Inverse problems for the study of climate-ecological processes in - PowerPoint PPT Presentation

Inverse problems for the study of climate-ecological processes in the conditions of anthropogenic impacts Пененко В.В. Цветова Е.А.

Report content: • What are inverse problems? • definitions • differences from direct problems • implementation specifics • How can we summarize the formulations of inverse problems? • variational approach • What is the meaning of the term climate-ecological? • How to take into account the different scales in climate & ecology? • How to consider and evaluate anthropogenic effects in inverse problems? • The main types of inverse problems on research topics 2

What are inverse problems? Direct problems are formulated as “In essence, the inverse of this task is the task in which the desired and the given are Initial- boundary- value problems. reversed” (Wikipedia, 1st grade of Goals defined; elementary school). • definitions What is known? • differences from The inverse problem is a type of tasks, often all model parameters; direct tasks arising in many sections of science, when sources of impact; • Specific the values of model parameters must be initial and boundary conditions. implementation: obtained from observable data. • Conditional well- Examples of inverse problems can be found posedness in the following areas: geophysics, Problems are solved “forward” in • Ill-posednes astronomy, medical imaging, computed time, they are mostly correct, • interrelated models tomography, remote sensing of the Earth, sometimes conditionally correct. and observational spectral analysis, scattering theory and non- data destructive testing tasks, etc. 3

What is the meaning of the term climate-ecological problems in the Earth system? • Earth system climate • Man • climatic zones of the globe • regional climate • Environment • climate of urban • Ecology agglomerations • meso-climate • microclimate 4

Features of mathematical modeling in environmental protection problems • The need to obtain and refine the • Nonlinearity, a variety of time forecast as measurement data scales and processes, high (up to becomes available . 1012) dimension of the considered Real time models operation Main pollutants: • Ozone (O3), carbon monoxide (CO), ammonia (NH3), hydrogen sulfide (H2S), sulfur The dioxide (SO2), nitrogen oxides (NO, NO2), formaldehyde difficulty (HCHO), ... of direct • Aerosols: PM 10, PM 2.5, nano modeling ... • radioactive ... High • biologically active ... uncertainty The uniqueness of • Unknown sources of impacts, initial and boundary conditions, rates of each situation transformations and other parameters of models, structure of models. • Lack of required measurement data 5

How can we present formulations of direct and inverse problems in a generalized form? Variational principle Formulation with weak constraints, combining • mathematical models of processes (in the form of an integral identity); • available observational data; • target (goal) functionals. For unification, we use • the classical method of Lagrange multipliers (1762); • adjoint functions • the concept of adjoint integrating factors 6

General structure of modeling systems             L X L ( U Y f r , , , ) G ( U Y , ) f r 0  t       0 0 t 0 a The vector of functional arguments of the modeling system     X U Y,f,r,ξ , Results and models of observations %       H ( ) , m 1, M m m m Target functionals of studies          ( ) F ( ) ( , ) x t dDdt ( F , ) , k 1, K . k k k k k D t D 7 t

Variational principle for combining all objects of the modeling system Integral identity         * * I X φ , L X ,φ      *      G ( U Y , ) f r φ , dDdt 0     t  D t   X   I , 0 system energy balance equation  * * ( φ Q D ) adjoint functions vector t 8

Adjoint functions 1. "Global": Distributed Lagrange multipliers for combining process models, observational data, and prediction target functionals within a variational principle; 2. "Local": In the modeling technology in the decomposition and splitting mode in combination with the methods of finite elements / volumes: Solutions of homogeneous adjoint problems for the construction of numerical schemes with conservation laws (adjoint integrating factors) 9

System of convection-diffusion-reaction equations in models of hydrothermodynamics and atmospheric chemistry                L ( ) X i div u div grad S ( ) φ ( f r ) 0, i 1, n   i i i i  t i i      S ( ) φ P ( ) φ ( ), φ i 1, n i i i i 10

Extended functional of the variational principle with weak constraints  ( r 0)      h  * h  * ( , X , ) η I ( , X )   h D t   h            h h 0.5 ( W r r , ) ( W ξ ξ , ) 0.5 ( W ) ( ) 2 2 h 3 3 h 1 1 m 4 k D D D t t 11

Decomposition and splitting for the construction of numerical schemes and algorithms with the properties of the total approximation   q   q φ φ        ; G ( U Y , ) φ ;       t t       1 1 q q q          f f ; r r ; 1; t t t     j j 1       1 1 1  φ          φ f r 0 ; 1, q       t q q                j j j 1 j 1 j 1 j 1 φ φ ; φ φ ; r r          1 1 12

Gateaux-variations for functionals. Algorithms for sensitivity relations    S  ( ), * S X φ , Definition of Gateaux derivatives for the functional    p    p        ( S ) S S , p 1     p     0 Taylor series    S 0 at For each structure of vectors   p                       p  S S S O , p 1  !   1 When p = 1: Euler-Lagrange equations: systems of basic and adjoint equations, equations for uncertainties; sensitivity relations for target functionals to parameter variations When p = 2: sensitivity ratios of the second order; Hessian 13

Variations of extended functional ( p=1)  %   %   %     h h h %   h   *     k k k (,..., ) , φ , φ , r       k    * φ φ r               % % % %  h  h  h  h         k k k k , ξ , Ψ , Y , U         m     ξ Ψ Y U         m Stationarity conditions Sensitivity relations  %    h %      h k , Y  % h % h % h % h         k  Y       k 0, k 0, k 0, k 0  φ  φ  r  ξ *  %    h %   h    k , U  k  U 14  

Basic algorithm for forward / inverse modeling %  h     h k φ G ( φ U Y ) - f - r , , 0 direct problem  t  φ %  h        T T k ( ) φ A ( φ U Y φ , , ) d 0, adjoint problem t k k k  φ     φ ( ) x 0, x D       h T d ( ( φ ) 0.5 ( M )),  k t t k k 1 1  φ      0 0 1 φ φ M φ ( x , 0 ), t 0 , Initial data with uncertainty a 3 k   1 * model uncertainty function r x ( , t ) M φ ( x , t ), 2 k      h    A ( φ U Y , , ) φ G ( φ φ U Y , , ) ,       0 � � � approximation of time derivatives 15

Modeling technology based on variational principle Five fundamental spaces for environmental forecasting and design: • State functions of the direct problem • Solutions of adjoint problems • Uncertainty functions of process models • Sensitivity functions of process models and target functionals to variations of model parameters, initial data, functions of boundary conditions and sources of impacts • Sensitivity functions of target functionals to variations in observational data 16