On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

Numerical Evaluation of Arbitrary Singular Domain Integrals Using Third-Degree B-Spline Basis Functions

A new approach is presented for the numerical evaluation of arbitrary singular domain integrals. In this method, singular domain integrals are transformed into a boundary integral and a radial integral which contains singularities by using the radial integration method. The analytical elimination of singularities condensed in the radial integral formulas can be accomplished by expressing the nonsingular part of the integration kernels as a series of cubic B-spline basis functions of the distancerand using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. A few numerical examples are provided to verify the correctness and robustness of the presented method.

Coefficients of radial integral in the electrostatic interaction and their applications

We have derived the equations to calculate the electrostatic interaction energy and the coefficients of radial integrals between electrons ln or l1m and l′ of the atomic state |ln[S1L1]l1m[S2L2][ScLc]l′ SL >, where ln[S1L1], l1m[S2L2], and l′ are three open shells. The expressions have been checked against the formulas in the literature by reducing them to those for the case of atoms having two open shells. We demonstrate our formulas by evaluating the coefficients of the radial integrals in the interaction between the 2s or 2p4 and 3p electrons of the 2s2p4(2,4P)3p(3S,3P,3D) state of oxygen. Using these coefficients the wave functions and photoionization cross sections of oxygen 2s has been evaluated and compared with previous results. PACS Nos.: 31.15.Ne, 31.10.+z, 32.80.Fb

Mutual Magnetic Interactions and Oscillator Strengths in the First Spectrum of Oxygen

ABSTRACTThe effects of the spin-other-orbit and spin-spin interactions on the energy levels of the 2p3 3s configuration of oxygen are investigated, and they are found to make an appreciable contribution to the fine structure. The interactions are shown to have a negligible effect on the strengths of all the stronger lines of the 2p4-2p3 3s array, and the more important of the weaker transitions are likewise little affected. The value to be adopted for the radial integral involved in the absolute oscillator strengths is discussed, and numerical values of the oscillator strengths are given.