Calibration: an essential Calibration: an essential inverse problem inverse problem Haroldo Fraga de Campos Velho Lab. Associado de Computação e Matemática Aplicada E-mail: haroldo@lac.inpe.br http://www.lac.inpe.br/~haroldo 3º Colóquio de Matemática da Região Sul – Florianópolis (SC), Brasil
Presentation outline � Inverse problem classification � Model calibration (or: parameter identification) � Model calibration: an optimization problem � Model calibration: non-linear mapping � Applications � Hydrological model � Fault diagnosis � Ocean colour (two approaches) � Data assimilaton � Final remarks
Classification of Inverse Problems 1. Regarding to mathematical nature of the method: Explicit Implicit 2. Regarding to statistical nature of the method: Deterministic Stochastic 3. Regarding to nature of the estimated property: Initial condition Boundary condition Source/sink term Properties of the system 4. Regarding to nature of the solution (Beck): Parameter estimation Function estimation 5. Silva Neto / Moura Neto: Type-1 (DP-f e IP- f) Type-2 (DP- ∞ e IP- f) Type-3 (DP- ∞ e IP- ∞ ) Type-4 (DP- f and IP- ∞ ) – does not apply
Model calibration (or: parameter identification) “ Solve an Inverse Problem is to determine unknown CAUSES from the observed or desired EFFECTS ” - H.W. Engl (1996). ℑ D M − 1 ℑ causes effects M space of parameters or models D space of data or observations ℑ 1 forward model − ℑ inverse model
Model calibration: an optimization problem � Inverse problem formulated as na optimization problem 2 Exp Mod = − + α Ω J ( f ) T T ( f ) ( f ) α 2 α : regulariza tion parameter Ω ( f ) : regulariza tion operator
Model calibration: non-linear mapping � Non-linear mapping by artificial neural network FORWARD MODEL K x y Parametrer Data INVERSE MODEL K -1
Model calibration: non-linear mapping � Non-linear mapping by artificial neural network ( ) a o f = f , (non - linear mapping) x x x n NN n n Training phase : determination of the connection weights, bias Activation phase : generating analized data.
Applications: Hydrological model � Inverse problem classification
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum � A , B , C, E f , F f = known matrices � Ѳ f = [ f a f p f s ] = faults � f a = parameter vector from actuator(s) � f p = parameter vector from the process � f s = parameter vector from sensor(s)
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum � Ns = number of samples (in time) = output vector measured by sensors � = output vector computed by the model �
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum One more equation introduced:
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum Problem structure (biadjacentcy matrix)
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum Problem structure (biadjacentcy matrix) Dulmage-Meldelsohn decomposition
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum � Dulmage-Meldelsohn decomposition: � Allow partioning the model M on three parts (graph): � Under determined ( M 0 ) � Just determined ( U j M j ) � Over determined ( M + ) � The D-M decomposition gives the order between connected components of the graph. � Measures on any variable x 1 , x 2 , ..., x 5 make the fault structurally detectable.
