Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure

Doctor of Philosophy

dissertationPartial differential equations (PDEs) are widely used in science and engineering to model phenomena such as sound, heat, and electrostatics. In many practical science and engineering applications, the solutions of PDEs require the tessellation of computational domains into unstructured meshes and entail computationally expensive and time-consuming processes. Therefore, efficient and fast PDE solving techniques on unstructured meshes are important in these applications. Relative to CPUs, the faster growth curves in the speed and greater power efficiency of the SIMD streaming processors, such as GPUs, have gained them an increasingly important role in the high-performance computing area. Combining suitable parallel algorithms and these streaming processors, we can develop very efficient numerical solvers of PDEs. The contributions of this dissertation are twofold: proposal of two general strategies to design efficient PDE solvers on GPUs and the specific applications of these strategies to solve different types of PDEs. Specifically, this dissertation consists of four parts. First, we describe the general strategies, the domain decomposition strategy and the hybrid gathering strategy. Next, we introduce a parallel algorithm for solving the eikonal equation on fully unstructured meshes efficiently. Third, we present the algorithms and data structures necessary to move the entire FEM pipeline to the GPU. Fourth, we propose a parallel algorithm for solving the levelset equation on fully unstructured 2D or 3D meshes or manifolds. This algorithm combines a narrowband scheme with domain decomposition for efficient levelset equation solving

A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES

This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton-Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512-2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers.open4

A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.Comment: 22 page