Mori-Zwanzig Approach to Uncertainty Quantification

  • Daniele Venturi
  • Heyrim Cho
  • George Em Karniadakis
Living reference work entry

Later version available View entry history


Determining the statistical properties of nonlinear random systems is a problem of major interest in many areas of physics and engineering. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, low regularity and random frequencies often exhibited by the system. The Mori-Zwanzig and the effective propagator approaches discussed in this chapter have the potential of overcoming some of these limitations, in particular the curse of dimensionality and the lack of regularity. The key idea stems from techniques of irreversible statistical mechanics, and it 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 microstructure (e.g., modeled atomistically in terms of particles). We develop the goal-oriented framework in two different, although related, mathematical settings: the first one is based on the Mori-Zwanzig projection operator method, and 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.


High-dimensional stochastic dynamical systems Probability density function equations Projection operator methods Dimension reduction 


  1. 1.
    Akkermans, R.L.C., Briels, W.J.: Coarse-grained dynamics of one chain in a polymer melt. J. Chem. Phys. 113(15), 620–630 (2000)CrossRefGoogle Scholar
  2. 2.
    Al-Mohy, A.H., Higham, N.J.: Computing the Fréchet derivative of the matrix exponential with an application to condition number estimation. SIAM J. Matrix Anal. Appl. 30(4), 1639–1657 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arai, T., Goodman, B.: Cumulant expansion and Wick theorem for spins. Application to the antiferromagnetic ground state. Phys. Rev. 155(2), 514–527 (1967)Google Scholar
  5. 5.
    Balescu, R.: Equilibrium and Non-equilibrium Statistical Mechanics. Wiley, New York (1975)MATHGoogle Scholar
  6. 6.
    Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A: Math. Gen. 14:L453–L457 (1981)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Billings, S.A.: Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley, Chichester (2013)MATHCrossRefGoogle Scholar
  8. 8.
    Bird, G.A.: Molecular Gas Dynamics and Direct Numerical Simulation of Gas Flows. Clarendon Press, Oxford (1994)Google Scholar
  9. 9.
    Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Blanes, S., Casas, F., Murua, A.: Splitting methods in the numerical integration of non-autonomous dynamical systems. RACSAM 106, 49–66 (2012)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bonatto, C., Gallas, J.A.C., Ueda, Y.: Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator. Phys. Rev. E 77, 026217(1–5) (2008)Google Scholar
  12. 12.
    Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Stat. 38(5), 2916–2957 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Breuer, H.P., Kappler, B., Petruccione, F.: The time-convolutionless projection operator technique in the quantum theory of dissipation and decoherence. Ann. Phys. 291, 36–70 (2001)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Broer, H., Simó, C., Vitolo, R.: Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity 15, 1205–1267 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Casas, F.: Solutions of linear partial differential equations by Lie algebraic methods. J. Comput. Appl. Math. 76, 159–170 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Cercignani, C., Gerasimenko, U.I., Petrina, D.Y. (eds.): Many Particle Dynamics and Kinetic Equations, 1st edn. Kluwer Academic, Dordrecht/Boston (1997)MATHGoogle Scholar
  17. 17.
    Chaturvedi, S., Shibata, F.: Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion. Z. Phys. B 35, 297–308 (1979)MathSciNetGoogle Scholar
  18. 18.
    Cheng, Y., Gamba, I.M., Majorana, A., Shu, C.W.: A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices. Comput. Methods Appl. Mech. Eng. 198, 3130–3150 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Cheng, Y., Gamba, I.M., Majorana, A., Shu, C.W.: A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations. SEMA J. 54, 47–64 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Chertock, A., Gottlieb, D., Solomonoff, A.: Modified optimal prediction and its application to a particle method problem. J. Sci. Comput. 37(2), 189–201 (2008)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Cho, H., Venturi, D., Karniadakis, G.E.: Adaptive discontinuous Galerkin method for response-excitation PDF equations. SIAM J. Sci. Comput. 5(4), B890–B911 (2013)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equations. J. Comput. Phys. Under Rev. (2014)Google Scholar
  23. 23.
