Alexander Vladimirsky

Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms

We introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static Hamilton-Jacobi equations with Dirichlet boundary conditions. Standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation. Our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation. These techniques are of complexity O(M log M), where M is the total number of points in the domain. We consider anisotropic test problems in optimal control, seismology, and paths on surfaces.

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Ordered upwind methods for static Hamilton–Jacobi equations

We introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static Hamilton–Jacobi equations with Dirichlet boundary conditions. Standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation. Our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation. These techniques are of complexity O(M log M), where M is the total number of points in the domain. We consider anisotropic test problems in optimal control, seismology, and paths on surfaces.

A Fast Method for Approximating Invariant Manifolds

The task of constructing higher-dimensional invariant manifolds for dynamical systems can be computationally expensive. We demonstrate that this problem can be locally reduced to solving a system of quasi-linear PDEs, which can be efficiently solved in an Eulerian framework. We construct a fast numerical method for solving the resulting system of discretized nonlinear equations. The efficiency stems from decoupling the system and ordering the computations to take advantage of the direction of information flow. We illustrate our approach by constructing two-dimensional invariant manifolds of hyperbolic equilibria in

Fast Two-scale Methods for Eikonal Equations

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Static PDEs for time-dependent control problems

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Ordered Upwind Methods for Hybrid Control

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Homogenization of Metric Hamilton–Jacobi Equations

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Label-Setting Methods for Multimode Stochastic Shortest Path Problems on Graphs

Fast Marching and Fast Sweeping are the two most commonly used methods for solving the eikonal equation. Each of these methods performs best on a different set of problems. Fast Sweeping, for example, will outperform Fast Marching on problems where the characteristics are largely straight lines. Fast Marching, on the other hand, is usually more efficient than Fast Sweeping on problems where characteristics frequently change their directions and on domains with complicated geometry. In this paper we explore the possibility of combining the best features of both approaches by using Marching on a coarser scale and sweeping on a finer scale. We present three new hybrid methods based on this idea and illustrate their properties in several numerical examples with continuous and piecewise-constant speed functions in

A PARALLEL TWO-SCALE METHOD FOR EIKONAL EQUATIONS ∗

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