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June  2019, 6(1): 39-68. doi: 10.3934/jcd.2019002

## Mori-Zwanzig reduced models for uncertainty quantification

 Pacific Northwest National Laboratory, Richland, WA 99354, USA

* Corresponding author: Panos Stinis

Published  February 2019

Fund Project: 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.

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.

Citation: Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2019, 6 (1) : 39-68. doi: 10.3934/jcd.2019002
##### References:
 [1] I. Babuska, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.  Google Scholar [2] D.A. Barajas-Solano and A.M. Tartakovsky, Probabilistic density function method for nonlinear dynamical systems driven by colored noise, Phys. Rev. E 93 (2016). doi: 10.1103/physreve.93.052121.  Google Scholar [3] A.J. Chorin, O.H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Nat. Acad. Sci. USA, 97 (2000), 2968-2973.  doi: 10.1073/pnas.97.7.2968.  Google Scholar [4] A.J. Chorin, O.H. Hald and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), 239-257.  doi: 10.1016/S0167-2789(02)00446-3.  Google Scholar [5] A.J. Chorin and P. Stinis, Problem reduction, renormalization and memory, Comm. App. Math. Comp. Sci., 1 (2005), 1-27.  doi: 10.2140/camcos.2006.1.1.  Google Scholar [6] A. Doostan A and H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230 (2011), 3015-3034.  doi: 10.1016/j.jcp.2011.01.002.  Google Scholar [7] J. Foo and G.E. Karniadakis, Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229 (2010), 1536-1557.  doi: 10.1016/j.jcp.2009.10.043.  Google Scholar [8] R. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-3094-6.  Google Scholar [9] D. Given, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity 17 (2004), R55–R127. doi: 10.1088/0951-7715/17/6/R01.  Google Scholar [10] M. Gupta and S.G. Narasimhan, Legendre polynomials Triple Product Integral and lower-degree approximation of polynomials using Chebyshev polynomials, Technical Report - CMU-RI-TR-07-22, Carnegie Mellon (2007). Google Scholar [11] O.H. Hald and P. Stinis, Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions, Proc. Nat. Acad. Sci. USA, 104 (2007), 6527-6532.  doi: 10.1073/pnas.0700084104.  Google Scholar [12] T.Y. Hou, W. Luo, B. Rozovskii and H.M. Zhou, Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comput. Phys. 216 (2006) 687–706. doi: 10.1016/j.jcp.2006.01.008.  Google Scholar [13] J.D. Jakeman and S.G. Roberts, Local and dimension adaptive stochastic collocation for uncertainty quantification, in Sparse grids and applications, Springer (2013), 181–203.  Google Scholar [14] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Publications, Philadelphia, 1972.  Google Scholar [15] G. Leonenko and T. Phillips, On the solution of the Fokker-Planck equation using a high-order reduced basis approximation, Comput. Methods Appl. Mech. Engrg., 199 (2009), 158-168.  doi: 10.1016/j.cma.2009.09.028.  Google Scholar [16] J. Li and P. Stinis, A unified framework for mesh refinement in random and physical space, J. Comp. Phys., 323 (2016), 243-264.  doi: 10.1016/j.jcp.2016.07.027.  Google Scholar [17] X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation method for the solution of stochastic differential equations, J. Comput. Phys., 228 (2009), 3084-3113.  doi: 10.1016/j.jcp.2009.01.006.  Google Scholar [18] L. Mathelin and M.Y. Hussaini, A stochastic collocation algorithm for uncertainty analysis, Technical Report NASA/CR-2003-212153, NASA Langley Research Center, (2003). Google Scholar [19] F. Nobile, R. Tempone and C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345.  doi: 10.1137/060663660.  Google Scholar [20] A. Nouy and O.P. Le Maître, Generalized spectral decomposition for stochastic nonlinear problems, J. Comp. Phys., 228 (2009), 202-235.  doi: 10.1016/j.jcp.2008.09.010.  Google Scholar [21] P. Stinis, A phase transition approach to detecting singularities of PDEs, Comm. App. Math. Comp. Sci., 4 (2009), 217-239.  doi: 10.2140/camcos.2009.4.217.  Google Scholar [22] P. Stinis, Renormalized reduced models for singular PDEs, Comm. App. Math. Comp. Sci., 8 (2013), 39-66.  doi: 10.2140/camcos.2013.8.39.  Google Scholar [23] P. Stinis, Renormalized Mori-Zwanzig reduced models for systems without scale separation, Proc. Roy. Soc. A 471 (2015). doi: 10.1098/rspa.2014.0446.  Google Scholar [24] D. Venturi, A fully symmetric nonlinear biorthogonal decomposition theory for random fields, Physica D, 240 (2011), 415-425.  doi: 10.1016/j.physd.2010.10.005.  Google Scholar [25] D. Venturi, H. Cho and G.E. Karniadakis, The Mori-Zwanzig approach to uncertainty quantification, in Handbook for Uncertainty Quantification, Springer (2016), 1–36. doi: 10.1007/978-3-319-11259-6_28-2.  Google Scholar [26] X. Wan and G.E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), 901-928.  doi: 10.1137/050627630.  Google Scholar [27] D. Xiu and J.S. Hesthaven, High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139.  doi: 10.1137/040615201.  Google Scholar [28] D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.  Google Scholar