    Cho, H., Venturi, D., Karniadakis, G.E.: Statistical analysis and simulation of random shocks in Burgers equation. Proc. R. Soc. A 2171(470), 1–21 (2014)MathSciNetGoogle Scholar
  24. 24.
    Chorin, A., Lu, F.: A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics, pp. 1–12. arXiv:submit/1219662 (2015)Google Scholar
  25. 25.
    Chorin, A.J., Stinis, P.: Problem reduction, renormalization and memory. Commun. Appl. Math. Comput. Sci. 1(1), 1–27 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Chorin, A.J., Tu, X.: Implicit sampling for particle filters. PNAS 106(41), 17249–17254 (2009)CrossRefGoogle Scholar
  27. 27.
    Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction and the Mori-Zwanzig representation of irreversible processes. Proc. Natl. Acad. Sci. U. S. A. 97(7), 2968–2973 (2000)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods, Vol. 11 of Lecture Notes in Computational Science and Engineering. Springer, New York (2000)Google Scholar
  29. 29.
    Darve, E., Solomon, J., Kia, A.: Computing generalized Langevin equations and generalized Fokker-Planck equations. Proc. Natl. Acad. Sci. U. S. A. 106(27), 10884–10889 (2009)CrossRefGoogle Scholar
  30. 30.
    Dekker, H.: Correlation time expansion for multidimensional weakly non-Markovian Gaussian processes. Phys. Lett. A 90(1–2), 26–30 (1982)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Dimarco, G., Paresci, L.: Numerical methods for kinetic equations. Acta Numer. 23(4), 369–520 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Edwards, S.F.: The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18, 239–273 (1964)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)MATHGoogle Scholar
  35. 35.
    Faetti, S., Grigolini, P.: Unitary point of view on the puzzling problem of nonlinear systems driven by colored noise. Phys. Rev. A 36(1), 441–444 (1987)CrossRefGoogle Scholar
  36. 36.
    Faetti, S., Fronzoni, L., Grigolini, P., Mannella, R.: The projection operator approach to the Fokker-Planck equation. I. Colored Gaussian noise. J. Stat. Phys. 52(3/4), 951–978 (1988)MathSciNetMATHGoogle Scholar
  37. 37.
    Faetti, S., Fronzoni, L., Grigolini, P., Palleschi, V., Tropiano, G.: The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise. J. Stat. Phys. 52(3/4), 979–1003 (1988)MathSciNetMATHGoogle Scholar
  38. 38.
    Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Filbet, F., Russo, G.: High-order numerical methods for the space non-homogeneous Boltzmann equations. J. Comput. Phys. 186, 457–480 (2003)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. Proc. Natl. Acad. Sci. U.S.A. 73(2), 309–353 (1988)MathSciNetMATHGoogle Scholar
  41. 41.
    Foias, C., Manley, O.P., Rosa, R., Temam, R.: Navier-Stokes equations and turbulence, 1st edn. Cambridge University Press (2001)Google Scholar
  42. 42.
    Foias, C., Jolly, M.S., Manley, O.P., Rosa, R.: Statistical estimates for the Navier-Stokes equations and Kraichnan theory of 2-D fully developed turbulence. J. Stat. Phys. 108(3/4), 591–646 (2002)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Foo, J., Karniadakis, G.E.: The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 227, 9572–9595 (2008)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Foo, J., Karniadakis, G.E.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229, 1536–1557 (2010)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Fox, R.F.: A generalized theory of multiplicative stochastic processes using Cumulant techniques. J. Math. Phys. 16(2), 289–297 (1975)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Fox, R.F.: Functional-calculus approach to stochastic differential equations. Phys. Rev. A 33(1), 467–476 (1986)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Fox, R.O.: Computational Models for Turbulent Reactive Flows. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  48. 48.
    Friedrich, R., Daitche, A., Kamps, O., Lülff, J., Voβkuhle, M., Wilczek, M.: The Lundgren-Monin-Novikov hierarchy: kinetic equations for turbulence. Comp. Rend. Phys. 13(9–10), 929–953 (2012)Google Scholar
  49. 49.
