Boris Rozovsky - US grants

The principal investigator will continue to develop the theory of stochastic partial differential equations (SPDE's). Random fields governed by SPDE's have been used intensively in physics, biology, chemistry, and electrical engineering as models for various random phenomena. Substantial progress in this field has been made in the last decade, but still the theory is in its infancy. The investigator plans to continue his research in several related areas of linear and non-linear SPDE's.

This grant will support an International Conference in Stochastic Partial Differential Equations (SPDE's), May 6-8, 1990, and a satellite School-Seminar in the same topic for new researchers, May 3-5. Both activities will be held at the University of North Carolina at Charlotte. The purpose of the conference is to bring together researchers who are working on different theoretical and applied aspects of SPDE's and to promote research in this field. The purpose of the School-Seminar is to bring the subject of SPDE's to the attention of new researchers (graduate students, postdocs and recent Ph.D.'s) and provide them with a unique opportunity to be taught by, meet, and have discussions with the outstanding scientists working in the area. The introductory sessions scheduled for the School-Seminar are linear elliptic and parabolic SPDE's, methods in non-linear SPDE's, and applications of SPDE's. The main part of the conference will cover recent theoretical developments in SPDE's and various applications, especially to oceanography and signal processing.

9802423 Rozovskii Each time an outfielder tracks and catches a fly ball, he intuitively solves a problem of target tracking. That problem has stymied engineers, mathematicians, and computer scientists for years. Two great mathematicians, American, Norbert Wiener and Russian, Andrey Kolmogorov, first approached the problem during World War II. Rather than catching fly balls, Kolmogorov and Wiener were trying, in the days before computers, to develop mathematical algorithms that would help to track enemy aircraft by radar. The research started by Kolmogorov and Wiener has developed into a thriving area of applied mathematics known as Filtering Theory. Filtering, estimation of a signal or an image from noisy data, is the basic component of the data assimilation in target tracking. It is of central importance in navigation, image and signal processing, control theory, automatic tracking systems and other areas of engineering and science. This research is a joint collaborative effort between researchers at the Institute of Mathematics and Informatics, Vilnius, Lithuania, and researchers at the University of Southern California. The project is devoted to applications of stochastic partial differential equations to nonlinear filtering. The focus of the research is twofold: (1) Cauchy- boundary problems for parabolic partial differential equations arising in nonlinear filtering of stochastic processes evolving under some constraints; and (2) nonlinear filtering with distributed observation and applications to tracking of low-observable targets in images. The research will also consider the numerical aspects of nonlinear filtering. It is expected that relatively simple nonlinear filtering algorithms which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal from a statistical viewpoint will be developed for a wide variety of applications. The applications include air traffic control, human-computer interfaces based on motion- capture, and advanced optical and magnetic registration systems.

This proposal is for the travel funds for American attendees of the
conference ``Kolmogorov and Contemporary Mathematics'', an International
scholarly meeting to commemorate the Centennial birthday of A. N.
Kolmogorov -- arguably one of the best mathematicians of the 20th
century. The conference will take place on June 16-21, 2003 in Moscow,
Russia and is expected to draw 600-800 participants from around the world.
Organized by the Russian Academy of Science and Moscow State University,
the Kolmogorov's Centennial Conference will be one of the largest and
most important gatherings of mathematicians in 2003.

The conference will cover many areas of mathematics and Plenary Talks
will be given by prominent mathematicians in these areas. It is
therefore, expected that the conference will have a great impact on many
areas of current active research in mathematics and its applications in
various areas of science.

The PDEs known as the Euler and Navier-Stokes equations are the most important
deterministic models for the motion of fluids and gases. It has long been
suspected that stochastic versions of Navier-Stokes or Euler equations could
be valuable models for turbulent motion. The research to be performed is
concerned with developing methods for specifying stochastic versions of
equations of fluid dynamics related to turbulence, investigating their
properties, and constructing numerical approximations and asymptotics.

