Stochastic perturbation approach to the wavelet‐based analysis

SUMMARYThe wavelet-based decomposition of random variables and elds is proposed here in the context ofapplication of the stochastic second order perturbation technique. A general methodology is employedfor the rst two probabilistic moments of a linear algebraic equations system solution, which are obtainedinstead of a single solution projection in the deterministic case. The perturbation approach applicationallows determination of the closed formulas for a wavelet decomposition of random elds. Next, theseformulas are tested by symbolic projection of some elementary random eld. Copyright ? 2004 JohnWiley & Sons, Ltd. KEY WORDS : multiresolutional analysis; wavelet decomposition; stochastic perturbation technique 1. DETERMINISTIC MULTIRESOLUTIONAL ALGORITHM FOR LINEARALGEBRAIC EQUATIONSLet us consider a multiresolutional wavelet-based algorithm and its application in the solutionof the linear algebraic equations system, being a basis for various discrete numerical techniques(as Finite, Boundary Element or Finite Dierence Methods, for instance References [1–3]). Asit is known [4,5], Equation (1) is frequently considered and solved in terms of the randomvariables or elds included in the right hand side (RHS) vector and =or in the left handside (LHS) operator. Then, this equation is solved for probabilistic moments of the outputincluded in the vector q. Numerous solutions have been demonstrated thanks to the applicationof the simulation, perturbation and spectral numerical techniques. Analogous modelling issuesappear when the random behaviour of K and=or f is evident in a few geometrical resolutionsis analysed and then, the wavelet representation with random coecients can be involved [6]to link the randomness with the multiscale character. Some elementary numerical illustrationsare available now, however without any application of the stochastic perturbation technique.