Mori-Zwanzig Approach to UQ

The Mori-Zwanzig (MZ) approach to uncertainty quantification is a technique from irreversible statistical mechanics that has the potential of overcoming some of the well-known difficulties encountered in numerical simulations of stochastic systems, in particular the curse of dimensionality.

The key idea of MZ  relies on developing exact evolution equations and corresponding numerical methods for quantities of interest, e.g., functionals of the solution to stochastic ordinary and partial differential equations. Such quantities of interest could be low-dimensional objects in infinite dimensional phase spaces, e.g., the lift of an airfoil in a turbulent flow, the local displacement of a structure subject to random loads (e.g., ocean waves loading on an offshore platform) or the macroscopic properties of materials with random micro-structure (e.g., modeled atomistically in terms of particles).

The MZ framework can be developed in two different, although related, mathematical settings: the first one is based on projection operators it yields exact reduced-order equations for the quantity of interest. The second approach relies on effective propagators, i.e., integrals of exponential operators with respect to suitable distributions. Both methods can be applied to nonlinear systems of stochastic ordinary and partial differential equations subject to random forcing terms, random boundary conditions or random initial conditions.

 

                                            

Figure 1. Coarse graining particle systems using the MZ formulation. The microscopic degrees of freedoom associated with each atom are condensed in a smaller number of degrees of freedom (those associated with the big green particles). Coarse graining is not unique and therefore fundamental questions regarding model inadeguacy, selection and validation have to be carefully addressed.

 

 

Publications

  1. D. Venturi, H. Cho and G. E. Karniadakis, ``The Mori-Zwanzig approach to uncertainty quantification'', Springer Handbook for Uncertainty Quantification, 2015. (PDF)
  2. H. Cho, D. Venturi and G. E. Karniadakis, ``Statistical analysis and simulation of random shocks in stochastic Burgers equation'', Proc. R. Soc. A, 2014, 470 (2171), pp. 1-21.  (PDF)
  3. D. Venturi and G. E. Karniadakis, ``Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems'', Proc. R. Soc. A, 2014, 470 (2166), pp. 1-20. (PDF)