WIAS Preprint No. 483, (1999)

Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem


  • Kurbanmuradov, Orazgeldy
  • Rannik, Ullar
  • Sabelfeld, Karl
  • Vesala, Timo

2010 Mathematics Subject Classification

  • 65C05 76F99 65C20


  • Lagrangian stochastic models, Footprint problem, backward stochastic algorithm, the well-mixed condition


Lagrangian stochastic models and algorithms are constructed and justified for solving the footprint problem, namely, the problem of calculation of the mean concentration and the flux of particles at a fixed point released from a source arbitrarily situated in the space. The direct and adjoint Monte Carlo algorithms are suggested, and rigorous justifications are given. Two different backward trajectory algorithms are considered: Thomson's method and a method based on probabilistic representations of the relevant initial value problem. The cost of the latter algorithm may increase with time, but it allows to treat the general situation when a set of reacting species is scattered by the flow. Thomson's approach is extended to general stochastic differental equations which is especially usefull when it is desired to find a solution at a fixed point, and for large time instances.

Appeared in

  • Monte Carlo Meth. Appl., 5 (1999), No. 2, pp. 85-112

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