Final Report Summary - SPARCS (Stochastic Parametrizations in Complex Systems) The climate is a complex and complicated system with many interacting components: the atmosphere, oceans, vegetation, ice mass,... Each of these components has an enormous number of degrees of freedom that evolve on a wide range of time scales, from milliseconds to decades. Simulating such systems in all detail is an impossible task. Hence, a number of relevant degrees of freedom needs to be selected, whereas the overall effect of the degrees of freedom that are not explicitly considered on those that are needs to be inserted in the calculations, making use of a so called parametrization scheme. The objectives of the 'Stochastic parameterization of Complex Systems' (SPARCS) project are to obtain a better understanding of the dynamics on large scales of complex dynamical systems, geophysical systems in particular.In this project the fellow has developed new methods for performing model reduction. These methods allow to perform model reduction on systems where it has previously not been possible before.The following significant results have been obtained:The demonstration of significant improvements in performance on various metrics with the fellow's stochastic reduced modeling approach. This includes a better reproduction of rare event statistics compared to other methods.Development of new numerical schemes to efficiently calculate the statistics and dynamics of rare event in complex numerical models. The fellow has initiated a collaboration to apply these methods to a state-of-the-art climate model.Development of a new method of model reduction for system with slow and fast processes, extending the classical theory of homogenization to more realistic cases.The results have been presented at a number of international conferences and have been published in scientific journals.The developments in this project will lead to a mathematically sound methods for performing model reduction of climate models. This is expected to lead to more accurate weather and climate forecasts, which will benefit many economic activities and help plan for climate change.