Reduced-order methods are an attractive model for physical simulation in computer graphics. They aim to reduce the computational cost of a full-rank simulation by project- ing onto a reduced set of bases. However, an improper application of such methods can be unstable and scale poorly, which prevents many methods from being used in practice.

In this work, we aim to improve the scalability and stability of reduced order methods for two such scenarios. For the Eigenfluids method of fluid simulation, we show that by carefully applying the discrete sine and cosine transforms, the prohibitive scaling of its memory usage can be reduced asymptotically. The simulation is further stabilized by using a variational approach with different basis functions. We also show that the basis functions can be made quite general by extending them to spherical and polar coordinate systems. By using an orthogonalization method, we show a wide variety of basis functions can be designed that maintain fast transformations. For stochastic structure optimization, which assesses whether a fabricated object will break under real- world conditions, we show that the computation can be made asymptotically faster by carefully exploiting certain tensor structures. We then stabilize the optimization by applying an alternate basis to the function gradients.