Volume 13, Issue 3

Conventional approaches to Monte Carlo experiments involve finding the performance measure of a system to a particular input. Inverse Monte Carlo experiment reverses this and attempts to find the control inputs required to achieve a particular performance measure. Extensive computer processing is needed to find a design parameter value given a desired target for the performance measure of a given system. The designer simulates the process numerically and obtains an approximation for that same output. The goal is to match the numerical and experimental results as closely as possible by varying the values of input parameters in the numerical Monte Carlo experiments. The most obvious difficulty in solving the design problem is that one cannot simply calculate a straightforward solution and be done. Since the output has to be matched by varying the input, an iterative method of solution is implied. This paper proposes a "stochastic approximation" algorithm to estimate the necessary controllable input parameters within a desired accuracy given a target value for the performance function. The proposed solution algorithm is based on Newton's methods using a single-run Monte Carlo experiments approach to estimate the needed derivative. The proposed approach may be viewed as an optimization scheme, where a loss function must be minimized. The solution algorithm properties and the validity of the estimates are examined by applying it to a reliability system with known analytical solutions.

Combined control variates and importance sampling variance reduction and its two-fold optimality are investigated. Two-time-scale stochastic approximation algorithm is applied in parameter search for the combination and almost sure convergence of the algorithm to the unique optimum is proved. The parameter search procedure is further incorporated into adaptive Monte Carlo simulation, and its law of large numbers and central limit theorem are proved to hold. An numerical example is provided to illustrate the effectiveness of the method.

The Δ 2 -distribution is a multivariate distribution, which plays an important role in variance reduction of Monte Carlo integral evaluation. Selecting the nodes of random cubature formulae according to Δ 2 ensures an unbiased and efficient estimate of the studied integral regardless of the region it is solved over. The Δ 2 distribution is also relevant in problems such as separating errors in regression analysis and constructing D-optimal designs in multidimensional regions. Inefficient simulation of Δ 2 prevented the application of the underlying theory in real problems. Ermakov and Missov [S.M. Ermakov and T.I. Missov, On Simulation of the Δ 2 - distribution . Vestnik St. Petersburg University, issue 4 (2005), 123–140.], proposed an algorithm which combines all rejection, inversion, and mixture techniques. Its complexity allows simulating Δ 2 vectors of big lengths. Moreover, it works in the most general settings of the problem of integral evaluation. This article presents a modification of the simulation algorithm as well as its illustration for a popular integral in Reliability Theory.

The paper is devoted to the theoretical study of the so-called spectral test for Linear Congruential Generators (briefly, LCGs) of pseudorandom numbers with moduli 2 p and full periods. Using the technique developed in V. Gerlovina and V. Nekrutkin, Monte Carlo Methods and Appl. 11:2, 2005 , we present several examples of admissible and asymptotical optimal sequences of multiplicators. These examples give rise to well-directed search methods for LCGs with good equidistribution properties.

A threshold secret sharing scheme is one in which a piece of information is shared among a group of t persons such that, any k number of them for k ≤ t pool in their shares to recover the secret. In this paper, we propose a suitable technique for a (2, t )-threshold scheme that is based on a cryptanalytic attack of the Elliptic Curve Discrete Logarithm Problem (ECDLP). This scheme can be used in applications where two or more persons among a group of t ≥ 2 may combine to carry out a critical action such as, opening the door to the bank vault. We illustrate the application of our scheme to ATM cards.