sensitivity operator inverse modeling framework for advection … · 2021. 4. 26. · sensitivity...
TRANSCRIPT
![Page 1: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/1.jpg)
1 Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2 Novosibirsk State University, Novosibirsk, Russia3 Siberian Regional Hydrometeorological Research Institute, Novosibirsk, Russia
Sensitivity Operator Inverse Modeling Framework
for Advection-Diffusion-Reaction Models with Heterogeneous Measurement Data
Alexey Penenko 1,2, Vladimir Penenko 1,2, Elena Tsvetova 1, Alexander Gochakov 3, Elza Pyanova 1, and
Viktoriia Konopleva 1,2
vEGU, 2021
E-mail: [email protected]
![Page 2: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/2.jpg)
•Unified approach to different measurement data including image-type measurement data
(large volume of data with unknown value w.r.t. the considered inverse modelling task):
•Pointwise and time-series concentrations data (in situ data)
•Concentration profiles (aircraft sensing, lidar profiles, etc).
•Satellite images of concentration fields.
Motivation
•The progress in parallel computations technologies: the algorithms should allow natural mapping to the parallel
computation architectures (ensemble algorithms, splitting, decomposition, etc.)
•Unified framework for inverse and data assimilation problems (briefly, inverse modeling problems) for non-
stationary advection-diffusion-reaction models in various applications. E.g.
•Air quality studies (environmental applications)
•Biological problems (morphogen theory & developmental biology)
•The progress in nonlinear ill-posed operator equation solution (different regularization methods, SVD,
convergence theory, etc.) and analysis methods.
2
Solve & Analyze in the same way
Combine various sources of data
![Page 3: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/3.jpg)
Measurement operators
Advection-diffusion-reaction model
( )( )diag ( , , ) = ( , , ) ,ll ll
l l l l lP t t ft
r
− − + + +
yu φ φ y
0= , , = 0,l l t x
The domain
( )( )diag = , ( , ) (0, ],R
l l l l l outt T + n x
[0, ]T T =
= , ( , ) (0, ],D
l l int T xBC:
: ,Q
q
→ φ
Direct problem
operator
advection-diffusion destruction-production
= [qH +( )I φ ] I,
Inverse problem
IC:
rectangular
Given NoiseTo find
• Pointwise concentrations
• 2D images
• vertical profiles,
• etc
Model scale: 0D,1D,2D,3D 1, , cl N= - number of species
{ , , }q r y=
3
H
, :l lP
• Polynomial
• Sigmoid
functions
• Integral
Uncertainty
function
specification
![Page 4: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/4.jpg)
Conventionally used to
evaluate misfit cost
function gradients
(2) (2) (2) (1) (1), , , ,[ , ] [ , ] ( , , , , ) ,yR
K t q qy
+ = + +h H φ h H φδ h φμ δ φr δ φH
Sensitivity relation
( ) = 0,T+Ψ
Lagrange type identity (sensitivity relation)
-divided difference operator
( ) ( ) ( )1 1( ( ) ) ( ( , , ) ) = ,l
l l l ldiag G tt
h
− − − +
2u μ φ φ Ψ
( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )1 1 1 1( , , ) , , , , , ,G t diag P t P t diag t
= + − 2 2 2 2
φ φφ φ φ φ φ φ φ φ
+ adjoint problem
boundary conditions
(( )) [ ]m mq=φ φ
Sensitivity functions
Adjoint problem: Given find:( ) ( )
, , , 1, 2,m m
m =h φ μ
Linear parametrizations: m m
m
rr = , = , [ ]m m Rm
r
h φ h
Ψ
(2) (1) = −φ φ φ
= , 1, ,m
m
t ll m MH=
=
4
( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( )1(2) (2) (1) (1)( , , , , ) , ; , ; ,y yK t q q t P t diag= −2 2 1 2 2 1
φ φ φ y ,y φ y ,y φ
![Page 5: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/5.jpg)
Sensitivity Operator Construction
5
![Page 6: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/6.jpg)
Sensitivity operator
Sensitivity relation (Lagrange type identity)
measU U
= (ξ)uGiven functions (functionals)
( )2 1 1 ( )( )[ ] [ ] [ ] [ ]UH
− = − ( ) ( ) (2) ( )φ q φ uq φ q φ q e
( )( )] [ ][ ] [Q
M
− = −(2) (1) (2) (1)(2) (1)qφ q φ q u qu qq
Image (model) to structure operator
[Dimet et al,2015]
Sensitivity operator ( ) ( )[ ][ ]
,U
R
R
MM
→
(2) (
(
1)
2) (1)
q q uq q
z z e
The inverse problem solution for any and satisfy( )
q q U
( ) ( )( )[ ] = [ , ] .PrU U UU
meas
H M H
− − + I φ q q q q q I
Parametric family of quasi-linear
operator equations
Parallel computation w.r.t.