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##### References:
 [1] I. Babuska, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.  Google Scholar [2] D.A. Barajas-Solano and A.M. Tartakovsky, Probabilistic density function method for nonlinear dynamical systems driven by colored noise, Phys. Rev. E 93 (2016). doi: 10.1103/physreve.93.052121.  Google Scholar [3] A.J. Chorin, O.H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Nat. Acad. Sci. USA, 97 (2000), 2968-2973.  doi: 10.1073/pnas.97.7.2968.  Google Scholar [4] A.J. Chorin, O.H. Hald and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), 239-257.  doi: 10.1016/S0167-2789(02)00446-3.  Google Scholar [5] A.J. Chorin and P. Stinis, Problem reduction, renormalization and memory, Comm. App. Math. Comp. Sci., 1 (2005), 1-27.  doi: 10.2140/camcos.2006.1.1.  Google Scholar [6] A. Doostan A and H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230 (2011), 3015-3034.  doi: 10.1016/j.jcp.2011.01.002.  Google Scholar [7] J. Foo and G.E. Karniadakis, Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229 (2010), 1536-1557.  doi: 10.1016/j.jcp.2009.10.043.  Google Scholar [8] R. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-3094-6.  Google Scholar [9] D. Given, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity 17 (2004), R55–R127. doi: 10.1088/0951-7715/17/6/R01.  Google Scholar [10] M. Gupta and S.G. Narasimhan, Legendre polynomials Triple Product Integral and lower-degree approximation of polynomials using Chebyshev polynomials, Technical Report - CMU-RI-TR-07-22, Carnegie Mellon (2007). Google Scholar [11] O.H. Hald and P. Stinis, Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions, Proc. Nat. Acad. Sci. USA, 104 (2007), 6527-6532.  doi: 10.1073/pnas.0700084104.  Google Scholar [12] T.Y. Hou, W. Luo, B. Rozovskii and H.M. Zhou, Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comput. Phys. 216 (2006) 687–706. doi: 10.1016/j.jcp.2006.01.008.  Google Scholar [13] J.D. Jakeman and S.G. Roberts, Local and dimension adaptive stochastic collocation for uncertainty quantification, in Sparse grids and applications, Springer (2013), 181–203.  Google Scholar [14] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Publications, Philadelphia, 1972.  Google Scholar [15] G. Leonenko and T. Phillips, On the solution of the Fokker-Planck equation using a high-order reduced basis approximation, Comput. Methods Appl. Mech. Engrg., 199 (2009), 158-168.  doi: 10.1016/j.cma.2009.09.028.  Google Scholar [16] J. Li and P. Stinis, A unified framework for mesh refinement in random and physical space, J. Comp. Phys., 323 (2016), 243-264.  doi: 10.1016/j.jcp.2016.07.027.  Google Scholar [17] X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation method for the solution of stochastic differential equations, J. Comput. Phys., 228 (2009), 3084-3113.  doi: 10.1016/j.jcp.2009.01.006.  Google Scholar [18] L. Mathelin and M.Y. Hussaini, A stochastic collocation algorithm for uncertainty analysis, Technical Report NASA/CR-2003-212153, NASA Langley Research Center, (2003). Google Scholar [19] F. Nobile, R. Tempone and C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345.  doi: 10.1137/060663660.  Google Scholar [20] A. Nouy and O.P. Le Maître, Generalized spectral decomposition for stochastic nonlinear problems, J. Comp. Phys., 228 (2009), 202-235.  doi: 10.1016/j.jcp.2008.09.010.  Google Scholar [21] P. Stinis, A phase transition approach to detecting singularities of PDEs, Comm. App. Math. Comp. Sci., 4 (2009), 217-239.  doi: 10.2140/camcos.2009.4.217.  Google Scholar [22] P. Stinis, Renormalized reduced models for singular PDEs, Comm. App. Math. Comp. Sci., 8 (2013), 39-66.  doi: 10.2140/camcos.2013.8.39.  Google Scholar [23] P. Stinis, Renormalized Mori-Zwanzig reduced models for systems without scale separation, Proc. Roy. Soc. A 471 (2015). doi: 10.1098/rspa.2014.0446.  Google Scholar [24] D. Venturi, A fully symmetric nonlinear biorthogonal decomposition theory for random fields, Physica D, 240 (2011), 415-425.  doi: 10.1016/j.physd.2010.10.005.  Google Scholar [25] D. Venturi, H. Cho and G.E. Karniadakis, The Mori-Zwanzig approach to uncertainty quantification, in Handbook for Uncertainty Quantification, Springer (2016), 1–36. doi: 10.1007/978-3-319-11259-6_28-2.  Google Scholar [26] X. Wan and G.E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), 901-928.  doi: 10.1137/050627630.  Google Scholar [27] D. Xiu and J.S. Hesthaven, High-Order Collocation Methods for Differential Equations with Random Inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139.  doi: 10.1137/040615201.  Google Scholar [28] D. Xiu and G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.  Google Scholar
Linear ODE: Evolution of the memory kernel $(Le^{tQL}QLu_1,h^{01})$ (see text for details)
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)
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)
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)
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)
Burgers equation: Evolution of the mean of the energy of the solution using only the first two Legendre polynomials
Burgers equation: Evolution of the standard deviation of the energy of the solution using only the first two Legendre polynomials
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
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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