Summation through stochastic drawing of addends under steered morphing

• A tool for stochastic evaluation of nested sums with many indexes is proposed.
• The Jarzynski equality is borrowed from the context of molecular thermodynamics.
• Addends build-up is seen as homologous of driven transformations in real systems.
• Numerical tests demonstrate the high performance of the method.

A stochastic methodology for the numerical estimation of sums over a very large number of addends with huge spread of magnitudes is presented as continuation of our recent work in the field of multidimensional integration. The approach is based on the employment of Jarzynski’s equality, borrowed from the physical context of thermodynamics of small systems (mainly macromolecular) subjected to driven transformations while all uncontrolled degrees of freedom freely fluctuate. An abstract interpretation of such an equality enables us to convert the sum into an exponential average over the “computational work” required to morph the addends from an initial set of values (taken all equal for simplicity) up to their actual values while the summation indexes are stochastically sampled by means of Importance Sampling Monte Carlo moves. A series of numerical tests reveals the high efficiency of the method in performing summations otherwise unfeasible.