U
To control the data considered and dimensionality
Idea: [Marchuk G. I., On the formulation of certain inverse problems, Dokl. Akad. Nauk SSSR, 156:3 (1964), 503–506], 6
![Page 7: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/7.jpg)
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]
• Using sensitivity function aggregates (supports, level sets, etc.): [Pudykiewicz, J. A. Application of adjoint tracer transport equations for
evaluating source parameters Atmospheric Environment, Elsevier BV, 1998, 32, 3039-3050], [Penenko, V. et al. Methods of sensitivity theory and
inverse modeling for estimation of source parameters, Future Generation Computer Systems, Elsevier BV, 2002, 18, 661-671], [Bieringer, P. E. et
al. Automated source term and wind parameter estimation for atmospheric transport and dispersion applications Atmospheric
Environment, Elsevier BV, 2015, 122, 206-219]
• 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],
• Passive tracer (linear) case: [Issartel, J.-P. Rebuilding sources of linear tracers after atmospheric concentration measurements //
Atmospheric Chemistry and Physics, Copernicus GmbH, 2003 , 3 , 2111-2125]
• Nonlinear case, pointwise sources: [Mamonov, A. V. & Tsai, Y.-H. R. Point source identification in nonlinear advection-diffusion-
reaction systems Inverse Problems, IOP Publishing, 2013 , 29 , 035009]
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]
Adjoint problem solution ensemblesin inverse problem algorithms
7
![Page 8: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/8.jpg)
Adjoint ensemble construction
= | [0, ], [0, ], [0, ], ,x x y y t t mesU L ηθ θ θx y t
e• Fourier cos-basis
• Wavelets, curvlets, etc. [Dimet et al.,2015]
«a priori» approach – ensemble for the class of problems (smoothness)
=1
1( , , ) ( , , ) ( , , ), =
= ,
0,
Nc
k
t x i y j
x y t
l
C T t C X x C Y y le
l
2 cos , > 01( , , ) = .
1, = 0
t
C T t TT
«a posteriori» approach – ensemble for the considered problem
• «Adaptive basis»: chose elements of with maximal projections
[Penenko, A. V.; Nikolaev, S. V.; Golushko, S. K.; Romashenko, A. V. & Kirilova, I. A. Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering //
Math. Biol. Bioinf., 2016 , 11 , 426-444 (In Russian)]
• «Informative basis»: use left singular vectors of the operator [ , ]Um
(0) (0)r r
U[ ] ,Pr
Umes
−(0)
ηθ θ θx y t
φ r I e
[Penenko, A.; Zubairova, U.; Mukatova, Z. & Nikolaev, S. Numerical algorithm for morphogen synthesis region identification with indirect image-type measurement data //
Journal of Bioinformatics and Computational Biology, World Scientific Pub Co Pte Lt, 2019 , 17 , 1940002]
(inverse source problem, 2D, components are measured )
Defined by the adjoint problem sources sets
8
![Page 9: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/9.jpg)
Inversion algorithm
( )( ) ( )( ) ( )( ), [ , ] [ ,, ]]Pr [=UU
m
UU
e s
UU
a
H m Hm m − − + + − −
* **qφ q q q qq qI q q q δIq
,
= .Pr,
T T
unknowns
UUT T
mesunknowns
m mm NH I
m m m N
+
+
−
δq φ q
C+
-truncated SVD inversion parametrized by conditional number
, inversesource problem
, inversecoefficient problem
src t x y
unknowns
coeff
L N N NN
N
=
Newton-
Kantorovich
-type update
[ , ]Um m q q
( )unknownsN
Nonlinearity:
sequential increase of
the conditional number
Admissible solutions:
projection regularization
Noise:
discrepancy
principle
Optional monotonicity:
monotonic decrease of the
discrepancy
Theoretical foundations: [Issartel, J.-P., 2003], [Cheverda V.A., Kostin V.I., 1995], [Kaltenbacher et al, 2008],
[Vainikko, Veretennikov, 1986] 9
![Page 10: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/10.jpg)
