Numerical Results of Nonlinear Filtering Problem from Yau-Yau Method

In the paper, we introduce a kind of method to solve the nonlinear filtering problem. Firstly, we review the basic filtering problem and the reduction from robust Duncan-Mortensen-Zakai equation to Kolmogorov equation. Then we use the difference discrete method to solve the Kolmogorov equation. The result is given to prove that the solution of the difference scheme convergences pointwise to the solution of the initial-value problem of the Kolmogorov equation. At last, the numerical results show that the numerical method can give the exact result.

[1]  G.-G. Hu.  Finite-dimensional filters with nonlinear drift. , 1997 .

[2]  Mohammad Reza Ganjali,et al.  Highly efficient removal and preconcentration of lead and cadmium cations from water and wastewater samples using ethylenediamine functionalized SBA-15 , 2011 .

[3]  S. Yau,et al.  Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system , 1997, IEEE Transactions on Aerospace and Electronic Systems.

[4]  Apurba K Bhattacharjee,et al.  Oxindole-based compounds are selective inhibitors of Plasmodium falciparum cyclin dependent protein kinases. , 2003, Journal of medicinal chemistry.

[5]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part III: Nonlinear estimation in white Gaussian noise , 1971 .

[6]  Lei Guo,et al.  Brockett’s problem on nonlinear filtering theory , 2000 .

[7]  G. Mcmahon,et al.  Synthesis and biological evaluations of 3-substituted indolin-2-ones: a novel class of tyrosine kinase inhibitors that exhibit selectivity toward particular receptor tyrosine kinases. , 1998, Journal of medicinal chemistry.

[8]  H. Akaike,et al.  Comment on "An innovations approach to least-squares estimation, part I: Linear filtering in additive white noise" , 1970 .

[9]  Shing-Tung Yau,et al.  Real Time Solution of the Nonlinear Filtering Problem without Memory II , 2008, SIAM J. Control. Optim..

[10]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[11]  K. Bahrami,et al.  SBA-15-Pr–SO3H as nanoreactor catalyzed oxidation of sulfides into sulfoxides , 2011 .

[12]  Xi Wu,et al.  Hessian matrix non-decomposition theorem , 1999 .

[13]  Stephen S.-T. Yau,et al.  Real time solution of nonlinear filtering problem without memory I , 2000 .

[14]  Roger W. Brockett,et al.  Nonlinear Systems and Nonlinear Estimation Theory , 1981 .

[15]  Didier Villemin,et al.  Potassium Fluoride on Alumina: Dry Synthesis of 3-Arylidene-1,3-Dihydro-indol-2-one Under Microwave Irradiation , 1998 .

[16]  S. S.-T. Yau *,et al.  Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture , 2005 .

[17]  Stephen S.-T. Yau,et al.  Finite Dimensional Filters with Nonlinear Drift, XI: Explicit Solution of the Generalized Kolmogorov Equation in Brockett–Mitter Program☆☆☆ , 1998 .

[18]  Jörg Steinbach,et al.  Synthesis and radiopharmacological investigation of 3-[4'-[(18)F]fluorobenzylidene]indolin-2-one as possible tyrosine kinase inhibitor. , 2009, Bioorganic & medicinal chemistry.

[19]  Simon J Coles,et al.  Seven 3-methylidene-1H-indol-2(3H)-ones related to the multiple-receptor tyrosine kinase inhibitor sunitinib. , 2010, Acta crystallographica. Section C, Crystal structure communications.

[20]  Eiichi Akaho,et al.  Synthesis and anti-tyrosine kinase activity of 3-(substituted-benzylidene)-1, 3-dihydro-indolin derivatives: investigation of their role against p60c-Src receptor tyrosine kinase with the application of receptor docking studies. , 2005, Farmaco.

[21]  Bradley F. Chmelka,et al.  Direct Syntheses of Ordered SBA-15 Mesoporous Silica Containing Sulfonic Acid Groups , 2000 .

