Multiresolution Analysis for Uncertainty Quantification

We survey the application of multiresolution analysis (MRA) methods in uncertainty propagation and quantification problems. The methods are based on the representation of uncertain quantities in terms of a series of orthogonal multiwavelet basis functions. The unknown coefficients in this expansion are then determined through a Galerkin formalism. This is achieved by injecting the multiwavelet representations into the governing system of equations and exploiting the orthogonality of the basis in order to derive suitable evolution equations for the coefficients. Solution of this system of equations yields the evolution of the uncertain solution, expressed in a format that readily affords the extraction of various properties. One of the main features in using multiresolution representations is their natural ability to accommodate steep or discontinuous dependence of the solution on the random inputs, combined with the ability to dynamically adapt the resolution, including basis enrichment and reduction, namely, following the evolution of the surfaces of steep variation or discontinuity. These capabilities are illustrated in light of simulations of simple dynamical system exhibiting a bifurcation and more complex applications to a traffic problem and wave propagation in gas dynamics.