A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations

We present a fast sweeping method for a class of Hamilton-Jacobi equations that arise from time-independent problems in optimal control theory. The basic method in two dimensions uses a four point stencil and is extremely simple to implement. We test our basic method against Eikonal equations in different norms, and then suggest a general method for rotating the grid and using additional approximations to the derivatives in different directions in order to more accurately capture characteristic flow. We display the utility of our method by applying it to relevant problems from engineering

A Model for Optimal Human Navigation with Stochastic Effects

We present a method for optimal path planning of human walking paths in mountainous terrain, using a control theoretic formulation and a Hamilton-Jacobi-Bellman equation. Previous models for human navigation were entirely deterministic, assuming perfect knowledge of the ambient elevation data and human walking velocity as a function of local slope of the terrain. Our model includes a stochastic component which can account for uncertainty in the problem, and thus includes a Hamilton-Jacobi-Bellman equation with viscosity. We discuss the model in the presence and absence of stochastic effects, and suggest numerical methods for simulating the model. We discuss two different notions of an optimal path when there is uncertainty in the problem. Finally, we compare the optimal paths suggested by the model at different levels of uncertainty, and observe that as the size of the uncertainty tends to zero (and thus the viscosity in the equation tends to zero), the optimal path tends toward the deterministic optimal path

Models for Human Navigation and Optimal Path Planning Using Level Set Methods and Hamilton-Jacobi Equations

We present several models for different physical scenarios which are centered around human movement or optimal path planning, and use partial differential equations and concepts from control theory. The first model is a game-theoretic model for environmental crime which tracks criminals' movement using the level set method, and improves upon previous continuous models by removing overly restrictive assumptions of symmetry. Next, we design a method for determining optimal hiking paths in mountainous regions using an anisotropic level set equation. After this, we present a model for optimal human navigation with uncertainty which is rooted in dynamic programming and stochastic optimal control theory. Lastly, we consider optimal path planning for simple, self-driving cars in the Hamilton-Jacobi formulation. We improve upon previous models which simplify the car to a point mass, and present a reasonably general upwind, sweeping scheme to solve the relevant Hamilton-Jacobi equation

Visual perspective and the characteristics of mind wandering

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An Efficient Semi-Real-Time Algorithm for Path Planning in the Hamilton-Jacobi Formulation

We present a semi-real-time algorithm for minimal-time optimal path planning based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ) equations. Partial differential equation (PDE) based optimal path planning methods are well-established in the literature, and provide an interpretable alternative to black-box machine learning algorithms. However, due to the computational burden of grid-based PDE solvers, many previous methods do not scale well to high dimensional problems and are not applicable in real-time scenarios even for low dimensional problems. We present a semi-real-time algorithm for optimal path planning in the HJ formulation, using grid-free numerical methods based on Hopf-Lax formulas. In doing so, we retain the intepretablity of PDE based path planning, but because the numerical method is grid-free, it is efficient and does not suffer from the curse of dimensionality, and thus can be applied in semi-real-time and account for realistic concerns like obstacle discovery. This represents a significant step in averting the tradeoff between interpretability and efficiency. We present the algorithm with application to synthetic examples of isotropic motion planning in two-dimensions, though with slight adjustments, it could be applied to many other problems.Comment: 6 pages, 2 figures, submitted to American Control Conference 202

Analysis of a Reaction-Diffusion SIR Epidemic Model with Noncompliant Behavior

Recent work from public health experts suggests that incorporating human behavior is crucial in faithfully modeling an epidemic. We present a reaction-diffusion partial differential equation SIR-type population model for an epidemic including behavioral concerns. In our model, the disease spreads via mass action, as is customary in compartmental models. However, drawing from social contagion theory, we assume that as the disease spreads and prevention measures are enacted, noncompliance with prevention measures also spreads throughout the population. We prove global existence of classical solutions of our model, and then perform R0-type analysis and determine asymptotic behavior of the model in different parameter regimes. Finally, we simulate the model and discuss the new facets which distinguish our model from basic SIR-type models

When Imagining Yourself in Pain, Visual Perspective Matters : The Neural and Behavioral Correlates of Simulated Sensory Experiences

© 2015 Massachusetts Institute of TechnologyPeer reviewedPublisher PD