[22]  Karen Lackey,et al.  Discovery and in vitro evaluation of potent TrkA kinase inhibitors: oxindole and aza-oxindoles. , 2004, Bioorganic & medicinal chemistry letters.

[23]  Ping Chen,et al.  Structure-activity-relationship studies of conformationally restricted analogs of combretastatin A-4 derived from SU5416. , 2006, Bioorganic & medicinal chemistry.

[24]  Stephen S.-T. Yau,et al.  Finite-dimensional filters with nonlinear drift X: explicit solution of DMZ equation , 2001, IEEE Trans. Autom. Control..

[25]  L. G. Chatten,et al.  Substituted oxindoles. Part VI. Polarographic reduction of substituted trans-3-benzylideneindol-2(3H)-ones , 1973 .

[26]  Fredrickson,et al.  Triblock copolymer syntheses of mesoporous silica with periodic 50 to 300 angstrom pores , 1998, Science.

[27]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[28]  Amid Rasoulian,et al.  Finite dimensional filters with non-linear drift IX: construction of finite dimensional estimation algebras of non-maximal rank , 1997 .

[29]  Bradley F. Chmelka,et al.  Nonionic Triblock and Star Diblock Copolymer and Oligomeric Surfactant Syntheses of Highly Ordered, Hydrothermally Stable, Mesoporous Silica Structures , 1998 .

[30]  Jie Chen,et al.  Finite-Dimensional Filters with Nonlinear Drift VII: Mitter Conjecture and Structure of $\eta$ , 1997 .

[31]  J. S. Beck,et al.  Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism , 1992, Nature.

[32]  Jie Chen,et al.  Finite-dimensional filters with nonlinear drift, VI: Linear structure of Ω , 1996, Math. Control. Signals Syst..

[33]  Wei Zhang,et al.  Functionalized 3-benzylidene-indolin-2-ones: inducers of NAD(P)H-quinone oxidoreductase 1 (NQO1) with antiproliferative activity. , 2009, Bioorganic & medicinal chemistry.

[34]  Mark H. A. Davis.  On a multiplicative functional transformation arising in nonlinear filtering theory , 1980 .

[35]  Barbara N. Weaver,et al.  Synthesis of New 3-(Pyridylmethylene)-, 3-(Pyridylmethyl)-, 3-(Piperidylmethyl)-, and 3-(β-Alkylaminoethyl)-2-indolinones. The Reduction of Isoindogenides, Nitro Compounds, and Pyridines in a Series of 2-Indolinones , 1965 .

[36]  James F. Eggler,et al.  Synthesis and in vitro profile of a novel series of catechol benzimidazoles. The discovery of potent, selective phosphodiesterase type IV inhibitors with greatly attenuated affinity for the [3H]rolipram binding site , 1995 .

[37]  Stephen S.-T. Yau.  Recent results on nonlinear filtering: new class of finite dimensional filters , 1990, 29th IEEE Conference on Decision and Control.

[38]  Stephen S.-T. Yau,et al.  Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation , 1998 .

[39]  H. Kunita,et al.  Stochastic differential equations for the non linear filtering problem , 1972 .

[40]  Rafael van Grieken,et al.  Etherification of benzyl alcohols with 1-hexanol over organosulfonic acid mesostructured materials , 2006 .

[41]  Joachim Jose,et al.  Synthesis and Biological Evaluation of 3-(Substituted-benzylidene)-1,3-dihydro-indolin Derivatives as Human Protein Kinase CK2 and p60c-Src Tyrosine Kinase Inhibitors , 2007 .

[42]  Stephen S.-T. Yau,et al.  New direct method for Yau filtering system with arbitrary initial conditions , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[43]  Jie Chen,et al.  Finite-Dimensional Filters with Nonlinear Drift VIII: Classification of Finite-Dimensional Estimation Algebras of Maximal Rank with State-Space Dimension 4 , 1996 .

[44]  Qian Liu,et al.  Ordered Crystalline Alumina Molecular Sieves Synthesized via a Nanocasting Route , 2006 .