An inverse problem of the Lanchester square law in estimating time-dependent attrition coefficients

This paper considers the inverse problem of estimating time-varying attrition coefficients in Lanchester's square law with reinforcements, using observed data on some or all of the battle's strength histories and the reinforcement schedules. The method employed is a nonparametric extension of the parametric conjugate gradient method (P-CGM). We use hypothetical strength histories and reinforcement schedules that are known to be without error at several points in time to illustrate the method. However, the method has application in other circumstances. The problem of estimating the time-dependent attrition coefficients that best fit a set of given strength histories is inherently a nonparametric inverse problem. In this paper we cast it into a nonlinear optimization problem, and show how to solve it numerically by using a nonparametric conjugate gradient method (NP-CGM). Two numerical test cases are provided to illustrate the application of the method.