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum ACO: Ant Colony Optimization
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum ACO: Ant Colony Optimization Best ant = min J(x) Pheromone deposit (bolt dotted curve)
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum ACO: Ant Colony Optimization
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum Fuzzy-ACO
Applications: Fault diagnosis � Fault diagnosis: inverted-pendulum ACO vs Fuzzy-ACO: results
Application: Ocean colour (function estimation) Hydrologic optics Hydrologic optics Radiative transfer process transfer process Radiative Inverse problem Inverse problem Ant colony optimization Ant colony optimization
Applications: Ocean colour azimuthal angle Light beam Boundary Conditions Polar angle Scattering Absorption angle Internal coefficient sources Scattering medium coefficient Optical depth Scattering phase function Boundary Conditions
Applications: Ocean colour Radiance :a measure of amount of light beam energy where Internal source
Applications: Ocean colour � Multi-spectral estimation N N ξ ∑∑ [ ] λ 2 [ ] Exp Mod = − + α Ω J ( p ) L L ( p ) p k , l k , l = = k 1 l 1
Applications: Ocean colour � Multi-spectral estimation
Applications: Ocean colour Absortion and scattering coefficients: ] [ ] [ − λ − 0 . 014 ( 440 ) 0 . 65 w c = + + g 0 . 06 ( ) 1 0 . 2 a a a C z e r , g g g [ ] 0 . 65 = λ b 550 0 . 30 C ( z ) r , g g h [ ] ( ) 2 − − 1 2 z z / s = + C ( z ) C e max 0 Chorophyll concentration: π s 2 2 − 1 1 f Θ = Phase function: p ( cos ) ( ) 3 2 π 4 2 + − Θ 1 f 2 f cos
Applications: Ocean colour Multispectral reconstruction:different seeds an avearge value for ACS
Applications: data assimilation � Data assimilation: the problem
Applications: data assimilation � Data assimilation: Methods � Newtonian relaxation (nudging) � Statistical (“optimal”) interpolation � Kalman filter � Variational method: 3D and 4D � New methods for data assimilation: � Ensemble Kalman filter � Particle filter � Artificial neural networks
Applications: data assimilation � Data assimilation: essencial issue Hybrid Methods in Engineering: (2000) 2(3): 291-310
Applications: data assimilation � Data assimilation: Kalman filter ∂ F [ ] 2 = ≈ + + ∆ ≈ x F x , t F x O ( t ) E x + n 1 n n n n n n ∂ x = t t n ?
Applications: data assimilation � Data assimilation: adaptive Kalman filter - Adaptive KF - Fokker-Planck - (L)EKF 1. Advance in time : f = q q + n n 1 f , q a , q = P P + n n 1 2 . Kalman gain 4 . Update error covariance [ ] [ ] − 1 q f , q o f , q a , q q f , q = + G P W P = − P I G P n n + + n 1 n 1 + + + n 1 n 1 n 1 3. Update estimation [ ] ( ) a f q f f f ν = + + − q q G q q + n 1 + + + + + n 1 n 1 n 1 n 1 n 1
Applications: data assimilation � Data assimilation: Neural Networks
Applications: data assimilation � Data assimilation: Neural Networks
Applications: data assimilation � Data assimilation: Non-extensive Particle Filter
Applications: data assimilation � Data assimilation: Non-extensive Particle Filter
Applications: data assimilation � Data assimilation: Non-extensive Particle Filter Helaine C. Morais Furtado Haroldo F. de Campos Velho Elbert E. Macau
Applications: data assimilation � Error: [Kalman, Particle, Variational] x Neural Network
Applications: data assimilation � Data assimilation: Ocean (shallow water 2D)
Applications: data assimilation � Data assimilation: Space weather Interaction Sun-Earth: Solar Propagation Impact on Activity magnetosphere ionosphere
Applications: data assimilation � Data assimilation: Space weather
Applications: data assimilation � How can we find a good architecture for an ANN? Standard approach: employing an empirical process: � � Some preliminaries topologies are defined and tested, � changing the ANN parameters, and the process is � re-started. Disadvantages: � � Requeriment of a continuos effort from an expert � This can require a long time
Applications: data assimilation � How can we find a good architecture for an ANN? Alternative approach : � � Formulating the problem as an optimization problem. The goal: � � Determine the best set of parameters for the ANN to optimize an objetive function.
Applications: data assimilation � Neural network: self-configuring � Best configuration for multi-layer perceptron neural network (MLP-NN): � Multi-Particle Collision Algorithm (MPCA) � The MLP network was configured to identify the NN applied to data assimilation. � Two data set used for training: NN emulating Kalman filter for data assimilation.
Applications: data assimilation � Neural network: self-configuring � PCA Algorithm (Introduced by prof. Wagner F. Sacco, UFOP) � Algorithm inspired from particle traveling inside of a nuclear reactor: � A bsorption � S cattering
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