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Comment: ICHEP 2002 talk
Comment: Invited talk presented at Mini-Workshop ``Electroweak Physics Data and the Higgs Mass'', DESY Zeuthen, Germany, February 28 - March 1, 2003
A general purpose, self-adapting, Monte Carlo (MC) event generator (simulator) is described. The high efficiency of the MC, that is small maximum weight or variance of the MC weight is achieved by means of dividing the integration domain into small cells. The cells can be $n$-dimensional simplices, hyperrectangles or Cartesian product of them. The grid of cells, called ``foam'', is produced in the process of the binary split of the cells. The choice of the next cell to be divided and the position/direction of the division hyper-plane is driven by the algorithm which optimizes the ratio of the maximum weight to the average weight or (optionally) the total variance. The algorithm is able to deal, in principle, with an arbitrary pattern of the singularities in the distribution. As any MC generator, it can also be used for the MC integra...
I show how to construct Monte Carlo algorithms (programs), prove that they are correct and document them. Complicated algorithms are build using a handful of elementary methods. This construction process is transparently illustrated using graphical representation in which complicated graphs consist of only several elementary building blocks. In particular I discuss the equivalent algorithms, that is different MC algorithms, with different arrangements of the elementary building blocks, which generate the same final probability distribution. I also show how to transform a given MC algorithm into another equivalent one and discuss advantages of the various ``architectures''.
A new general purpose Monte Carlo event generator with self-adapting grid consisting of simplices is described. In the process of initialization, the simplex-shaped cells divide into daughter subcells in such a way that: (a) cell density is biggest in areas where integrand is peaked, (b) cells elongate themselves along hyperspaces where integrand is enhanced/singular. The grid is anisotropic, i.e. memory of the axes directions of the primary reference frame is lost. In particular, the algorithm is capable of dealing with distributions featuring strong correlation among variables (like ridge along diagonal). The presented algorithm is complementary to others known and commonly used in the Monte Carlo event generators. It is, in principle, more effective then any other one for distributions with very complicated patterns of singulariti...
We present the exact and precise (~0.1%) numerical solution of the QCD evolution equations for the parton distributions in a wide range of $Q$ and $x$ using Monte Carlo (MC) method, which relies on the so-called Markovian algorithm. We point out certain advantages of such a method with respect to the existing non-MC methods. We also formulate a challenge of constructing non-Markovian MC algorithm for the evolution equations for the initial state QCD radiation with tagging the type and $x$ of the exiting parton. This seems to be within the reach of the presently available computer CPUs and the sophistication of the MC techniques.
We present the constrained Monte Carlo (CMC) algorithm for the QCD evolution. The constraint resides in that the total longitudinal energy of the emissions in the MC and in the underlying QCD evolution is predefined (constrained). This CMC implements exactly the full DGLAP evolution of the parton distributions in the hadron with respect to the logarithm of the energy scale. The algorithm of the CMC is referred to as the non-Markovian type. The non-Markovian MC algorithm is defined as the one in which the multiplicity of emissions is chosen randomly as the first variable and not the last one, as in the Markovian MC algorithms. The former case resembles that of the fixed-order matrix element calculations. The CMC algorithm can serve as an alternative to the so-called backward evolution Markovian algorithm of Sjostrand, which is used fo...
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