This book addresses the problem of accurate state estimation in nonlinear continuous-time stochastic models with additive noise and discrete measurements. Its main focus is on numerical aspects of computation of the expectation and covariance in Kalman-like filters rather than on statistical properties determining a model of the system state. Nevertheless, it provides the sound theoretical background and covers all contemporary state estimation techniques beginning at the celebrated Kalman filter, including its versions extended to nonlinear stochastic models, and till the most advanced universal Gaussian filters with deterministically sampled mean and covariance. In particular, the authors demonstrate that, when applying such filtering procedures to stochastic models with strong nonlinearities, the use of adaptive ordinary differential equation solvers with automatic local and global error control facilities allows the discretization error―and consequently the state estimation error―to be reduced considerably. For achieving that, the variable-stepsize methods with automatic error regulation and stepsize selection mechanisms are applied to treating moment differential equations arisen. The implemented discretization error reduction makes the self-adaptive nonlinear Gaussian filtering algorithms more suitable for application and leads to the novel notion of accurate state estimation.
The book also discusses accurate state estimation in mathematical models with sparse measurements. Of special interest in this regard, it provides a means for treating stiff stochastic systems, which often encountered in applied science and engineering, being exemplified by the Van der Pol oscillator in electrical engineering and the Oregonator model of chemical kinetics. Square-root implementations of all Kalman-like filters considered and explored in this book for state estimation in Ill-conditioned continuous–discrete stochastic systems attract the authors’ particular attention.
This book covers both theoretical and applied aspects of numerical integration methods, including the concepts of approximation, convergence, stiffness as well as of local and global errors, suitably for applied scientists and engineers. Such methods serve as a basis for the development of accurate continuous–discrete extended, unscented, cubature and many other Kalman filtering algorithms, including the universal Gaussian methods with deterministically sampled expectation and covariance as well as their mixed-type versions. The state estimation procedures in this book are presented in the fashion of complete pseudo-codes, which are ready for implementation and use in MATLAB® or in any other computation platform. These are examined numerically and shown to outperform traditional variants of the Kalman-like filters in practical prediction/filtering tasks, including state estimations of stiff and/or ill-conditioned continuous–discrete nonlinear stochastic systems.
Gennady Yu. Kulikov graduated in Mathematics from the Faculty of Mechanics and Mathematics of the Moscow State University in 1988 (Diploma Cum Laude), and earned his Ph.D. (Russian degree "Candidate of Sciences in Physics and Mathematics") in computational mathematics from the Computer Engineering Center at the Russian Academy of Sciences in 1994. He obtained his Habilitation (Russian degree "Doctor of Sciences in Physics and Mathematics") in 2002. G. Yu. Kulikov worked at the Faculty of Mechanics and Mathematics of the Ulyanovsk State University in Russia from 1993 till his relocation to South Africa in 2004, where he became a senior lecturer and, then, a reader at the University of the Witwatersrand. In 2009, he immigrated to Portugal and became a full-time researcher at Centro de Matemática Computacional e Estocástica (CEMAT), Instituto Superior Técnico, Universidade de Lisboa.
Kulikov’s research interests are twofold. First, these focus on numerical methods for differential equations with special emphasis to global error estimation and control strategies. Second, his research topics include applications of such methods with global error control in fluid mechanics, nonlinear Kalman filtering and mathematical neuroscience. He has published widely in quality peer-reviewed journals (about 150 articles in journals, book chapters, and conference proceedings) and gained 16 research grants. In addition, G. Yu. Kulikov has served as a referee for various national and international peer reviewed publications and as a reviewer for the Mathematical Reviews of the American Mathematical Society (AMS). Over the years, he taught several undergraduate and graduate courses in computational mathematics, numerical methods for differential equations, computational linear algebra, prepared a number of M.Sc. and Ph.D. students, and supervised postdoctoral research projects in the area of his expertise. In recent years, G. Yu. Kulikov has been recognized as a TOP 2% cited researcher in the world according to Scopus' data.
Maria V. Kulikova graduated from the Faculty of Mechanics and Mathematics of the Ulyanovsk State University in 2001, and earned her Ph.D. (degree "Candidate of Sciences in Physics and Mathematics") in applied mathematics in 2006. She worked (2007-2009) as a post-doctoral fellow at the University of the Witwatersrand, South Africa till her relocation to Portugal in 2009, where she held a six-year full-time researcher position (2010-2015) granted by the Portuguese Research Fund (FCT). Since 2016 she has been an integrated research member at Centro de Matemática Computacional e Estocástica (CEMAT), Instituto Superior Técnico, Universidade de Lisboa, Portugal. In 2022, M. V. Kulikova became a full-time researcher at the mentioned institution.
Her main research interests include Kalman filtering and nonlinear Bayesian filtering methods, numerical stability and robust estimation with applications in target tracking, econometrics and mathematical neuroscience. She has published widely in national and international peer-reviewed journals and has received individual research grants from the University of the Witwatersrand (South Africa), CEMAT (Portugal) as well as from the FCT (Portugal). She has served as a referee for various international peer reviewed journals and as a reviewer for the Mathematical Reviews of the American Mathematical Society (AMS). In addition, M. V. Kulikova has an experience to be a part of evaluation panels of the higher degrees committees and she is a research associate of the "African Collaboration for Quantitative Finance & Risk Research" (ACQuFRR). Dr. Kulikova has contributed to teaching undergraduate and graduate courses in computational mathematics and numerical methods in finance. She has also supervised a number of M.Sc. students in the area of her expertise. In recent years, M. V. Kulikova has been recognized as a TOP 2% cited researcher in the world according to Scopus' data.