Turbulence is prominent in numerous contexts of significance for science and
technology, including oceanography, climatology, flight dynamics, navigation,
combustion and propulsion, etc. The proposed research deals with mathematical
models of turbulence that capture its essential behavior, and also yield
computationally tractable representations. Related numerical algorithms will
be developed, coded, and tested.

Predictive understanding of the dynamics of seismicity is one of the greatest challenges of modern geophysics. To accomplish such an understanding requires a combination of exploratory data analysis and relevant mathematical modeling. This project will provide a framework for modeling and analyzing regional seismicity based on the stochastic quantization approach and the use of non-linear filtering for marked point processes. The resulting description will allow one to represent regional fault networks, including details such as complex geometry, memory, interactions, and structural heterogeneities. The proposed research will expand upon ideas captured in existing geophysical models. In particular, one of the most important goals is to enhance Epidemic Type Aftershock Sequence (ETAS) approach to short-term earthquake modeling and forecasts and improve the model performance on larger spatial and temporal scales by incorporating known aspects of regional fault systems. The models constructed in the project will be tested on the observed seismicity of southern California and will leverage high-quality data from Southern California Earthquake Data Center (SCEDC) and the Southern California Earthquake Center Community Fault Model (SCEC CFM).

Natural hazards pose a threat to society, and earthquakes are probably the greatest danger to the built environment in tectonically active areas. This project will contribute to the predictive understanding of the seismicity dynamics by merging the state-of-the-art mathematical methods and seismicity data. Ultimately, the proposed work will be used to formulate earthquake forecasting strategies, which will be tested within the international Collaboratory for the Study of Earthquake Predictability (CSEP), recently organized by SCEC. The collaborative and cross-disciplinary approach of this project makes it an ideal training ground for graduate students and young scientists.

Dynamics and stochastics, both interpreted in their broadest sense, are important mathematical areas that have made many contributions to applications. There are many natural links between these two areas that lead to a better appreciation of techniques and methods used in either field. In addition, an approach that combines stochastic modelling with a dynamical-systems analysis often has the best chances of tackling a problem successfully: three examples are nonlinear optics, where noise from various sources plays an important role in laser and fiber dynamics; stochastic networks, where stability of the network can, in many cases, be determined by the analysis of a related fluid limit that is characterized by an ordinary or partial differential equation; and cell physiology, where stochastic models of ion channel gates often provide better agreement with experiments. A similar trend occurs on the theoretical level: probabilistic methods are an important tool that aids in the analysis of PDEs, for instance in the derivation and validation of scaling laws and the dynamics of fronts in heterogeneous media; conversely, dynamical-systems methods provide insight into the behavior of PDEs with noise. The goal of this project is to broaden and enhance the scope and quality of the educational and research training provided to graduate students and postdoctoral fellows by integrating research and education in the fields of dynamics, stochastics, and their applications and to involve more undergraduate students in courses and research experiences in applied mathematics.

Dynamical systems and stochastic processes are highly active and exciting fields of research that make important contributions to many applications in economics and the natural and social sciences, whilst also being of intrinsic mathematical interest. Dynamical-systems theory is concerned with time-dependent processes, while stochastics deals with nondeterministic, random processes. Examples where the interplay between these two fields is important are noise fluctuations in the design of high-power lasers, the long-time behavior of random interacting systems such as consensus formation in social networks, self-assembly of micro- and nanostructures for drug delivery, random fluctuations in biological cell processes, and importance sampling of rare events in finance. Providing more systematic and integrated training in dynamics and stochastics for graduate students and postdoctoral fellows will prepare these groups better for careers in academia and industry. It will also lead to an increase in the number of US mathematicians trained in stochastic systems, and therefore make the US better able to compete with Europe, which has a larger community in this area. The initiatives for undergraduate students will increase the number of students who are exposed to applied mathematics and are engaged in summer research experiences.