Monte Carlo and quasi-Monte Carlo algorithms for a linear integro-differential equation
We consider a linear Boltzmann-type transport equation in the s-dimensional unit cube I s = [0,1) s. A quasi-Monte Carlo simulation algorithm is described. The equation is approximated by Euler’s method and the simulation makes use of a (0, 2s + 1)-sequence. In addition, we use a technique involving renumbering the simulated particles at every time step. The convergence of the quasi-Monte Carlo simulation is studied. Experimental results are presented for a model problem whose solution can be found analytically. The results show that quasi-Monte Carlo algorithms can produce more efficient solutions than standard Monte Carlo algorithms, for s < 3. Of primary importance appears to be the way to do the renumbering.
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- E. Hlawka, F. Firneis and P. Zinterhof Zahlentheoretische Methoden in der Numerischen Mathematik (R. Oldenburg Verlag, Wien, 1981).Google Scholar
- C. Lécot and I. Coulibaly, A quasi-Monte Carlo scheme using nets for a linear Boltzmann equation, SIAM J. Numer. Anal., to appear.Google Scholar