Abstract: We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing high-resolution fluid simulation. We show how to analyze the results of an existing high-resolution simulation, discover an efficient reduced approximation, and use it to quickly "re-simulate" novel variations of the original dynamics. Prior subspace methods have had difficulty re-simulating the original input dynamics because they lack efficient means of handling semi-Lagrangian advection methods. We show that multi-dimensional cubature schemes can be applied to this and other advection methods, such as MacCormack advection. The remaining pressure and diffusion stages can be written as a single matrix-vector multiply, so as with previous subspace methods, no matrix inversion is needed at runtime. We additionally propose a novel importance sampling-based fitting algorithm that asymptotically accelerates the precomputation stage, and show that the Iterated Orthogonal Projection method can be used to elegantly incorporate moving internal boundaries into a subspace simulation. In addition to efficiently producing variations of the original input, our method can produce novel, abstract fluid motions that we have not seen from any other solver.