Multigrid
Direct multi-grid methods for linear systems with harmonic aliasing patterns
IEEE Transactions on Signal Processing
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We provide an overview of models of vehicular motion that are based on continuous-time Markov chains. Of these models, we concentrate on the subset represented by correlated random walks because they are general enough to capture essential patterns of the mobility of vehicles and simple enough to allow the analytical study of special but still realistic cases. We review the analytical techniques available to obtain stochastic properties of correlated random walks in simple configurations and introduce the general problem of computing statistics of absorbing times. The numerical problem reduces to the solution of sparse linear systems for which we configure and evaluate an algebraic multi-grid technique. We apply the numerical method in a simple 2D example of a correlated random walk that models the mobility of vehicles on a grid of city streets. We consider different configurations of absorbing states and obtain approximations of the expected value of absorbing times for arbitrary initial conditions. The approach works directly with the analytic expression for expected values and thus does not rely on ergodic assumptions.