Graham, C.; Meleard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob.25, 115-132 (1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Monte Carlo or spectral methods. Using stochastic or Monte Carlo methods, the distribution function is represented by pseudo particles. The numerical method is based on 7 a time splitting: one time step solving the transport part, one time step for collisions. In the Boltzmann case (see [3, 9]) the method consists in performing a change of the particles velocity (choosing a post collision velocity uniformly on the sphere) according to a random variable related to the collision time. It can be checked that the probability P of a particle exiting the domain without changing its velocity ....

Graham, C. and Meleard, S. (1995). Stochastic particle approximations for generalized Boltzmann models and convergence estimates. The Annals of Probability 5, N.3, 666-680.

....convergence of the correlation functions is proven in 47 L1 norms. On the other hand, the derivation of the Vlasov equation is based on the use of the variation norm and we have not been able to find a norm suited for both terms. The only related result, as far as we know, has been obtained in [GM] and is about a stochastic particle systems converging to a Vlasov Boltzmann equation with a modified Boltzmann kernel (Povzner) The proof is based on martingale methods. In the linear case of a Lorentz gas with a Kac potential term it is possible to prove the convergence to a Boltzmann equation ....

C. Graham, S. Meleard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, The Annals of probability 25, 115--132 (1997).

....viewpoint of their computational complexity. Note that these results are valid for more general non homogeneous Boltzmann like equations, where particles are moving between jumps with trajectories guided by general Markov families. As for functional analogies of (1. 2) they may be found in [14] [11] and [15] where the strong convergence in the space of signed measures on the corresponding Skorohod space as well as the rate of such a convergence is investigated. The polynomial structure of B is considerably used by all these approaches. For linear equations, we may gather up the independent ....

Graham, C. and M'el'eard, S. (1997) Stochastic particle approximation for generalized Boltzmann models and convergence estimates. Ann. Prob., 25, 115--132.

....homogeneous Boltzmann equation of Maxwellian molecules without cuto is related to a Poisson driven stochastic di erential equation. Using this tool, the convergence to fP t g t of solutions fP l t g t of approximating Boltzmann equations with cuto is proved. Then, a result of Graham M el eard, [6] is used, and allows to approximate fP l t g t with the empirical measure f l;n t g t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper. Key words : Boltzmann equations without cuto , Stochastic ....

....by this approach the existence and regularity of a function valued solution, as in dimension 1 or 2. Next we approximate the law of the stochastic process by simulable interacting particle systems, proving a generalized law of large numbers on a path space. We use a result of Graham M el eard [6] who discuss this problem in a general context, but in a cuto case. Thus we consider rst cuto approximations of our model and associate with each cuto model some cuto approximating interacting particle systems. We prove the convergence of the cuto model to the model without cuto . We obtain ....

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Graham, C.; Meleard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob.25, 115-132 (1997).

....In [18] the jump measure has a one order moment and one can directly use Poisson point processes. As in [4] we approximate the law of the stochastic process by simulable interacting particle systems, proving a generalized law of large numbers on a path space. We use a result of Graham M el eard [9] who discuss this problem in a general context, but in a cuto case. We consider rst cuto approximations of our model and associate with each cuto equation some cuto approximating interacting particle systems. The cuto model converges to the model without cuto with an easily computable rate ....

....be found in M el eard [12] Section 4, and as it will be developed below. The convergence is then understood as a convergence in law, in the path space ID( 0; T ] IR 2 ) Here, because the dynamics is just a jump dynamics, one can prove a stronger approximation result, due to Graham M el eard [9] Theorem 3.1. For a given T 0, let us denote by j:j T the total variation norm in the space of signed measures on ID( 0; T ] IR 2 ) Then we have a propagation of chaos result in variation norm. Theorem 3.1 Let (V i 0 ) i 1 be i.i.d. P 0 distributed random variables. For given T 0 and ....

Graham, C.; Meleard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob.25, 115-132 (1997).

....above derivation rigorous is to establish property (3. 3) For the case of standard DSMC (Y j1 ; oe =0) this was done in [16] General results for stochastic systems with Boltzmann type interaction were obtained in [12] Some results covering Vlasov type terms (like the Y factor) can be found in [7]. We refer to [17] concerning historical comments and an extended reference list. 4. Transformation of the limiting equation Assume the measures have densities P (t; dx; dv) p(t; x; v) dx dv ; and denote fi(x) Y Z R 3 g(x; u) t; u) du ; 4.1) where (t; x) Z R 3 p(t; x; ....

C. Graham and S. M'el'eard. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab., 25(1):115--132, 1997.

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Graham, C.; Meleard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob.25, 115-132 (1997).

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Graham, C.; Meleard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Prob. 25, 115-132 (1997).

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C. Graham and S. Meleard, Stochastic Particle Approximations for Generalized Boltzmann Models and Convergence Estimates, The Annals of Probability, vol. 25, no. 1, pp. 115--132, 1997.

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C. Graham and S. Meleard. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. The Annals of Probability, 25(1):115--132, 1997. 30

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