The quantitative analysis of risks is important to reveal and identify the permitted levels of chemical, biological, radioactive agents/pollutants, etc. in the atmospheric environment and for population as well as change of climatic conditions for population's protection. In the air quality modeling for risk assessments there is an increased interest to quantitative methods of analysis of environmental processes in combination with cost-effective methods and methods from a point of view of economic and social development. These methods are required for analysis of environment quality, and they are connected with problems of population health. The interest is also related to methods of comparative analysis, strategy to reduce risks and expenses for practical realization of such approaches. It is important for the comparative analysis of management strategy of the current behavior of environment in order to reach the final aims and in the quantitative estimates of cumulative risks. These depend on the individual pollutants with multiple ways of impact on the environment and population health, and joint influence of multiple effects.
The problem to formulate such general metrics as well as targeted applications will require additional studies which should be done in collaboration with multidisciplinary specialists. Moreover, such studies need to be done in parallel and in cooperation with research in development of mathematical models and methods oftheir realization required to achieve the optimal estimates of air pollutant concentrations.
Several approaches are considered for the tasks of risk assessment and control theory:
• Methods of forward modeling based on analysis of ensembles of scenarios for different variants of input data and existing factors. These methods can be implemented with deterministic and stochastic (e.g., Monte Carlo method) algorithms.
• Methods using the adjoint equations generated for evaluation of linear functions such as scalar inner products defined in spaces of both the state functions of models and the weight functions.
• Variational methods for linear and nonlinear dynamical systems and functions in combination with methods of the control theory, risk theory, and sensitivity theory. These methods can be realized using a combination of the forward and inverse modeling approaches taking into account the uncertainties of models, parameters, and observational data.
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