On the convergence of the wavelet-Galerkin method for nonlinear filtering
Łukasz D. Nowak ; Monika Pasławska-Południak ; Krystyna Twardowska
International Journal of Applied Mathematics and Computer Science, Tome 20 (2010), p. 93-108 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:207981 
@article{bwmeta1.element.bwnjournal-article-amcv20i1p93bwm,
     author = {\L ukasz D. Nowak and Monika Pas\l awska-Po\l udniak and Krystyna Twardowska},
     title = {On the convergence of the wavelet-Galerkin method for nonlinear filtering},
     journal = {International Journal of Applied Mathematics and Computer Science},
     volume = {20},
     year = {2010},
     pages = {93-108},
     zbl = {1300.93169},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i1p93bwm}
}

Łukasz D. Nowak; Monika Pasławska-Południak; Krystyna Twardowska. On the convergence of the wavelet-Galerkin method for nonlinear filtering. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 93-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i1p93bwm/

[000] Ahmed, N. U. and Radaideh, S. M. (1997). A powerful numerical technique solving Zakai equation for nonlinear filtering, Dynamics and Control 7(3): 293-308. | Zbl 0880.93052

[001] Bennaton, J. F. (1985). Discrete time Galerkin approximations to the nonlinear filtering solution, Journal of Mathematical Analysis and Applications 110: 364-383. | Zbl 0591.65096

[002] Beuchler, S., Schneider, R. and Schwab, C. (2004). Multiresolution weighted norm equivalences and applications, Numerische Mathematik 98(2): 67-97. | Zbl 1058.65149

[003] Bramble, J. H., Cohen, A. and Dahmen, W. (2003). Multiscale Problems and Methods in Numerical Simulations. Lectures given at the C.I.M.E. Summer School, held in Martina Franca, Italy, September 9-15, 2001, Lecture Notes in Mathematics, Vol. 1825, Springer, Berlin.

[004] Ciesielski, Z. (1961). Hölder condition for realizations of Gaussian processes, Transactions of American Mathematical Society 99: 403-413. | Zbl 0133.10502

[005] Cohen, A. (2003). Numerical Analysis of Wavelet Methods, North-Holland, Amsterdam. | Zbl 1038.65151

[006] Cohen, A., Daubechies, I. and Feauveau, J.-C. (1992). Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics 45(5): 485-560. | Zbl 0776.42020

[007] Crisan, D., Gaines, J. and Lyons, T. (1998). Convergence of a branching particle method to the solution of the Zakai equation, SIAM Journal on Applied Mathematics 58(5): 1568-1590. | Zbl 0915.93060

[008] Dahmen, W. (1997). Wavelet and multiscale methods for operator equations, Acta Numerica 6: 55-228. | Zbl 0884.65106

[009] Dahmen, W. and Schneider, R. (1999). Composite wavelet bases for operator equations, Mathematics of Computation 68(228): 1533-1567. | Zbl 0932.65148

[010] Dai, X. and Larson, D. R. (1998). Wandering vectors for unitary systems and orthogonal wavelets, Memoirs of the American Mathematical Society 134(640). | Zbl 0990.42022

[011] Daubechies, I. (1992). Ten Lectures on Wavelets, CBMSNSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA. | Zbl 0776.42018

[012] Eisenstat, S. C., Elman, H. C. and Schultz, M. H. (1983). Variational iterative methods for nonsymmetric systems of linear equations, SIAM Journal on Numerical Analysis 20: 345-357. | Zbl 0524.65019

[013] Elliott, R. J. and Glowinski, R. (1989). Approximations to solutions of the Zakai filtering equation, Stochastic Analysis and Applications 7(2): 145-168. | Zbl 0685.60044

[014] Germani, A. and Picconi, M. (1984). A Galerkin approximation for the Zakai equation, in P. Thoft-Christensen (Ed.), System Modelling and Optimization (Copenhagen, 1983), Lecture Notes in Control and Information Sciences, Vol. 59, Springer-Verlag, Berlin, pp. 415-423.

[015] Hilbert, N., Matache, A.-M. and Schwab, C. (2004). Sparse wavelet methods for option pricing under stochastic volatility, Technical Report 2004-07, Seminar für angewandte Mathematik, Eidgenössische Technische Hochschule, Zürich.

[016] Itô, K. (1996). Approximation of the Zakai equation for nonlinear filtering, SIAM Journal on Control and Optimization 34(2): 620-634. | Zbl 0847.93061

[017] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin. | Zbl 0752.60043

[018] Krylov, N. V. and Rozovski˘i, B. L. (1981). Stochastic evolution equations, Journal of Soviet Mathematics 14: 1233-1277. | Zbl 0462.60060

[019] Kurtz, T. G. and Ocone, D. L. (1988). Unique characterization of conditional distributions in nonlinear filtering, The Annals of Probability 16(1): 80-107. | Zbl 0655.60035

[020] Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes. I. General Theory, Springer-Verlag, New York, NY.

[021] McKean, H. P. (1969). Stochastic Integrals, Academic Press, New York, NY. | Zbl 0191.46603

[022] Pardoux, E. (1991). Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, École d'Eté de Probabilités de Saint-Flour XIX, 1989, Lecture Notes in Mathematics, Vol. 1464, Springer-Verlag, Berlin, pp. 67-163.

[023] Rozovskiĭ, B. L. (1991). A simple proof of uniqueness for Kushner and Zakai equations, in E. Mayer-Wolf, E. Merzbach and A. Shwartz (Eds), Stochastic Analysis, Academic Press, Boston, MA, pp. 449-458. | Zbl 0732.60055

[024] Thomée, V. (1997). Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin. | Zbl 0884.65097

[025] Twardowska, K., Marnik, T. and Pasławska-Południak, M. (2003). Approximation of the Zakai equation in a nonlinear problem with delay, International Journal of Applied Mathematics and Computer Science 13(2): 151-160. | Zbl 1052.93058

[026] von Petersdorff, T. and Schwab, C. (1996). Wavelet approximations for first kind boundary integral equations on polygons, Numerische Mathematik 74(4): 479-519. | Zbl 0863.65074

[027] von Petersdorff, T. and Schwab, C. (2003). Wavelet discretizations of parabolic integrodifferential equations, SIAM Journal on Numerical Analysis 41(1): 159-180. | Zbl 1050.65134

[028] Wang, J. (2002). Spline wavelets in numerical resolution of partial differential equations, in D. Deng, D. Huang, R.-Q. Jia, W. Lin and J. Wand (Eds), Wavelet Analysis and Applications. Proceedings of an International Conference, Guangzhou, China, November 15-20, 1999, AMS/IP Studies in Advanced Mathematics, Vol. 25, American Mathematical Society, Providence, RI, pp. 257-277. | Zbl 1001.65121

[029] Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts, Vol. 37, Cambridge University Press, Cambridge. | Zbl 0865.42026

[030] Yau, S.-T. and Yau, S. S.-T. (2000). Real time solution of nonlinear filtering problem without memory I, Mathematical Research Letters 7(5-6): 671-693. | Zbl 0967.93089

[031] Yau, S.-T. and Yau, S. S.-T. (2008). Real time solution of nonlinear filtering problem without memory II, SIAM Journal on Control and Optimization 47: 163-195. | Zbl 1172.35411

[032] Yserentant, H. (1990). Two preconditioners based on the multilevel splitting of finite element spaces, Numerische Mathematik 58(2): 163-184. | Zbl 0708.65103