Rate of convergence of Local Linearization schemes for random differential equations

Springer Science and Business Media LLC - Tập 49 - Trang 357-373 - 2009
Juan Carlos Jimenez1, Felix Carbonell2
1Instituto de Cibernética, Matemática y Física, Departamento de Matemática Interdisciplinaria, Havana, Cuba
2Montreal Neurological Institute, McGill University, Montreal, Canada

Tóm tắt

Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.

Tài liệu tham khảo

Arnold, L.: Random Dynamical Systems. Springer, Heidelberg (1998)

Berland, H., Skaflestad, B., Wright, W.M.: EXPINT—A MATLAB package for exponential integrators. ACM Trans. Math. Softw. 33, 2–17 (2007)

Carbonell, F., Jimenez, J.C., Biscay, R., de la Cruz, H.: The local linearization method for numerical integration of random differential equations. BIT 45, 1–14 (2005)

Carbonell, F., Biscay, R.J., Jimenez, J.C., de la Cruz, H.: Numerical simulation of nonlinear dynamical systems driven by commutative noise. J. Comput. Phys. 226, 1219–1233 (2007)

Cartan, H.: Calcul Differentiel. Hermann, Paris (1967)

Cortés, J., Jódar, L., Villafuerte, L.: Numerical solution of random differential equations: a mean square approach. Math. Comput. Model. 45, 757–765 (2007)

Cortés, J., Jódar, L., Villafuerte, L.: A random Euler method for solving differential equations with uncertainties. Math. Ind. 12, 944–948 (2008)

Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Press, Baltimore (1996)

Grune, L., Kloeden, P.: Pathwise approximation of random ordinary differential equations. BIT 41, 711–721 (2001)

Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (1993)

Hasminskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff Noordhoff, Rockville (1980)

Hurewicz, W.: Lectures on Ordinary Differential Equations. Edicion Revolucionaria, La Habana (1966)

Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM Numer. Anal. 34, 1911–1925 (1997)

Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)

Imkeller, P., Schmalfuss, B.: The conjugacy of stochastic and random differential equations and the existence of global attractors. J. Dyn. Differ. Equ. 13, 215–249 (2001)

Imkeller, P., Lederer, C.: On the cohomology of flows of stochastic and random differential equations. Probab. Theory Relat. Fields 120, 209–235 (2001)

Jimenez, J.C., Carbonell, F.: Rate of convergence of local linearization schemes for initial-value problems. Appl. Math. Comput. 171, 1282–1295 (2005)

Kloeden, P.E., Keller, H., Schmalfuß, B.: Towards a theory of random numerical dynamics. In: Crauel, H., Gundlach, V.M. (eds.) Stochastic Dynamics, pp. 259–282. Springer, Berlin (1999)

Kloeden, P.E., Jentzen, A.: Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. A 463, 2929–2944 (2007)

Moler, C., Van Loan, C.F.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801–836 (1978)

Sidje, R.B.: EXPOKIT: software package for computing matrix exponentials. AMC Trans. Math. Softw. 24, 130–156 (1998)

Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, Dordrecht (1991)

Sotero, R.C., Trujillo, N.J., Carbonell, F., Jimenez, J.C.: Realistically coupled neural mass models can generate EEG rhythms. Neural Comput. 19, 478–512 (2007)

Van Loan, C.F.: Computing integrals involving the matrix exponential. IEEE Trans. Autom. Contr. AC-23, 395–404 (1978)

Vom Scheidt, J., Starkloff, H.J., Wunderlich, R.: Random transverse vibrations of a one-sided fixed beam and model reduction. Z. Angew. Math. Mech. 82, 847–859 (2002)

Weba, M.: Simulation and approximation of stochastic processes by spline functions. SIAM J. Sci. Stat. Comput. 13, 1085–1096 (1992)

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