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Mori-Zwanzig Approach to Uncertainty Quantification

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Handbook of Uncertainty Quantification

Abstract

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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Balescu, R.: Equilibrium and Non-equilibrium Statistical Mechanics. Wiley, New York (1975)

    MATH  Google Scholar 

  6. Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A: Math. Gen. 14:L453–L457 (1981)

    Article  MathSciNet  Google Scholar 

  7. Billings, S.A.: Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley, Chichester (2013)

    Book  MATH  Google Scholar 

  8. Bird, G.A.: Molecular Gas Dynamics and Direct Numerical Simulation of Gas Flows. Clarendon Press, Oxford (1994)

    Google Scholar 

  9. Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)

    Article  MathSciNet  Google Scholar 

  10. Blanes, S., Casas, F., Murua, A.: Splitting methods in the numerical integration of non-autonomous dynamical systems. RACSAM 106, 49–66 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Stat. 38(5), 2916–2957 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Casas, F.: Solutions of linear partial differential equations by Lie algebraic methods. J. Comput. Appl. Math. 76, 159–170 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cercignani, C., Gerasimenko, U.I., Petrina, D.Y. (eds.): Many Particle Dynamics and Kinetic Equations, 1st edn. Kluwer Academic, Dordrecht/Boston (1997)

    MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equations. J. Comput. Phys. Under Rev. (2014)

    MATH  Google Scholar 

  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)

    Google Scholar 

  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. Chorin, A.J., Stinis, P.: Problem reduction, renormalization and memory. Commun. Appl. Math. Comput. Sci. 1(1), 1–27 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chorin, A.J., Tu, X.: Implicit sampling for particle filters. PNAS 106(41), 17249–17254 (2009)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  30. Dekker, H.: Correlation time expansion for multidimensional weakly non-Markovian Gaussian processes. Phys. Lett. A 90(1–2), 26–30 (1982)

    Article  MathSciNet  Google Scholar 

  31. Dimarco, G., Paresci, L.: Numerical methods for kinetic equations. Acta Numer. 23(4), 369–520 (2014)

    Article  MathSciNet  Google Scholar 

  32. Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Edwards, S.F.: The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18, 239–273 (1964)

    Article  MathSciNet  Google Scholar 

  34. Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  38. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  39. Filbet, F., Russo, G.: High-order numerical methods for the space non-homogeneous Boltzmann equations. J. Comput. Phys. 186, 457–480 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  41. Foias, C., Manley, O.P., Rosa, R., Temam, R.: Navier-Stokes equations and turbulence, 1st edn. Cambridge University Press (2001)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Foo, J., Karniadakis, G.E.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229, 1536–1557 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Fox, R.F.: A generalized theory of multiplicative stochastic processes using Cumulant techniques. J. Math. Phys. 16(2), 289–297 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  46. Fox, R.F.: Functional-calculus approach to stochastic differential equations. Phys. Rev. A 33(1), 467–476 (1986)

    Article  MathSciNet  Google Scholar 

  47. Fox, R.O.: Computational Models for Turbulent Reactive Flows. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  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. Frisch, U.: Turbulence: the legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  50. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1998)

    MATH  Google Scholar 

  51. Hänggi, P.: Correlation functions and master equations of generalized (non-Markovian) Langevin equations. Z. Phys. B 31, 407–416 (1978)

    Article  MathSciNet  Google Scholar 

  52. Hänggi, P.: On derivations and solutions of master equations and asymptotic representations. Z. Phys. B 30, 85–95 (1978)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  56. Herring, J.R.: Self-consistent-field approach to nonstationary turbulence. Phys. Fluids 9(11), 2106–2110 (1966)

    Article  MATH  Google Scholar 

  57. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. Izvekov, S.: Microscopic derivation of particle-based coarse-grained dynamics. J. Chem. Phys. 138, 134106(1–16) (2013)

    Google Scholar 

  62. Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, (2003)

