Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T23:27:47.528Z Has data issue: false hasContentIssue false
  • English
  • Français

NEWTON’S METHOD FOR STOCHASTIC FUNCTIONAL EVOLUTION EQUATIONS IN HILBERT SPACES

Part of: Numerical methods in calculus of variations and optimal control Miscellaneous topics - Partial differential equations Stochastic analysis

Published online by Cambridge University Press:  25 March 2019

Monika Wrzosek
Monika Wrzosek*
Affiliation:
Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland email mwrzosek@mat.ug.edu.pl
Get access

Abstract

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

MSC classification

Primary: 60H15: Stochastic partial differential equations 35R60: Partial differential equations with randomness, stochastic partial differential equations 49M15: Newton-type methods
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 542 - 556
Copyright
Copyright © University College London 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by grant BW-UG 538-5100-B151-13 from the University of Gdańsk.

References

Amano, K., A note on Newton’s method for stochastic differential equations and its error estimate. Proc. Japan Acad. 84 2008, 1–3.Google Scholar
Amano, K., Newton’s method for stochastic differential equations and its probabilistic second-order error estimate. Electron. J. Differential Equations 2012 2012, 1–8.Google Scholar
Bessaih, H., Brzeźniak, Z. and Millet, A., Splitting up method for the 2D stochastic Navier–Stokes equations. Stoch. PDE: Anal. Comp. 2 2014, 433–470.Google Scholar
Bessaih, H., Hausenblas, E., Randrianasolo, I. A. and Razafimandimby, P. A., Numerical approximation of stochastic evolution equations: convergence in scale of Hilbert spaces. J. Comput. Appl. Math. 343 2018, 250–274.Google Scholar
Bessaih, H. and Millet, A., Strong $L^{2}$ convergence of time numerical schemes for the stochastic 2D Navier–Stokes equations. IMA J. Numer. Anal. (2018), doi:10.1093/imanum/dry058.Google Scholar
Brzeźniak, Z., Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4 1995, 1–45.Google Scholar
Brzeźniak, Z., Carelli, E. and Prohl, A., Finite-element-based discretizations of the incompressible Navier–Stokes equations with multiplicative random forcing. IMA J. Numer. Anal. 33 2013, 771–824.Google Scholar
Carelli, E., Hausenblas, E. and Prohl, A., Time-splitting methods to solve the stochastic incompressible Stokes equation. SIAM J. Numer. Anal. 50 2012, 2917–2939.Google Scholar
Chow, P. L., Stochastic partial differential equations in turbulence. In Probabilistic Analysis and Related Topics, Vol. 1 (ed. Bharucha-Reid, A. T.), Academic Press (New York, 1978), 1–43.Google Scholar
Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press (Cambridge, 1992).Google Scholar
Da Prato, G. and Zabczyk, J., Ergodicity for Infnite-Dimensional Systems (London Mathematical Society Lecture Note Series 229 ), Cambridge University Press (Cambridge, 1996).Google Scholar
Deck, T., Kruse, S., Potthoff, J. and Watanabe, H., White noise approach to s.p.d.e.’s. In Stochastic Partial Differential Equations and Applications V (Trento, 2002) (Lecture Notes in Pure and Appl. Math. 227 ) (eds Da Prato, G. and Tubaro, L.), Dekker (New York, 2002), 183–195.Google Scholar
Fleming, W. H., Distributed parameter stochastic systems in population biology. In Control Theory, Numerical Methods and Computer System Modelling (Lecture Notes in Economy and Mathematical Systems 107 ) (eds Bensoussan, A. and Lions, J. L.), Springer (Berlin, 1975), 179–191.Google Scholar
Frisch, U., Wave propagation in random media. In Probabilistic Methods in Applied Mathematics (ed. Bharucha-Reid, A. T.), Academic Press (New York, 1968), 76–198.Google Scholar
Fujisaki, M., Kallianpur, G. and Kunita, H., Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9 1972, 19–40.Google Scholar
Govindan, T. E., Existence and stability of solutions of stochastic semilinear functional differential equations. Stoch. Anal. Appl. 20 2002, 1257–1280.Google Scholar
Govindan, T. E., Stability of mild solutions of stochastic evolution equations with variable delay. Stoch. Anal. Appl. 21 2003, 1059–1077.Google Scholar
Gyöngy, I. and Millet, A., On discretization schemes for stochastic evolution equations. Potential Anal. 23 2005, 99–134.Google Scholar
Hausenblas, E., Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147 2002, 485–516.Google Scholar
Holden, H., Øksendal, B., Ubøe, J. and Zhang, T., Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, Birkhäuser (Boston, 1996).Google Scholar
Itô, K., Differential equations determining Markov processes. Zenkoku Shijo Sugaku Danwakai 244(1077) 1942, 1352–1400 (in Japanese).Google Scholar
Jahanipur, R., Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions. J. Differential Equations 248 2010, 1230–1255.Google Scholar
Kawabata, S. and Yamada, T., On Newton’s method for stochastic differential equations. In Seminaire de Probabilites XXV (Lecture Notes in Mathematics 1485 ), Springer (Berlin, 1991), 121–137.Google Scholar
Kovács, M., Larsson, S. and Lindgren, F., Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise. BIT Numer. Math. 52 2012, 85–108.Google Scholar
Kunita, K., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press (Cambridge, 1990).Google Scholar
Kushner, H. J., On the optimal control of a system governed by a linear parabolic equation with white noise inputs. SIAM J. Control Optim. 6 1978, 596–614.Google Scholar
Nagase, N., Remarks on nonlinear stochastic partial differential equations: an application of the splitting-up method. SIAM J. Control Optim. 33 1995, 1716–1730.Google Scholar
van Neerven, J. M. A. M., Veraar, M. C. and Weis, L., Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255 2008, 940–993.Google Scholar
Pardoux, E., Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 1979, 127–167.Google Scholar
Ren, J., Röckner, M. and Wang, F.-Y., Stochastic generalized porous media and fast diffusion equations. J. Differential Equations 238 2007, 118–152.Google Scholar
Taniguchi, T., Liu, K. and Truman, A., Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differential Equations 181 2002, 72–91.Google Scholar
Walsh, J. B., An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour XIV (Lecture Notes in Mathematics 1180 ), Springer (Berlin, 1986), 265–439.Google Scholar
Wrzosek, M., Newton’s method for stochastic functional differential equations. Electron. J. Differential Equations 2012 2012, 1–10.Google Scholar
Zhang, X., On stochastic evolution equations with non-Lipschitz coefficients. Stoch. Dyn. 9 2009, 549–595.Google Scholar