    Frisch, U.: Turbulence: the legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  50. 50.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1998)MATHGoogle Scholar
  51. 51.
    Hänggi, P.: Correlation functions and master equations of generalized (non-Markovian) Langevin equations. Z. Phys. B 31, 407–416 (1978)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Hänggi, P.: On derivations and solutions of master equations and asymptotic representations. Z. Phys. B 30, 85–95 (1978)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Hänggi, P.: The functional derivative and its use in the description of noisy dynamical systems. In: Pesquera, L., Rodriguez, M. (eds.) Stochastic Processes Applied to Physics, pp. 69–95. World Scientific, Singapore (1985)Google Scholar
  54. 54.
    Hänggi, P., Jung, P.: Colored noise in dynamical systems. In: Prigogine, I., Rice, S.A. (eds.) Advances in Chemical Physics, vol. 89, pp. 239–326. Wiley-Interscience, New York (1995)CrossRefGoogle Scholar
  55. 55.
    Hegerfeldt, G.C., Schulze, H.: Noncommutative cumulants for stochastic differential equations and for generalized Dyson series. J. Stat. Phys. 51(3/4), 691–710 (1988)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Herring, J.R.: Self-consistent-field approach to nonstationary turbulence. Phys. Fluids 9(11), 2106–2110 (1966)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)MATHCrossRefGoogle Scholar
  58. 58.
    Hijón, C., nol, P.E., Vanden-Eijnden, E., Delgado-Buscalioni, R.: Mori-Zwanzig formalism as a practical computational tool. Faraday Discuss 144, 301–322 (2010)CrossRefGoogle Scholar
  59. 59.
    Hosokawa, I.: Monin-Lundgren hierarchy versus the Hopf equation in the statistical theory of turbulence. Phys. Rev. E 73, 067301(1–4) (2006)Google Scholar
  60. 60.
    Hughes, K.H., Burghardt, I.: Maximum-entropy closure of hydrodynamic moment hierarchies including correlations. J. Chem. Phys. 136, 214109(1–18) (2012)Google Scholar
  61. 61.
    Izvekov, S.: Microscopic derivation of particle-based coarse-grained dynamics. J. Chem. Phys. 138, 134106(1–16) (2013)Google Scholar
  62. 62.
    Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, (2003)MATHCrossRefGoogle Scholar
  63. 63.
    Jensen, R.V.: Functional integral approach to classical statistical dynamics. J. Stat. Phys. 25(2), 183–210 (1981)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Kampen, N.G.V.: A cumulant expansion for stochastic linear differential equations. II. Physica 74, 239–247 (1974)MathSciNetGoogle Scholar
  65. 65.
    Kampen, N.G.V.: Elimination of fast variables. Phys. Rep. 124(2), 69–160 (1985)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Kampen, N.G.V.: Stochastic Processes in Physics and Chemistry, 3rd edn. North Holland, Amsterdam (2007)MATHGoogle Scholar
  67. 67.
    Kampen, N.G.V., Oppenheim, I.: Brownian motion as a problem of eliminating fast variables. Physica A 138, 231–248 (1986)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Kanwal, R.P.: Generalized Functions: Theory and Technique, 2nd edn. Birkhäuser, Boston (1998)MATHGoogle Scholar
  69. 69.
    Karimi, A., Paul, M.R.: Extensive chaos in the Lorenz-96 model. Chaos 20(4), 043105(1–11) (2010)Google Scholar
  70. 70.
    Kato, T.: Perturbation Theory for Linear Operators, 4th edn. Springer, New York (1995)MATHGoogle Scholar
  71. 71.
    Khuri, A.I.: Applications of Dirac’s delta function in statistics. Int. J. Math. Educ. Sci. Technol. 35(2), 185–195 (2004)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Kraichnan, R.H.: Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155–175 (1975)MATHCrossRefGoogle Scholar
  73. 73.