    Book  MATH  Google Scholar 

  63. Jensen, R.V.: Functional integral approach to classical statistical dynamics. J. Stat. Phys. 25(2), 183–210 (1981)

    Article  MathSciNet  Google Scholar 

  64. Kampen, N.G.V.: A cumulant expansion for stochastic linear differential equations. II. Physica 74, 239–247 (1974)

    MathSciNet  Google Scholar 

  65. Kampen, N.G.V.: Elimination of fast variables. Phys. Rep. 124(2), 69–160 (1985)

    Article  MathSciNet  Google Scholar 

  66. Kampen, N.G.V.: Stochastic Processes in Physics and Chemistry, 3rd edn. North Holland, Amsterdam (2007)

    MATH  Google Scholar 

  67. Kampen, N.G.V., Oppenheim, I.: Brownian motion as a problem of eliminating fast variables. Physica A 138, 231–248 (1986)

    Article  MathSciNet  Google Scholar 

  68. Kanwal, R.P.: Generalized Functions: Theory and Technique, 2nd edn. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  69. Karimi, A., Paul, M.R.: Extensive chaos in the Lorenz-96 model. Chaos 20(4), 043105(1–11) (2010)

    Google Scholar 

  70. Kato, T.: Perturbation Theory for Linear Operators, 4th edn. Springer, New York (1995)

    Book  MATH  Google Scholar 

  71. Khuri, A.I.: Applications of Dirac’s delta function in statistics. Int. J. Math. Educ. Sci. Technol. 35(2), 185–195 (2004)

    Article  MathSciNet  Google Scholar 

  72. Kraichnan, R.H.: Statistical dynamics of two-dimensional flow. J. Fluid Mech. 67, 155–175 (1975)

    Article  MATH  Google Scholar 

  73. Kubo, R.: Generalized cumulant expansion method. J. Phys. Soc. Jpn. 17(7), 1100–1120 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  74. Kubo, R.: Stochastic Liouville equations. J. Math. Phys. 4(2), 174–183 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  79. Lorenz, E.N.: Predictability – a problem partly solved. In: ECMWF Seminar on Predictability, Reading, vol. 1, pp. 1–18 (1996)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  81. Lundgren, T.S.: Distribution functions in the statistical theory of turbulence. Phys. Fluids 10(5), 969–975 (1967)

    Article  Google Scholar 

  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. Ma, X., Karniadakis, G.E.: A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  85. Mattuck, R.D.: A Guide to Feynman Diagrams in the Many-Body Problem. Dover, New York (1992)

    Google Scholar 

  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. McComb, W.D.: The Physics of Fluid Turbulence. Oxford University Press, Oxford (1990)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  89. Monin, A.S.: Equations for turbulent motion. Prikl. Mat. Mekh. 31(6), 1057–1068 (1967)

    MATH  Google Scholar 

  90. Montgomery, D.: A BBGKY framework for fluid turbulence. Phys. Fluids 19(6), 802–810 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  91. Mori, H.: Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33(3), 423–455 (1965)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Mukamel, S., Oppenheim, I., Ross, J.: Statistical reduction for strongly driven simple quantum systems. Phys. Rev. A 17(6), 1988–1998 (1978)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  96. Nakajima, S.: On quantum theory of transport phenomena – steady diffusion. Prog. Theor. Phys. 20(6), 948–959 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  97. Neu, P., Speicher, R.: A self-consistent master equation and a new kind of cumulants. Z. Phys. B 92, 399–407 (1993)

    Article  MathSciNet  Google Scholar 

  98. Español, P., Warren, P.: Statistical mechanics of dissipative particle dynamics. EuroPhys. Lett. 30(4), 191–196 (1995)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  101. Nouy, A., Maître, O.P.L.: Generalized spectral decomposition for stochastic nonlinear problems. J. Comput. Phys. 228, 202–235 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  102. Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  103. Novati, P.: Solving linear initial value problems by Faber polynomials. Numer. Linear Algebra Appl. 10, 247–270 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Orszag, S.A., Bissonnette, L.R.: Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10(12), 2603–2613 (1967)

    Article  MATH  Google Scholar 

  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. Pesquera, L., Rodriguez, M.A., Santos, E.: Path integrals for non-Markovian processes. Phys. Lett. 94(6–7), 287–289 (1983)

    Article  MathSciNet  Google Scholar 

  109. Pope, S.B.: A Monte Carlo method for the PDF equations of turbulent reactive flow. Combust Sci. Technol. 25, 159–174 (1981)

    Article  Google Scholar 

  110. Pope, S.B.: Lagrangian PDF methods for turbulent flows. Ann. Rev. Fluid Mech. 26, 23–63 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  111. Pope, S.B.: Simple models of turbulent flows. Phys. Fluids 23(1), 011301(1–20) (2011)

    Google Scholar 

  112. Rabitz, H., Aliş ÖF, Shorter, J., Shim, K.: Efficient input–output model representations. Comput. Phys. Commun. 117(1–2), 11–20 (1999)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  116. Rjasanow, S., Wagner, W.: Stochastic Numerics for the Boltzmann Equation. Springer, Berlin/New York (2004)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Stinis, P.: A comparative study of two stochastic mode reduction methods. Physica D 213, 197–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  120. Stinis, P.: Mori-Zwanzig-reduced models for systems without scale separation. Proc. R. Soc. A 471, 20140446(1–13) (2015)

    Google Scholar 

  121. Stratonovich, R.L.: Topics in the Theory of Random Noise, vols. 1 and 2. Gordon and Breach, New York (1967)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  123. Suzuki, M.: General decomposition theory of ordered exponentials. Proc. Jpn. Acad. B 69(7), 161–166 (1993)

    Article  Google Scholar 

  124. Suzuki, M.: Convergence of general decompositions of exponential operators. Commun. Math. Phys. 163, 491–508 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  126. Terwiel, R.H.: Projection operator method applied to stochastic linear differential equations. Physica 74, 248–265 (1974)

    Article  MathSciNet  Google Scholar 

  127. Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  128. Turkington, B.: An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics. J. Stat. Phys. 152, 569–597 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  130. Venkatesh, T.G., Patnaik, L.M.: Effective Fokker-Planck equation: Path-integral formalism. Phys. Rev. E 48(4), 2402–2412 (1993)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  132. Venturi, D.: A fully symmetric nonlinear biorthogonal decomposition theory for random fields. Physica D 240(4–5), 415–425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  133. Venturi, D.: Conjugate flow action functionals. J. Math. Phys. 54, 113502(1–19) (2013)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  138. Venturi, D., Wan, X., Karniadakis, G.E.: Stochastic bifurcation analysis of Rayleigh-Bénard convection. J. Fluid Mech. 650, 391–413 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Viswanath, D.: The fractal property of the lorentz attractor. Physica D 190, 115–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  146. Wan, X., Karniadakis, G.E.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wei, J., Norman, E.: Lie algebraic solutions of linear differential equations. J. Math. Phys. 4(4), 575–581 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  149. Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  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. Wilcox, R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 399–407 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  154. Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  155. Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Zwanzig, R.: Ensemble methods in the theory of irreversibility. J. Chem. Phys. 33(5), 1338–1341 (1960)

    Article  MathSciNet  Google Scholar 

  159. Zwanzig, R.: Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992 (1961)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, University of California Santa Cruz, 1156 High Street, 95064, Santa Cruz, CA, USA

    Daniele Venturi

  2. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Heyrim Cho

  3. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    George Em Karniadakis

Authors
  1. Daniele Venturi
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  2. Heyrim Cho
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  3. George Em Karniadakis
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Corresponding authors

Correspondence to Daniele Venturi or George Em Karniadakis .

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Editors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA

    Roger Ghanem

  2. Social and Decision Analytics Laboratory, Virginia Bioinformatics Institute, Virginia Tech University, Arlington, Virginia, USA

    David Higdon

  3. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, USA

    Houman Owhadi

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Venturi, D., Cho, H., Karniadakis, G.E. (2017). Mori-Zwanzig Approach to Uncertainty Quantification. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_28

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