A New Adaptive Method for Geometric Convergence

Summary

Conventional wisdom in solving transport problems is to identify unbiased, low variance (or low variation) estimators to estimate unknown functionals of the solution by Monte Carlo (or quasi-Monte Carlo) algorithms. Our adaptive implementations using this approach have involved the iterative improvement of either an approximate solution obtained through correlated sampling (SCS) or of an approximate importance function (AIS) for the problem. Each of these methods has some drawbacks: for SCS, the (required) estimation of the residual creates various problems and for AIS, sampling from the complex expressions that result from the use of an importance function can be extremely costly. In both of these cases, substantial loss of precision may result. A new adaptive method — generalized weighted analog sampling (GWAS) - combines many of the best features of SCS (simple sampling functions) and AIS (rapid error reduction) and makes use of biased (but asymptotically unbiased) estimators in a very flexible and efficient algorithm. In this work we sketch the needed theory and present numerical results that confirm the potential of the new method, at least for some model transport problems.