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Abstract The numerical analysis of stochastic parabolic partial differential equations of the form$$\begin{aligned} du + A(u)\, dt = f \,dt + g \, dW, \end{aligned}$$ $\begin{array}{c}du+A\left(u\right)dt=fdt+gdW,\end{array}$is surveyed, whereAis a nonlinear partial operator andWa Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory. 

This article presents a retrospective account of our group’s heterobinuclear (NHC)Cu-[MCO] catalyst design concept (NHC = N-heterocyclic carbene, [MCO] = metal carbonyl anion), the discovery of its application towards UV-light-induced dehydrogenative borylation of unactivated arenes, and the subsequent pursuit of thermal reaction conditions through structural modifications of the catalysts. The account highlights advantages of using a hypothesis-driven catalyst design approach that, while often fruitless with regard to the target transformation in this case, nonetheless vastly expanded the set of heterobinuclear catalysts available for other applications. In other words, curiosity-driven research conducted in a rational manner often provides valuable products with unanticipated applications, even if the primary objective is viewed to have failed. 1 Introduction to Heterobinuclear Catalysts for C–H Borylation 2 Pursuit of Thermal Borylation Conditions 3 Catalysts beyond Copper Carbenes 4 Conclusions