Mori-Zwanzig reduced models for uncertainty quantification

Jing Li; Panos Stinis

doi:10.3934/jcd.2019002

Article Contents

2019, Volume 6, Issue 1: 39-68. Doi: 10.3934/jcd.2019002

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Mori-Zwanzig reduced models for uncertainty quantification

Jing Li and
Panos Stinis^,

Pacific Northwest National Laboratory, Richland, WA 99354, USA

^* Corresponding author: Panos Stinis
^* Corresponding author: Panos Stinis

Published: August 2019

This research at Pacific Northwest National Laboratory (PNNL) was partially supported by the U.S. Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4), under Award Number DE-SC0009280 and partially by the U.S. DOE ASCR project "Uncertainty Quantification For Complex Systems Described by Stochastic Partial Differential Equations". PNNL is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.

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Abstract

In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions.

Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which is better suited for reduced models with long memory. The reformulation can be used as a starting point for approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.

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Mathematics Subject Classification: Primary: 65C20, 65M99; Secondary: 41A10.

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Figure 1. Linear ODE: Evolution of the memory kernel $ (Le^{tQL}QLu_1,h^{01}) $ (see text for details)

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Figure 2. Linear ODE: Evolution of the resolved variables $ u_0,u_1 $ predicted by the full model (black line), the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)

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Figure 3. Linear ODE: Relative error with respect to the true solution for the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)

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Figure 4. Nonlinear ODE: Evolution of the resolved variables $ u_0,u_1 $ predicted by the full model (black line), the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)

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Figure 5. Nonlinear ODE: Logarithimic scale relative error for $ u_0,u_1 $ with respect to the true solution for the (Markovian) reduced model without memory (blue line) and the (non-Markovian) reduced model with memory (red line)

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Figure 6. Burgers equation: Evolution of the mean of the energy of the solution using only the first two Legendre polynomials

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Figure 7. Burgers equation: Evolution of the standard deviation of the energy of the solution using only the first two Legendre polynomials

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Figure 8. Burgers equation: Evolution of the mean of the squared $ l_2 $ norm of the gradient of the solution calculated using only the first two Legendre polynomials

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Figure 9. Burgers equation: Evolution of the standard deviation of the squared $ l_2 $ norm of the gradient of the solution calculated using only the first two Legendre polynomials

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