Disentangling Complexity in Bayesian Automatic Adaptive Quadrature

Abstract

The paper describes a Bayesian automatic adaptive quadrature (BAAQ) solution for numerical integration which is simultaneously robust, reliable, and efficient. Detailed discussion is provided of three main factors which contribute to the enhancement of these features: (1) refinement of the m-panel automatic adaptive scheme through the use of integration-domain-length-scale-adapted quadrature sums; (2) fast early problem complexity assessment – enables the non-transitive choice among three execution paths: (i) immediate termination (exceptional cases); (ii) pessimistic – involves time and resource consuming Bayesian inference resulting in radical reformulation of the problem to be solved; (iii) optimistic – asks exclusively for subrange subdivision by bisection; (3) use of the weaker accuracy target from the two possible ones (the input accuracy specifications and the intrinsic integrand properties respectively) – results in maximum possible solution accuracy under minimum possible computing time.