    Kubo, R.: Generalized cumulant expansion method. J. Phys. Soc. Jpn. 17(7), 1100–1120 (1962)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Kubo, R.: Stochastic Liouville equations. J. Math. Phys. 4(2), 174–183 (1963)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Kullberg, A., del Castillo-Negrete, D.: Transport in the spatially tempered, fractional Fokker-Planck equation. J. Phys. A: Math. Theor. 45(25), 255101(1–21) (2012)Google Scholar
  76. 76.
    Li, G., Wang, S.W., Rabitz, H., Wang, S., Jaffé, P.: Global uncertainty assessments by high dimensional model representations (HDMR). Chem. Eng. Sci. 57(21), 4445–4460 (2002)CrossRefGoogle Scholar
  77. 77.
    Li, Z., Bian, X., Caswell, B., Karniadakis, G.E.: Construction of dissipative particle dynamics models for complex fluids via the Mori-Zwanzig formulation. Soft. Matter. 10, 8659–8672 (2014)CrossRefGoogle Scholar
  78. 78.
    Lindenberg, K., West, B.J., Masoliver, J.: First passage time problems for non-Markovian processes. In: Moss, F., McClintock, P.V.E. (eds.) Noise in Nonlinear Dynamical Systems, vol. 1, pp. 110–158. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  79. 79.
    Lorenz, E.N.: Predictability – a problem partly solved. In: ECMWF Seminar on Predictability, Reading, vol. 1, pp. 1–18 (1996)Google Scholar
  80. 80.
    Luchtenburg, D.M., Brunton, S.L., Rowley, C.W.: Long-time uncertainty propagation using generalized polynomial chaos and flow map composition. J. Comput. Phys. 274, 783–802 (2014)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Lundgren, T.S.: Distribution functions in the statistical theory of turbulence. Phys. Fluids 10(5), 969–975 (1967)CrossRefGoogle Scholar
  82. 82.
    Luo, X., Zhu, S.: Stochastic resonance driven by two different kinds of colored noise in a bistable system. Phys. Rev. E 67(3/4), 021104(1–13) (2003)Google Scholar
  83. 83.
    Ma, X., Karniadakis, G.E.: A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190 (2002)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation method for the solution of stochastic differential equations. J. Comput. Phys. 228, 3084–3113 (2009)MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    Mattuck, R.D.: A Guide to Feynman Diagrams in the Many-Body Problem. Dover, New York (1992)Google Scholar
  86. 86.
    McCane, A.J., Luckock, H.C., Bray, A.J.: Path integrals and non-Markov processes. 1. General formalism. Phys. Rev. A 41(2), 644–656 (1990)Google Scholar
  87. 87.
    McComb, W.D.: The Physics of Fluid Turbulence. Oxford University Press, Oxford (1990)MATHGoogle Scholar
  88. 88.
    Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Monin, A.S.: Equations for turbulent motion. Prikl. Mat. Mekh. 31(6), 1057–1068 (1967)MATHGoogle Scholar
  90. 90.
    Montgomery, D.: A BBGKY framework for fluid turbulence. Phys. Fluids 19(6), 802–810 (1976)MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    Mori, H.: Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33(3), 423–455 (1965)MATHCrossRefGoogle Scholar
  92. 92.
    Mori, H., Morita, T., Mashiyama, K.T.: Contraction of state variables in non-equilibrium open systems. I. Prog. Theor. Phys. 63(6), 1865–1883 (1980)MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Moss, F., McClintock, P.V.E. (eds.): Noise in Nonlinear Dynamical Systems. Volume 1: Theory of Continuous Fokker-Planck Systems. Cambridge University Press, Cambridge (1995)Google Scholar
  94. 94.
    Mukamel, S., Oppenheim, I., Ross, J.: Statistical reduction for strongly driven simple quantum systems. Phys. Rev. A 17(6), 1988–1998 (1978)MathSciNetCrossRefGoogle Scholar
  95. 95.
