×

Numerical analysis of semilinear stochastic evolution equations in Banach spaces. (English) Zbl 1026.65005

The author considers the numerical approximation of solutions of stochastic partial differential equation \[ du(t)= (Au(t)+ f(t, u(t))) dt+ \sum_j \sigma_j(t, u(t)) dw^j(t), \] where \(A\) is an infinitesimal generator of a \(C_0\)-semigroup.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banks, H.; Burns, J., Hereditary control problems, SIAM J. Control Optim., 16, 169-208 (1978) · Zbl 0379.49025
[2] J. Bergh, J. Löfström, Interpolation spaces: an introduction, Die Grundlehren der mathematischen Wissenschaften, Vol. 223, Springer, Berlin, 1976.; J. Bergh, J. Löfström, Interpolation spaces: an introduction, Die Grundlehren der mathematischen Wissenschaften, Vol. 223, Springer, Berlin, 1976.
[3] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Vol. 15 (1994), Springer: Springer Berlin · Zbl 0804.65101
[4] Brzezniak, Z., Stochastic partial differential equations in \(M\)-type 2 Banach spaces, Potential Anal., 4, 1-45 (1995) · Zbl 0831.35161
[5] Brzezniak, Z., On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61, 245-295 (1997) · Zbl 0891.60056
[6] Brzezniak, Z.; Peszat, S., Space-time continuous solutions to SPDE’s driven by a homogeneous Wiener process, Stud. Math., 137, 261-299 (1999) · Zbl 0944.60075
[7] Buckwar, E., Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math., 125, 297-307 (2000) · Zbl 0971.65004
[8] Chernoff, P. R., Note on product formulas for operator semigroups, J. Funct. Anal., 2, 238-242 (1968) · Zbl 0157.21501
[9] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Vol. 44 (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0761.60052
[10] Dahmen, W., Wavelet and multiscale methods for operator equations, Acta Numer., 6, 55-228 (1997) · Zbl 0884.65106
[11] Dahmen, W.; Kunoth, A.; Urban, K., Biorthogonal spline wavelets on the interval-stability and moment conditions, Appl. Comput. Harmon. Anal., 6, 132-196 (1999) · Zbl 0922.42021
[12] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, CBMS, (Washington, DC,) 1992.; I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, CBMS, (Washington, DC,) 1992. · Zbl 0776.42018
[13] Delfour, M. C., State theory of linear hereditary differential systems, J. Math. Anal. Appl., 60, 1, 8-35 (1977) · Zbl 0358.34071
[14] Dettweiler, E., On the martingale problem for Banach space valued stochastic differential equations, J. Theoret. Probab., 2, 2, 159-191 (1989) · Zbl 0674.60056
[15] E. Dettweiler, Representation of Banach space valued martingales as stochastic integrals, in: Probability in Banach spaces, Birkhäuser, Basel, 1990, pp. 43-62.; E. Dettweiler, Representation of Banach space valued martingales as stochastic integrals, in: Probability in Banach spaces, Birkhäuser, Basel, 1990, pp. 43-62. · Zbl 0701.60049
[16] Dettweiler, E., Stochastic integration relative to Brownian motion on a general Banach space, Doga, Turk. J. Math., 15, 2, 58-97 (1991) · Zbl 0970.60517
[17] Engel, K.; Nagel, R., One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, Vol. 194 (2000), Springer: Springer Berlin · Zbl 0952.47036
[18] Greksch, W.; Kloeden, P., Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc., 54, 79-85 (1996) · Zbl 0880.35143
[19] Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise: I, Potential Anal., 9, 1-25 (1998) · Zbl 0915.60069
[20] Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise: II, Potential Anal., 11, 1-37 (1999) · Zbl 0944.60074
[21] E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal. (2002), in press.; E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal. (2002), in press. · Zbl 1026.65005
[22] Ichikawa, A., Equivalence of \(L_p\) stability and exponential stability for a class of nonlinear semigroups, Nonlinear Anal. Theory Methods Appl., 8, 805-815 (1984) · Zbl 0547.47041
[23] P. Kloeden, W. Shott, Linear-implicit strong schemes for Ito-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stochastic. Anal. (2000), to appear.; P. Kloeden, W. Shott, Linear-implicit strong schemes for Ito-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stochastic. Anal. (2000), to appear. · Zbl 0988.60066
[24] A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa—Appunti, 1999.; A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa—Appunti, 1999.
[25] Mao, X., Stochastic differential equations and their applications, Ellis Horwood Series in Mathematics and its Applications (1997), Horwood Publishing: Horwood Publishing Chichester
[26] Miklavcic, M., Applied Functional Analysis and Partial Differential Equations (1998), World Scientific: World Scientific Singapore · Zbl 0913.35002
[27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, Springer, Berlin, New York, 1983.; A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, Springer, Berlin, New York, 1983. · Zbl 0516.47023
[28] Runst, T.; Sickel, W., Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, Vol. 3 (1996), de Gruyter: de Gruyter Berlin · Zbl 0873.35001
[29] Sasai, H., Approximation of optimal control problems governed by non-linear evolution equations, Int. J. Control, 28, 313-324 (1978) · Zbl 0441.49012
[30] Seidler, J., Da Prato-Zabczyk’s maximal inequality revisited, Math. Bohem., 118, 67-106 (1993) · Zbl 0785.35115
[31] Shardlow, T., Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20, 121-145 (1999) · Zbl 0919.65100
[32] Sickel, W.; Triebel, H., Hoelder inequalities and sharp embeddings in function spaces of \(B_{ pq }^s\) and \(F_{ pq }^s\) type, Z. Anal. Angew., 14, 105-140 (1995) · Zbl 0820.46030
[33] Stevenson, R., Piecewise linear (pre-)wavelets on non-uniform meshes, (Hackbusch, W.; etal., Multigrid methods V, Proceedings of the Fifth European Multigrid Conference. Multigrid methods V, Proceedings of the Fifth European Multigrid Conference, Lecture Notes in Computer Science Engineering, Vol. 3 (1998), Springer: Springer Berlin), 306-319 · Zbl 0926.65110
[34] H. Tanabe, Equations of evolution. Monographs and Studies in Mathematics, Vol. 6, Pitman, London, 1979 (translated from Japanese by N. Mugibayashi, H. Haneda).; H. Tanabe, Equations of evolution. Monographs and Studies in Mathematics, Vol. 6, Pitman, London, 1979 (translated from Japanese by N. Mugibayashi, H. Haneda). · Zbl 0417.35003
[35] Thomee, V., Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, Vol. 25 (1997), Springer: Springer Berlin · Zbl 0884.65097
[36] Triebel, H., Theory of function spaces, Monographs in Mathematics, Vol. 78 (1983), Birkhauser Verlag: Birkhauser Verlag Basel · Zbl 0546.46028
[37] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1995), Barth: Barth Leipzig · Zbl 0830.46028
[38] Wojtaszczyk, P., A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts, Vol. 37 (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0865.42026
[39] Yoo, H., Semi-discretization of stochastic partial differential equations on \(r^1\) by a finite-difference method, Math. Comput., 69, 653-666 (2000) · Zbl 0942.65006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.