A wavelet-based numerical scheme for stochastic mechanics

Uncertainty is an inherent part of many physical systems. This is often ignored to simplify mathematical models thereby leading to a deterministic treatment of the system. Incorporation of the uncertainty into the model, particularly in the presence of strong correlation across scales is a difficult task for the conventional modeling techniques. This work studies a biorthogonal wavelet framework for the representation of random fields. It is shown that such a representation scheme leads to significantly decorrelated wavelet coefficients. The amount of decorrelation obtained is an improvement over that achieved with orthonormal wavelet basis functions. It is shown that a biorthogonal dual wavelets with sufficient number of vanishing moments and corresponding to a low primal order perform better than Daubechies wavelets at this task. These observations are used in pursuing the development of Wavelet based Galerkin and Petrov-Galerkin schemes for one-dimensional and two-dimensional stochastic mechanics problems