A one-time truncate and encode multiresolution stochastic framework

Abstract

In this work a novel adaptive strategy for stochastic problems, inspired to the classical Harten's framework, is presented. The proposed algorithm allows building, in a very general manner, stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems, from unsteady to discontinuous solutions. Its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach, but with an extension to uncertainty quantification problems. The interest of the present strategy is demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account, such as discontinuous and unsteady custom-defined probability density functions. In addition to algebraic and ordinary differential equations, numerical results for the challenging 1D Kraichnan-Orszag are reported in terms of accuracy and convergence. Finally, a two degree-of-freedom aeroelastic model for a subsonic case is presented. Though quite simple, the model allows recovering some physical key aspect, on the fluid/structure interaction, thanks to the quasi-steady aerodynamic approximation employed. The injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix. All the numerical results are compared with respect to classical Monte Carlo solution and with a non-intrusive Polynomial Chaos method.

Stochastic Discrete Equation Method (sDEM) for two-phase flows

“…the probability density function p(ξ). This distribution is considered uniform in this work, without loss of generality of the approach as already demonstrated in [29,42]. The proposed stochastic method belongs to the so-called intrusive class of method for UQ.…”

“…The fundamental brick is the Truncate and Encode (TE) multiresolution framework presented in [42,48]. The TE framework is employed in order to perform a hierarchical refinement of the stochastic space employing the wavelets as error estimators.…”

“…The TE framework is employed in order to perform a hierarchical refinement of the stochastic space employing the wavelets as error estimators. The TE framework [42,48] has been successfully extended to the analysis of differential equations in presence of uncertainty [49][50][51][52] when the point-value multiresolution setting is applied. The cell-average setting can be also applied.…”

“…In this work the law of variation of the threshold is defined as ε k = ε/2 L−k . Convergence properties of the TE scheme, even if in the point-value MR setting, are provided in [42].…”

An adaptive multiresolution semi‐intrusive scheme for UQ in compressible fluid problems

“…In Section 2, the mathematical setting for the stochastic differential equation, here of interest, is given. Section 3 illustrates the multi‐resolution framework adopted, the TE , where a cell‐average setting is chosen and the non‐linear high‐order reconstruction is performed. In particular, the TE algorithm is presented in Section 3.2, where the representation of the discrete data is obtained, moving from the coarsest level towards the finest.…”