We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.

Cellular complexity is unraveled at nanometer resolution using electron microscopy (EM), but interpretation of macromolecular functionality is hampered by the difficulty in interpreting grey-scale images and the unidentified molecular content. We perform large-scale EM on mammalian tissue complemented with energy-dispersive X-ray analysis (EDX) to allow EM-data analysis based on elemental composition. Endogenous elements, labels (gold and cadmium-based nanoparticles) as well as stains are analyzed at ultrastructural resolution. This provides a wide palette of colors to paint the traditional grey-scale EM images for composition-based interpretation. Our proof-of-principle application of EM-EDX reveals that endocrine and exocrine vesicles exist in single cells in Islets of Langerhans. This highlights how elemental mapping reveals unbiased biomedical relevant information. Broad application of EM-EDX will further allow experimental analysis on large-scale tissue using endogenous elements, multiple stains, and multiple markers and thus brings nanometer-scale 'color-EM' as a promising tool to unravel molecular (de)regulation in biomedicine.