Monte Carlo Techniques for Parametric Finite Multidimensional Integral Equations

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.