    Muradoglu, M., Jenny, P., Pope, S.B., Caughey, D.A.: A consistent hybrid finite-volume/particle method for the PDF equations of turbulent reactive flows. J. Comput. Phys. 154, 342–371 (1999)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Nakajima, S.: On quantum theory of transport phenomena – steady diffusion. Prog. Theor. Phys. 20(6), 948–959 (1958)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Neu, P., Speicher, R.: A self-consistent master equation and a new kind of cumulants. Z. Phys. B 92, 399–407 (1993)MathSciNetCrossRefGoogle Scholar
  98. 98.
    Español, P., Warren, P.: Statistical mechanics of dissipative particle dynamics. EuroPhys. Lett. 30(4), 191–196 (1995)CrossRefGoogle Scholar
  99. 99.
    Noack, B.R., Niven, R.K.: A hierarchy of maximum entropy closures for Galerkin systems of incompressible flows. Comput. Math. Appl. 65(10), 1558–1574 (2012)MathSciNetCrossRefGoogle Scholar
  100. 100.
    Nouy, A.: Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Appl. Mech. Eng. 17, 403–434 (2010)MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Nouy, A., Maître, O.P.L.: Generalized spectral decomposition for stochastic nonlinear problems. J. Comput. Phys. 228, 202–235 (2009)MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Novati, P.: Solving linear initial value problems by Faber polynomials. Numer. Linear Algebra Appl. 10, 247–270 (2003)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Nozaki, D., Mar, D.J., Grigg, P., Collins, J.J.: Effects of colored noise on stochastic resonance in sensory neurons. Phys. Rev. Lett 82(11), 2402–2405 (1999)CrossRefGoogle Scholar
  105. 105.
    O’Brien, E.E.: The probability density function (pdf) approach to reacting turbulent flows. In: Topics in Applied Physics. Turbulent Reacting Flows, vol. 44, pp. 185–218. Springer, Berlin/New York (1980)Google Scholar
  106. 106.
    Orszag, S.A., Bissonnette, L.R.: Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10(12), 2603–2613 (1967)MATHCrossRefGoogle Scholar
  107. 107.
    Pereverzev, A., Bittner, E.R.: Time-convolutionless master equation for mesoscopic electron-phonon systems. J. Chem. Phys. 125, 144107(1–7) (2006)Google Scholar
  108. 108.
    Pesquera, L., Rodriguez, M.A., Santos, E.: Path integrals for non-Markovian processes. Phys. Lett. 94(6–7), 287–289 (1983)MathSciNetCrossRefGoogle Scholar
  109. 109.
    Pope, S.B.: A Monte Carlo method for the PDF equations of turbulent reactive flow. Combust Sci. Technol. 25, 159–174 (1981)CrossRefGoogle Scholar
  110. 110.
    Pope, S.B.: Lagrangian PDF methods for turbulent flows. Ann. Rev. Fluid Mech. 26, 23–63 (1994)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Pope, S.B.: Simple models of turbulent flows. Phys. Fluids 23(1), 011301(1–20) (2011)Google Scholar
  112. 112.
    Rabitz, H., Aliş ÖF, Shorter, J., Shim, K.: Efficient input–output model representations. Comput. Phys. Commun. 117(1–2), 11–20 (1999)MATHCrossRefGoogle Scholar
  113. 113.
    Remacle, J.F., Flaherty, J.E., Shephard, M.S.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45(1), 53–72 (2003)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Remacle, J.F., Flaherty, J.E., Shephard, M.S.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45(1), 53–72 (2003)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Richter, M., Knorr, A.: A time convolution less density matrix approach to the nonlinear optical response of a coupled system-bath complex. Ann. Phys. 325, 711–747 (2010)MATHCrossRefGoogle Scholar
  116. 116.
    Rjasanow, S., Wagner, W.: Stochastic Numerics for the Boltzmann Equation. Springer, Berlin/New York (2004)MATHGoogle Scholar
  117. 117.
    Sapsis, T.P., Lermusiaux, P.F.J.: Dynamically orthogonal field equations for continuous stochastic dynamical systems. Physica D 238(23–24), 2347–2360 (2009)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    Snook, I.: The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems, 1st edn. Elsevier, Amsterdam/Boston (2007)Google Scholar
  119. 119.