Adjoint ensemble inverse modeling framework
Inverse Problem Statement(PDE/ODE)
Adjointproblem
Lagrange-type Identity
Family of quasi-linear ill-posed operator
equations
Generalensemble
Inverse problem solution algorithm
Sensitivity function
Finite ensemble
Ill-posed nonlinear
operator equation
methods
# computation
threads &
memory available
Data assimilation mode
Sensitivity operator analysis
measUProcess model
10
«Illumination»functions
Singular spectrum
Measurement
data
specification
Uncertainty function
specification
Information quantity
Projection to the kernel
![Page 11: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/11.jpg)
Heterogeneous Measurement Data
11
![Page 12: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/12.jpg)
Heterogeneous measurement data
12
• «Timeseries»
• «Pointwise»
• «Integral»
• «Snapshot»
0=1
, = ( , ) ( , ) .
Nc T
l lHl
a b a t b t d dt
x x x
( )
( ) ( ) ( )( ) ( ) ( , ), 0, , , ,m m m
m measl
x t t T x l L ( ) ( ) ( ) ( )= ( , , ) ( ) ( );h C T t x x l l
− −
( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( , ), , , ,m m m m m
Tm measl
x t x t l L ( ) ( ) ( ) ( )= ( ) ( ) ( );h x x t t l l
− − −
( )
( ) ( ) ( )( ) ( ) 0( , ) , , ,
T m m m
m measl
x t dt x l L ( ) ( ) ( )= ( ) ( ).h x x l l
− −
( )
( ) ( ) ( )( ) ( ) ( , ), , 0, , ,m m m
m measlx t t l T L x ( ) ( ) ( ) ( ) ( )
= ( , , ) ( , , ) ( ) ( );x yh C X x C Y y t t l l
− −
In variational approach we sum the misfit functions
for different measurements, in the adjoint ensemble-
based approach we join the corresponding
ensembles
A. Penenko, V. Penenko, E. Tsvetova, A. Gochakov, E. Pyanova, V. Konopleva Sensitivity Operator-Based Approach to the Interpretation
of Heterogeneous Air Quality Monitoring Data // submitted
![Page 13: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/13.jpg)
Specific measurement types
13
O3 Monitoring sites locations
(b): time series (red crosses),
pointwise measurements in
space and time (blue circles),
integrals over a time interval
(magenta triangles).
Pointwise (100) Time-series (60)
Integral (5) Snapshot (20x20)
3.5 day-long
summer
scenario,
source-term
uncertainty,
NOx-O3 cycle
NO Emission sources
![Page 14: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/14.jpg)
«Composite»measurements
14
Composite
Concentration field
«сcontinuation» error
Uncertainty (source-term)
identification error
![Page 15: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/15.jpg)
15
Estimating the inverse modeling result prior to solution
Inverse problem solution times measured
in direct problem solution times
Relative error vs computation time
![Page 16: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/16.jpg)
Sensitivity operator analysis
16
( )( ) ( )( )M q q H I A q− = −
( )( ) ( )( )T T T TM MM M q q M MM H I A q+ +
− = −
«Illumination» function characteristics: [ ] T T
l l lE M M MM M+
=
( )1
NT T T T
l ll
M MM M diag M MM M+ +
=
=
Projector to the orthogonalcomplement of the sensitivity
operator kernel
Equivalent to the generalized Sensitivity functions
Issartel, J.-P. Emergence of a tracer source from air concentration measurements, a new strategy
for linear assimilation // Atmospheric Chemistry and Physics, 2005, 5, 249-273
( )T Tdiag M M M M+
Kappel, F. & Munir, M. Generalized sensitivity functions for multiple output systems // Journal of Inverse and Ill-posed Problems, 2017, 25
Pseudo-inverse
( )( )T TM MM M q q+
−
Singular spectrum decay
![Page 17: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/17.jpg)
17
Sensitivity operator analysis«Illumination»
function Reconstructed
solution (~500 iters.)
