor-ligand systems: A level-set variational implicit solvent approach.

The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of these wave propagations. In particular, our method is able to accurately reproduce the quasi-transverse, or quasi-S, waves with cusps, which form a class of multi-valued solutions. The level set formulation we use is borrowed from one for moving curves in three spatial dimensions, with the velocity fields for evolution following from the method of characteristics on the anisotropic eikonal equation. We present here our derivation of the algorithm and numerical results to illustrate its accuracy in different cases of anisotropic wave propagations related to seismic imaging.

Variational Implicit Solvation with Poisson-Boltzmann Theory.

We report on a combined atomistic molecular dynamics simulation and implicit solvent analysis of a generic hydrophobic pocket-ligand (host-guest) system. The approaching ligand induces complex wetting-dewetting transitions in the weakly solvated pocket. The transitions lead to bimodal solvent fluctuations which govern magnitude and range of the pocket-ligand attraction. A recently developed implicit water model, based on the minimization of a geometric functional, captures the sensitive aqueous interface response to the concave-convex pocket-ligand configuration semiquantitatively.

Evaluation of Hydration Free Energy by Level-Set Variational Implicit-Solvent Model with Coulomb-Field Approximation.

Construction of signed distance to a given interface is a topic of special interest to level set methods. There are currently, however, few algorithms that can efficiently produce highly accurate solutions. We introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations. We provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy. Comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction, which, for example, are critical in local level set methods. We further apply the same ideas to extension of values off interfaces. Together, our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems.