Spectral Methods for Multiscale Stochastic Differential Equations
Abstract
This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.
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References
1.
A. Abdulle, W. E, B. Engquist, and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer., 21 (2012), pp. 1--87.
2.
A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231 (2012), pp. 2482--2497, https://doi.org/10.1016/j.jcp.2011.11.039.
3.
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of $N$-Body Schrödinger Operators, Math. Notes 29, Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo, 1982.
4.
G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge, UK, 1999, https://doi.org/10.1017/CBO9781107325937.
5.
A. Arnold, J.-P. Bartier, and J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., 5 (2007), pp. 971--979, http://projecteuclid.org/euclid.cms/1199377560.
6.
W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), pp. 397--400, https://doi.org/10.2307/2046956.
7.
D. Blömker, M. Hairer, and G. A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), pp. 1721--1744, https://doi.org/10.1088/0951-7715/20/7/009.
8.
V. I. Bogachev, N. V. Krylov, M. Röckner, and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Math. Surveys Monogr. 207, American Mathematical Society, Providence, RI, 2015, https://doi.org/10.1090/surv/207.
9.
V. Bonnaillie-Noël, J. A. Carrillo, T. Goudon, and G. A. Pavliotis, Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations, IMA J. Numer. Anal., 36 (2016), pp. 1536--1569, https://doi.org/10.1093/imanum/drv066.
10.
J. P. Boyd, Asymptotic coefficients of Hermite function series, J. Comput. Phys., 54 (1984), pp. 382--410, https://doi.org/10.1016/0021-9991(84)90124-4.
11.
C.-E. Bréhier, Analysis of an HMM time-discretization scheme for a system of stochastic PDEs, SIAM J. Numer. Anal., 51 (2013), pp. 1185--1210, https://doi.org/10.1137/110853078.
12.
J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010, pp. 297--336, https://doi.org/doi:10.1007/978-0-8176-4946-3_12.
13.
R. R. Coifman, I. G. Kevrekidis, S. Lafon, M. Maggioni, and B. Nadler, Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems, Multiscale Model. Simul., 7 (2008), pp. 842--864, https://doi.org/10.1137/070696325.
14.
R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), pp. 5--30, https://doi.org/10.1016/j.acha.2006.04.006.
15.
R. R. Coifman and S. Lafon, Geometric harmonics: a novel tool for multiscale out-of-sample extension of empirical functions, Appl. Comput. Harmon. Anal., 21 (2006), pp. 31--52, https://doi.org/10.1016/j.acha.2005.07.005.
16.
C. Cuthbertson, G. A. Pavliotis, A. Rafailidis, and P. Wiberg, Asymptotic analysis for foreign exchange derivatives with stochastic volatility, Int. J. Theor. Appl. Finance, 13 (2010), pp. 1131--1147, https://doi.org/10.1142/S0219024910006145.
17.
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, UK, 2014, https://doi.org/10.1017/CBO9781107295513.
18.
A. B. Duncan, S. Kalliadasis, G. A. Pavliotis, and M. Pradas, Noise-induced transitions in rugged energy landscapes, Phys. Rev. E (3), 94 (2016), 032107, https://doi.org/10.1103/PhysRevE.94.032107.
19.
W. E, Principles of Multiscale Modeling, Cambridge University Press, Cambridge, UK, 2011.
20.
W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden, Heterogeneous multiscale methods: a review, Commun. Comput. Phys., 2 (2007), pp. 367--450.
21.
W. E, D. Liu, and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), pp. 1544--1585, https://doi.org/10.1002/cpa.20088.
22.
L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, AMS, Providence, RI, 2010, https://doi.org/10.1090/gsm/019.
23.
J. C. M. Fok, B. Guo, and T. Tang, Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp., 71 (2002), pp. 1497--1528, https://doi.org/10.1090/S0025-5718-01-01365-5.
24.
G. Froyland, G. A. Gottwald, and A. Hammerlindl, A computational method to extract macroscopic variables and their dynamics in multiscale systems, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1816--1846, https://doi.org/10.1137/130943637.
25.
G. Froyland, G. A. Gottwald, and A. Hammerlindl, A trajectory-free framework for analysing multiscale systems, Phys. D, 328/329 (2016), pp. 34--43, https://doi.org/10.1016/j.physd.2016.04.010.
26.
J. Gagelman and H. Yserentant, A spectral method for Schrödinger equations with smooth confinement potentials, Numer. Math., 122 (2012), pp. 383--398, https://doi.org/10.1007/s00211-012-0458-8.
27.
T. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms, 18 (1998), pp. 209--232, https://doi.org/10.1023/A:1019129717644.
28.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
29.
V. Golp̧rimedshtein and A. Ukhlov, Weighted Sobolev spaces and embedding theorems, Trans. Amer. Math. Soc., 361 (2009), pp. 3829--3850, https://doi.org/10.1090/S0002-9947-09-04615-7.
30.
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM, Philadelphia, 1977.
31.
B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Math. 1862, Springer-Verlag, Berlin, 2005, https://doi.org/10.1007/b104762.
32.
D. J. Higham, X. Mao, and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), pp. 1041--1063, https://doi.org/10.1137/S0036142901389530.
33.
P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory: With Applications to Schr|"odinger Operators, Appl. Math. Sci. 113, Springer-Verlag, New York, 1996, https://doi.org/10.1007/978-1-4612-0741-2.
34.
V. Kaarnioja, Smolyak Quadrature, (2013).
35.
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Grad. Texts in Math. 113, Springer-Verlag, New York, 1991, https://doi.org/10.1007/978-1-4612-0949-2.
36.
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. Kevrekidis, O. Runborg, and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), pp. 715--762, http://projecteuclid.org/euclid.cms/1119655353.
37.
P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Appl. Math. (N.Y.) 23, Springer-Verlag, Berlin, 1992, https://doi.org/10.1007/978-3-662-12616-5.
38.
J. C. Latorre, G. A. Pavliotis, and P. R. Kramer, Corrections to Einstein's relation for Brownian motion in a tilted periodic potential, J. Stat. Phys., 150 (2013), pp. 776--803, https://doi.org/10.1007/s10955-013-0692-1.
39.
B. Leimkuhler, C. Matthews, and G. Stoltz, The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics, IMA J. Numer. Anal., 36 (2016), pp. 13--79, https://doi.org/10.1093/imanum/dru056.
40.
T. Lelièvre, M. Rousset, and G. Stoltz, Free Energy Computations: A Mathematical Perspective, Imperial College Press, London, 2010, https://doi.org/10.1142/9781848162488.
41.
T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numer., 25 (2016), pp. 681--880, https://doi.org/10.1017/S0962492916000039.
42.
T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numer., 25 (2016), pp. 681--880, https://doi.org/10.1017/S0962492916000039.
43.
T. Li, A. Abdulle, and W. E, Effectiveness of implicit methods for stiff stochastic differential equations, Commun. Comput. Phys., 3 (2008), pp. 295--307.
44.
L. Lorenzi and M. Bertoldi, Analytical Methods for MArkov Semigroups, Pure Appl. Math. (Boca Raton) 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.
45.
A. Majda, I. Timofeyev, and E. V. Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), pp. 891--974.
46.
X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood, Chichester, 2008, https://doi.org/10.1533/9780857099402.
47.
P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Mat. Contemp., 19 (2000), pp. 1--29.
48.
G. Metafune, D. Pallara, and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), pp. 40--60, https://doi.org/10.1006/jfan.2002.3978.
49.
G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Scientific Computation, Springer-Verlag, Berlin, 2004, https://doi.org/10.1007/978-3-662-10063-9.
50.
M. Ottobre and G. A. Pavliotis, Asymptotic analysis for the generalized Langevin equation, Nonlinearity, 24 (2011), pp. 1629--1653, https://doi.org/10.1088/0951-7715/24/5/013.
51.
E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab., 29 (2001), pp. 1061--1085, https://doi.org/10.1214/aop/1015345596.
52.
E. Pardoux and A. Y. Veretennikov, On Poisson equation and diffusion approximation. II, Ann. Probab., 31 (2003), pp. 1166--1192, https://doi.org/10.1214/aop/1055425774.
53.
E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. III, Ann. Probab., 33 (2005), pp. 1111--1133, https://doi.org/10.1214/009117905000000062.
54.
G. A. Pavliotis, Asymptotic analysis of the Green-Kubo formula, IMA J. Appl. Math., 75 (2010), pp. 951--967, https://doi.org/10.1093/imamat/hxq039.
55.
G. A. Pavliotis, Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, Texts Appl. Math. 60, Springer, New York, 2014, https://doi.org/10.1007/978-1-4939-1323-7.
56.
G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts Appl. Math. 53, Springer, New York, 2008.
57.
G. A. Pavliotis and A. Vogiannou, Diffusive transport in periodic potentials: Underdamped dynamics, Fluct. Noise Lett., 8 (2008), pp. L155--173.
58.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.
59.
J. Shen and L.-L. Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), pp. 195--241.
60.
T. Tang, The Hermite spectral method for Gaussian-type functions, SIAM J. Sci. Comput., 14 (1993), pp. 594--606, https://doi.org/10.1137/0914038.
61.
U. Vaes, Numerical Solution of Multiscale Stochastic Equations, master's thesis, Department of Mathematics, Imperial College London, 2014.
62.
E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects, Commun. Math. Sci., 1 (2003), pp. 385--391.
63.
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), 1--141, https://doi.org/10.1090/S0065-9266-09-00567-5.
64.
H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Math. 2000, Springer-Verlag, Berlin, 2010, https://doi.org/10.1007/978-3-642-12248-4.
Information & Authors
Information
Published In
SIAM/ASA Journal on Uncertainty Quantification
Pages: 720 - 761
DOI: 10.1137/16M1094117
ISSN (online): 2166-2525
Copyright
© 2017, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 19 September 2016
Accepted: 8 February 2017
Published online: 8 August 2017
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Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L020564, EP/L024926, EP/L025159
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