    Stinis, P.: A comparative study of two stochastic mode reduction methods. Physica D 213, 197–213 (2006)MathSciNetMATHCrossRefGoogle Scholar
  120. 120.
    Stinis, P.: Mori-Zwanzig-reduced models for systems without scale separation. Proc. R. Soc. A 471, 20140446(1–13) (2015)Google Scholar
  121. 121.
    Stratonovich, R.L.: Topics in the Theory of Random Noise, vols. 1 and 2. Gordon and Breach, New York (1967)Google Scholar
  122. 122.
    Suzuki, M.: Decomposition formulas of exponential operators and Lie exponentials with applications to quantum mechanics and statistical physics. J. Math. Phys. 26(4), 601–612 (1985)MathSciNetMATHCrossRefGoogle Scholar
  123. 123.
    Suzuki, M.: General decomposition theory of ordered exponentials. Proc. Jpn. Acad. B 69(7), 161–166 (1993)CrossRefGoogle Scholar
  124. 124.
    Suzuki, M.: Convergence of general decompositions of exponential operators. Commun. Math. Phys. 163, 491–508 (1994)MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Tartakovsky, D.M., Broyda, S.: PDF equations for advective-reactive transport in heterogeneous porous media with uncertain properties. J. Contam. Hydrol. 120–121, 129–140 (2011)CrossRefGoogle Scholar
  126. 126.
    Terwiel, R.H.: Projection operator method applied to stochastic linear differential equations. Physica 74, 248–265 (1974)MathSciNetCrossRefGoogle Scholar
  127. 127.
    Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)MathSciNetMATHCrossRefGoogle Scholar
  128. 128.
    Turkington, B.: An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics. J. Stat. Phys. 152, 569–597 (2013)MathSciNetMATHCrossRefGoogle Scholar
  129. 129.
    Valino, L.: A field Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent flow. Flow Turbul. Combust. 60(2), 157–172 (1998)MATHCrossRefGoogle Scholar
  130. 130.
    Venkatesh, T.G., Patnaik, L.M.: Effective Fokker-Planck equation: Path-integral formalism. Phys. Rev. E 48(4), 2402–2412 (1993)CrossRefGoogle Scholar
  131. 131.
    Venturi, D.: On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate. J. Fluid Mech. 559, 215–254 (2006)MathSciNetMATHCrossRefGoogle Scholar
  132. 132.
    Venturi, D.: A fully symmetric nonlinear biorthogonal decomposition theory for random fields. Physica D 240(4–5), 415–425 (2011)MathSciNetMATHCrossRefGoogle Scholar
  133. 133.
    Venturi, D.: Conjugate flow action functionals. J. Math. Phys. 54, 113502(1–19) (2013)Google Scholar
  134. 134.
    Venturi, D., Karniadakis, G.E.: Differential constraints for the probability density function of stochastic solutions to the wave equation. Int. J. Uncertain. Quantif. 2(3), 131–150 (2012)MathSciNetMATHCrossRefGoogle Scholar
  135. 135.
    Venturi, D., Karniadakis, G.E.: New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs. J. Comput. Phys. 231, 7450–7474 (2012)MathSciNetMATHCrossRefGoogle Scholar
  136. 136.
    Venturi, D., Karniadakis, G.E.: Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems. Proc. R. Soc. A 470(2166), 1–20 (2014)MathSciNetCrossRefGoogle Scholar
  137. 137.
    Venturi, D., Wan, X., Karniadakis, G.E.: Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder. J. Fluid Mech. 606, 339–367 (2008)MathSciNetMATHCrossRefGoogle Scholar
  138. 138.
    Venturi, D., Wan, X., Karniadakis, G.E.: Stochastic bifurcation analysis of Rayleigh-Bénard convection. J. Fluid Mech. 650, 391–413 (2010)MathSciNetMATHCrossRefGoogle Scholar
  139. 139.