Projection of the
«Exact»
solutions to the
orthogonal
complement
of the sensitivity
operator kernel
(~1 iter.) Reconstruction error & its
predictions by projection
Singular spectrum
![Page 18: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/18.jpg)
Thank you for your attention!
18
The work has been supported by
Ministry of Science and Higher Education of the Russian Federation Large Scientific Project
No. 075-15-2020-787 (heterogeneous measurement systems)
Penenko A.V. Penenko V.V.
Pianova E.A.Gochakov A.V.
Tsvetova E.A.
Konopleva V.S.
![Page 19: Sensitivity Operator Inverse Modeling Framework for Advection … · 2021. 4. 26. · Sensitivity relation ,T Lagrange type identity (sensitivity relation)-divided difference operator](https://reader035.vdocument.in/reader035/viewer/2022071608/61472177f4263007b1359f77/html5/thumbnails/19.jpg)
Sensitivity Operator & Adjoint Ensemble References
19
1. Penenko, A. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-
type measurements // Inverse Problems & Imaging (AIMS), 2020, 14, 757-782 doi:10.3934/ipi.2020035
2. Penenko, A. V.; Mukatova, Z. S. & Salimova, A. B. Numerical study of the coefficient identification algorithm based on ensembles of adjoint problem
solutions for a production-destruction model // International Journal of Nonlinear Sciences and Numerical Simulation, accepted doi:
10.1515/ijnsns-2019-0088
3. Penenko, A. V. & Salimova, A. B. Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions // Numerical
Analysis and Applications, 2020, 13, 152-164 doi: 10.1134/s1995423920020068
4. Penenko, A.; Zubairova, U.; Mukatova, Z. & Nikolaev, S. Numerical algorithm for morphogen synthesis region identification with indirect image-type
measurement data // Journal of Bioinformatics and Computational Biology, 2019 , 17 , 1940002 doi:10.1142/s021972001940002x
5. V. V. Penenko A. V. Penenko, E. A. Tsvetova and A. V. Gochakov Methods for studying the sensitivity of atmospheric quality models and inverse
problems of geophysical hydrothermodynamics // Journal of Applied Mechanics and Technical Physics, 2019, Vol. 60, No. 2, pp. 392–399 doi:
10.1134/S0021894419020202
6. Penenko, A. V. A Newton–Kantorovich Method in Inverse Source Problems for Production-Destruction Models with Time Series-Type Measurement
Data // Numerical Analysis and Applications, 2019 , 12 , P. 51-69 doi: 10.1134/s1995423919010051
7. Penenko, A. V. Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–
Kantorovich Methods // Numerical Analysis and Applications, 2018 , 11 , P.73-88 doi:10.1134/s1995423918010081
8. Penenko, A. V.; Nikolaev, S. V.; Golushko, S. K.; Romashenko, A. V. & Kirilova, I. A. Numerical Algorithms for Diffusion Coefficient Identification in
Problems of Tissue Engineering // Math. Biol. Bioinf., 2016 , 11 , 426-444 doi: 10.17537/2016.11.426 (In Russian)
9. Penenko, A. On a solution of the inverse coefficient heatconduction problem with the gradient projection method // Siberian electronic
mathematical reports, 2010, 23 , 178-198. (in Russian)