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We will introduce Euler-Maruyama approximations given by an orthogonal systemin $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensionalapproximations of SPDEs. In general, the higher the dimension is, the more oneneeds to generate uniform random numbers at every time step in numericalsimulation. The schemes proposed in this paper, in contrast, can deal with thisproblem by generating very few uniform random numbers at every time step. Theschemes save time in the simulation of very high dimensional SDEs. Inparticular, we conclude that an Euler-Maruyama approximation based on the Walshsystem is efficient in high dimensions.

An efficient weak Euler-Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables