Stochastic computation based on orthogonal expansion of random fields

In solving stochastic differential equations, recently a random variable based Polynomial Chaos (rv-PC) method has been developed as a major numerical solver. For many realistic random media problems the rv-PC method however confronts a critical challenge of curse-of-dimensionality. Since a random field is represented by random variables, the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse-of-dimensionality. Distinguished from all the random variable based methods, in this study a novel Random Field based Orthogonal Expansion (RF-OE) method is proposed in aim to circumvent the curse-of-dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes, e.g. seismic/ocean wave, wind load, shock wave, and geophysical media. Multiscale modeling of random media problems is selected as the benchmark problem to test the RF-OE method. Especially, the RF-OE method provides a perfect matching with the higher-order Mehler’s formula. By replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field/process, the RF-OE method reduces the number of dimensions of a stochastic differential equation exponentially. In the first example the RF-OE method is verified with Monte Carlo simulation on a lognormal random media flow transport problem. In the second example the RF-OE method is applied to a time domain problem involving orthogonal expansion of random excitations. In the conclusion the items for further development of the RF-OE method are identified.

► Formulate novel random field based expansions.
► Tackle the high dimensional issue.
► Applicable to both spatial and temporal processes.