Abstract
Multidimensional integrals arise in many problems of physics. For example, moments of the distribution function in the problems of transport of various particles (photons, neutrons, etc.) are 6-dimensional integrals. When calculating the coefficients of electrical conductivity and thermal conductivity, scattering integrals arise, the dimension of which is equal to 12. There are also problems with a significantly large number of variables. The Monte Carlo method is the most effective method for calculating integrals of such a high multiplicity. However, the efficiency of this method strongly depends on the choice of a sequence that simulates a set of random numbers. A large number of pseudo-random number generators are described in the literature. Their quality is checked using a battery of formal tests. However, the simplest visual analysis shows that passing such tests does not guarantee good uniformity of these sequences. The magic Sobol points are the most effective for calculating multidimensional integrals. In this paper, an improvement of these sequences is proposed: the shifted magic Sobol points that provide better uniformity of points distribution in a multidimensional cube. This significantly increases the cubature accuracy. A significant difficulty of the Monte Carlo method is a posteriori confirmation of the actual accuracy. In this paper, we propose a multigrid algorithm that allows one to find the grid value of the integral simultaneously with a statistically reliable accuracy estimate. Previously, such estimates were unknown. Calculations of representative test integrals with a high actual dimension up to 16 are carried out. The multidimensional Weierstrass function, which has no derivative at any point, is chosen as the integrand function. These calculations convincingly show the advantages of the proposed methods.
Highlights
Integrals of multivariate functions occur in many areas of physics
The local Monte Carlo method is used for high dimensions (p > 3)
For the local Monte Carlo method, N random points xj are selected in the cube V; in this case, the number N can be arbitrary, in contrast to cubature formulae on regular grids
Summary
Integrals of multivariate functions occur in many areas of physics. Here are some examples. The medium is described by the equation for the distribution function; this function depends on three coordinates of the medium and three components of the particle velocity vector, that is, the number of variables is six. In order to obtain acceptable accuracy, more and more points have to be taken, which makes the calculations exorbitantly laborious and very time consuming. Due to this fact, the local Monte Carlo method is used for high dimensions (p > 3). Calculations of the representative test integrals show that to obtain good accuracy the most important is the uniformity of the points’ distribution and not its randomness. The advantages of the proposed algorithms are illustrated with representative test examples
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