Algorithms based on adjoint function ensembles for inverse modeling - PowerPoint PPT Presentation

Algorithms based on adjoint function ensembles for inverse modeling of transport and transformation of atmospheric pollutants A.V. Penenko Institute of Computational Mathematics and Mathematical Geophysics SB RAS Novosibirsk State University CITES 2019, Moscow, 3-6 June 2019

Motivation • The progress in the parallel computations technologies: the speedup is achieved trough the intensive parallelization (ensemble algorithms, splitting, decomposition, etc.) • The progress in nonlinear ill-posed operator equation solution (different regularization methods, SVD, convergence theory, etc.) and analysis methods • Variety of applications for the inverse and data assimilation problems for advection-diffusion-reaction models. E.g. • Air quality studies (environmental applications) • Morphogen theory (developmental biology) • Unified approach to different measurement data including image-type measurement data in air quality applications (large volume of data with unknown value w.r.t. the considered inverse modelling task): • Time-series ( in situ ) • Vertical concentration profiles (aircraft sensing, lidar profiles, etc). • Satellite images (total column 2D images). 2

Advection-diffusion-reaction model    [0, ] T  rectangular in (0D,)1D,2D The domain T                    l diag u P t ( , , ) φ y = ( , , ) t φ y f r , l l l l l l l l  t destruction-production advection-diffusion  K l 1, , N - number of species Model scale: 0D,1D,2D c                R n diag = , ( , ) x t (0, ], T l l l l l out BC:      D = , ( , ) x t (0, ], T l l in    0 = , x , = 0, t IC: l l Linear measurement Inverse problem Direct problem operators, e.g.   operator   ( ) ( ) I = φ [ r , y ] I, Pr • Pointwise concentrations Umes     R Y • Total column 2D images  : ,    • Vertical profiles r , y φ a  To find (or) Given Noise Subspace SpanU mes 3

Adjoint problem Lagrange type identity (sensitivity relation)         2 1  (2) (1) " h , φ = δr , [ ] h y , K t ( , φ , y , φ , y ) [ ] h  R Y  ( m ) ( m ) ( m ) φ φ r [ , y ] Sensitivity functions           2 1   2   2 1 (2) (1) " (2) (1) " (2) (1) " K t ( , φ , y , φ , y ) = ( , t φ ; y , y ) P t ( , φ ; y , y ) diag ( φ ), y y m m      m Adjoint problem: Given find : h φ , , y , 1,2, Ψ                 2 2 1 1 l u ( diag ( ) ) ( ( , G t φ , y , φ , y ) Ψ ) = h , l l l l  t                 2 2 1 1  2 2  G t ( , φ , y , φ , y ) diag P t , φ , y  -divided difference operator                        2 1 1 1  2 1 1 P t , φ , φ ; y diag φ t , φ , φ ; y , + adjoint problem boundary conditions TC: Ψ ( ) = 0, T           r r h , φ = r , [ ] h Linear parametrizations m m m m  R m m 4

Gradient algorithms (inverse source problem) Given the cost function    2    J r = [ ] r I . l l l  L ( ) 2 T  l Lmes if the parameters are smooth enough, then Nc         2 [ ] r I , l L      φ r [ ] l l mes J ( ) r [ , , ], r r h h =    0, l L      mes l =1 Penenko, V. V. & Obraztsov, N. N. A variational initialization method for the fields of the meteorological uncertainty elements // English translations Soviet Meteorology and Hydrology, 1976 , 11 , 3-16 Пененко, В. В. Методы численного моделирования атмосферных процессов Гидрометеоиздат, 1981 4DVAR Dimet, F.-X. L. & Talagrand, O. Variational algorithms for analysis and assimilation of meteorological state observations: theoretical aspects // Tellus, 1986 , 38A , 97-110 E.g. Polak-Ribiere conjugate gradient algorithm implemented in GSL                 k+1 k k k k k k        r := r s = arg min J r s  >0             k k  k-1  g , g g k   k k-1 g s , k >1          k  k k  k s = , = , g = J ( r ).        r k-1 k-1 k g , g  g , k =1  5

Sequential Data Assimilation at the Splitting Stages    j 1 j t t t Сonsider a splitting scheme on the interval ([Gordeziani,Meladze, 1974], [Samarski, Vabishevich, 2003]), • Transport Variational Direct Algorithm   r   DA WRT         l L f r       t Next step        j 1 j {x t} [0 X ] [ t t ]       j j ( ) t ( ). t   Measurements I           j 1 j 1 t t       { , , , } x y z c             { , , } x y z 1.  c     j 1 t     { , , } x y z   • Transformation Variational   r   DA WRT r   l c     l P t ( , ) c l c c  l t   r          ( , t ) f r l c c l c l        j 1   j 1  t t     c     Penenko, A. et al. Sequential Variational Data Assimilation Algorithms at the Splitting Stages of a Numerical Atmospheric Chemistry Model // Large-Scale Scientific Computing, Springer International Publishing, 2018 , 536-543 6 Penenko, A.; Mukatova, Z. S.; Penenko, V. V.; Gochakov, A. & Antokhin, P. N. Numerical study of the direct variational algorithm of data assimilation in urban conditions // Atmospheric and ocean optics, 2018 , 31 , 456-462

Adjoint problem solution ensembles in inverse problem algorithms Cost function based • Cost functional gradients with adjoint problem solution (single element ensemble for the discrepancy) • Gradient computation with adjoint ensemble when adjoint is independent of direct solution [Karchevsky, A., Eurasian journal of mathematical and computer applications, 2013 , 1 , 4-20] • Representer method (optimality system decomposition, ensemble generated for discrepancies for each measurement data) [Bennett, A. F. Inverse Methods in Physical Oceanography (Cambridge Monographs on Mechanics) Cambridge University Press, 1992] Sensitivity relation based • Coarse-fine mesh method (Sequential solution refinement with sequential adjoint problems solving) [Hasanov, A.; DuChateau, P. & Pektas, B. An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation// Journal of Inverse and Ill-Posed Problems, 2006 , 14 , 1-29] • Adjoint function for each measurement datum with the solution of the resulting operator equation [Marchuk G. I., On the formulation of certain inverse problems, Dokl. Akad. Nauk SSSR, 156:3 (1964), 503–506], [Issartel, J.-P. Rebuilding sources of linear tracers after atmospheric concentration measurements // Atmospheric Chemistry and Physics, Copernicus GmbH, 2003 , 3 , 2111-2125] 7

Sensitivity operator (inverse source problem)    Given functions   (ξ) U u SpanU Image (model) to meas   structure          (2) (1) (2) (1) ( ) ( ) H φ r [ ] φ r [ ] φ r [ ] φ r [ ] u e operator [Dimet et al,2015] U   Sensitivity relation           (2) (1) ( ) ( ) φ r [ ] φ r [ ] u [ r r u ] r r (2) (1) (2) ( ) 1 (Lagrange type identity)    R   Sensitivity    M [ r r ] , Parallel w.r.t. U (2) (1)       ( ) ( ) z Ψ r [ r u ] z e U a (2) (1) operator        The inverse problem solution for any r and satisfy r U   Parametric family of quasi-            H  H I φ r [ ] = M [ r , ] r r r I . Pr   linear operator equations U U U   Umes 8