    Venturi, D., Choi, M., Karniadakis, G.E.: Supercritical quasi-conduction states in stochastic Rayleigh-Bénard convection. Int. J. Heat Mass Transf. 55(13–14), 3732–3743 (2012)CrossRefGoogle Scholar
  140. 140.
    Venturi, D., Sapsis, T.P., Cho, H., Karniadakis, G.E.: A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems. Proc. R. Soc. A 468(2139), 759–783 (2012)MathSciNetCrossRefGoogle Scholar
  141. 141.
    Venturi, D., Tartakovsky, D.M., Tartakovsky, A.M., Karniadakis, G.E.: Exact PDF equations and closure approximations for advective-reactive transport. J. Comput. Phys. 243, 323–343 (2013)MathSciNetCrossRefGoogle Scholar
  142. 142.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of mathematical fluid dynamics, Vol I, North-Holland, Amsterdam, pp 73–258 (2002)Google Scholar
  143. 143.
    Viswanath, D.: The fractal property of the lorentz attractor. Physica D 190, 115–128 (2004)MathSciNetMATHCrossRefGoogle Scholar
  144. 144.
    Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209(2), 617–642 (2005)MathSciNetMATHCrossRefGoogle Scholar
  145. 145.
    Wan, X., Karniadakis, G.E.: Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Eng. 195, 5582–5596 (2006)MathSciNetMATHCrossRefGoogle Scholar
  146. 146.
    Wan, X., Karniadakis, G.E.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)MathSciNetMATHCrossRefGoogle Scholar
  147. 147.
    Wang, C.J.: Effects of colored noise on stochastic resonance in a tumor cell growth system. Phys. Scr. 80, 065004 (5pp) (2009)Google Scholar
  148. 148.
    Wei, J., Norman, E.: Lie algebraic solutions of linear differential equations. J. Math. Phys. 4(4), 575–581 (1963)MathSciNetMATHCrossRefGoogle Scholar
  149. 149.
    Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge (2002)Google Scholar
  150. 150.
    Wiebe, N., Berry, D., Høyer, P., Sanders, B.C.: Higher-order decompositions of ordered operator exponentials. J. Phys. A: Math. Theor. 43, 065203(1–20) (2010)Google Scholar
  151. 151.
    Wilcox, R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 399–407 (1967)MathSciNetMATHCrossRefGoogle Scholar
  152. 152.
    Wilczek, M., Daitche, A., Friedrich, R.: On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity. J. Fluid Mech. 676, 191–217 (2011)MathSciNetMATHCrossRefGoogle Scholar
  153. 153.
    Wio, H.S., Colet, P., San Miguel M, Pesquera, L., Rodríguez, M.A.: Path-integral formulation for stochastic processes driven by colored noise. Phys. Rev. A 40(12), 7312–7324 (1989)MathSciNetCrossRefGoogle Scholar
  154. 154.
    Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetMATHCrossRefGoogle Scholar
  155. 155.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)MathSciNetMATHCrossRefGoogle Scholar
  156. 156.
    Yang, Y., Shu, C.W.: Discontinuous Galerkin method for hyperbolic equations involving δ-singularities: negarive-order norma error estimate and applications. Numer. Math. 124, 753–781 (2013)MathSciNetCrossRefGoogle Scholar
  157. 157.
    Yoshimoto, Y., Kinefuchi, I., Mima, T., Fukushima, A., Tokumasu, T., Takagi, S.: Bottom-up construction of interaction models of non-Markovian dissipative particle dynamics. Phys. Rev. E 88, 043305(1–12) (2013)Google Scholar
  158. 158.
    Zwanzig, R.: Ensemble methods in the theory of irreversibility. J. Chem. Phys. 33(5), 1338–1341 (1960)MathSciNetCrossRefGoogle Scholar
  159. 159.
    Zwanzig, R.: Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992 (1961)MATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Daniele Venturi
    • 1
  • Heyrim Cho
    • 2
  • George Em Karniadakis
    • 3
  1. 1.Department of Applied Mathematics and StatisticsUniversity of California Santa CruzSanta CruzUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations