8 research outputs found
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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
Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk
Environmental management and restoration should be designed such that the
risk and uncertainty owing to nonlinear stochastic systems can be successfully
addressed. We apply the robustified dynamic Orlicz risk to the modeling and
analysis of environmental management and restoration to consider both the risk
and uncertainty within a unified theory. We focus on the control of a
jump-driven hybrid stochastic system that represents macrophyte dynamics. The
dynamic programming equation based on the Orlicz risk is first obtained
heuristically, from which the associated Hamilton-Jacobi-Bellman (HJB) equation
is derived. In the proposed Orlicz risk, the risk aversion of the
decision-maker is represented by a power coefficient that resembles a certainty
equivalence, whereas the uncertainty aversion is represented by the
Kullback-Leibler divergence, in which the risk and uncertainty are handled
consistently and separately. The HJB equation includes a new state-dependent
discount factor that arises from the uncertainty aversion, which leads to a
unique, nonlinear, and nonlocal term. The link between the proposed and
classical stochastic control problems is discussed with a focus on
control-dependent discount rates. We propose a finite difference method for
computing the HJB equation. Finally, the proposed model is applied to an
optimal harvesting problem for macrophytes in a brackish lake that contains
both growing and drifting populations
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS
We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite
element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is
accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement