The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O ( M log M ) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in R ⁿ and on manifolds and connections to more general static Hamilton–Jacobi equations.

Significance: Popular neural networks for image-processing problems often contain many different operations, multiple layers of connections, and a large number of trainable parameters, often exceeding several million. They are typically tailored to specific applications, making it difficult to apply a network that is successful in one application to different applications. Here, we introduce a neural network architecture that is less complex than existing networks, is easy to train, and achieves accurate results with relatively few trainable parameters. The network automatically adapts to a specific problem, allowing the same network to be applied to a wide variety of different problems.

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.

We review recent work on level set methods for following the evolution of complex interfaces. These techniques are based on solving initial value partial differential equations for level set functions, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The methodology results in robust, accurate, and efficient numerical algorithms for propagating interfaces in highly complex settings. We review the basic theory and approximations, describe a hierarchy of fast methods, including an extremely fast marching level set scheme for monotonically advancing fronts, based on a stationary formulation of the problem, and discuss extensions to multiple interfaces and triple points. Finally, we demonstrate the technique applied to a series of examples from geometry, material science and computer vision, including mean curvature flow, minimal surfaces, grid generation, fluid mechanics, combustion, image processing, computer vision, and etching, deposition and lithography in the microfabrication of electronic components.

In many applications of tomography, the acquired data are limited in one or more ways due to unavoidable experimental constraints. In such cases, popular direct reconstruction algorithms tend to produce inaccurate images, and more accurate iterative algorithms often have prohibitively high computational costs. Using machine learning to improve the image quality of direct algorithms is a recently proposed alternative, for which promising results have been shown. However, previous attempts have focused on using encoder–decoder networks, which have several disadvantages when applied to large tomographic images, preventing wide application in practice. Here, we propose the use of the Mixed-Scale Dense convolutional neural network architecture, which was specifically designed to avoid these disadvantages, to improve tomographic reconstruction from limited data. Results are shown for various types of data limitations and object types, for both simulated data and large-scale real-world experimental data. The results are compared with popular tomographic reconstruction algorithms and machine learning algorithms, showing that Mixed-Scale Dense networks are able to significantly improve reconstruction quality even with severely limited data, and produce more accurate results than existing algorithms.

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method’s accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann’s law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces.