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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. BabuskaF. 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.

[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.

[3]

A.J. ChorinO.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.

[4]

A.J. ChorinO.H. Hald and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), 239-257.  doi: 10.1016/S0167-2789(02)00446-3.

[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.

[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.

[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.

[8]

R. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-3094-6.

[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.

[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).

[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.

[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.

[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.

[14]

P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Publications, Philadelphia, 1972.

[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.

[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.

[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.

[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).

[19]

F. NobileR. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

I. BabuskaF. 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.

[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.

[3]

A.J. ChorinO.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.

[4]

A.J. ChorinO.H. Hald and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), 239-257.  doi: 10.1016/S0167-2789(02)00446-3.

[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.

[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.

[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.

[8]

R. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-3094-6.

[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.

[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).

[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.

[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.

[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.

[14]

P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Publications, Philadelphia, 1972.

[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.

[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.

[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.

[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).

[19]

F. NobileR. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  Linear ODE: Evolution of the memory kernel $ (Le^{tQL}QLu_1,h^{01}) $ (see text for details)
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)
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)
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)
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)
Figure 6.  Burgers equation: Evolution of the mean of the energy of the solution using only the first two Legendre polynomials
Figure 7.  Burgers equation: Evolution of the standard deviation of the energy of the solution using only the first two Legendre polynomials
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
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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