On the Use of Analytical Techniques for Parameter Identification in Radiation and Particle Transport Models L. B. Barichello † † Instituto de Matem´ atica e Estat´ ıstica Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil New Trends in Parameter Identification for Mathematical Models New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Particle Transport Problems: Parameters Identification Nuclear Safety: source reconstruction Optical Thomography : absorption coefficients reconstruction Solution of the forward problems: analytical approaches (K. Rui, Programa de P´ os Gradua¸ c˜ ao em Engenharia Mecˆ anica, UFRGS) Inverse techniques (C. Pazinatto, Programa de P´ os Gradua¸ c˜ ao em Matem´ atica Aplicada, UFRGS) New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Techniques Source Reconstruction On the use of Adjoint Operator to the solution of an Inverse Problem ; Medium (1D) where physical properties and geometry are known Relevant Issues of Interest: Analytical Discrete Ordinates Method (ADO) ; 1 Adjoint flux: explicit solutions for spatial variable [9] 2 Computational time; 3 General source term: particular solutions 4 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Techniques Source Reconstruction On the use of Adjoint Operator to the solution of an Inverse Problem ; Medium (1D) where physical properties and geometry are known Relevant Issues of Interest: Analytical Discrete Ordinates Method (ADO) ; 1 Adjoint flux: explicit solutions for spatial variable [9] 2 Computational time; 3 General source term: particular solutions 4 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Techniques Source Reconstruction On the use of Adjoint Operator to the solution of an Inverse Problem ; Medium (1D) where physical properties and geometry are known Relevant Issues of Interest: Analytical Discrete Ordinates Method (ADO) ; 1 Adjoint flux: explicit solutions for spatial variable [9] 2 Computational time; 3 General source term: particular solutions 4 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Techniques Source Reconstruction On the use of Adjoint Operator to the solution of an Inverse Problem ; Medium (1D) where physical properties and geometry are known Relevant Issues of Interest: Analytical Discrete Ordinates Method (ADO) ; 1 Adjoint flux: explicit solutions for spatial variable [9] 2 Computational time; 3 General source term: particular solutions 4 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Inverse Techniques Source Reconstruction On the use of Adjoint Operator to the solution of an Inverse Problem ; Medium (1D) where physical properties and geometry are known Relevant Issues of Interest: Analytical Discrete Ordinates Method (ADO) ; 1 Adjoint flux: explicit solutions for spatial variable [9] 2 Computational time; 3 General source term: particular solutions 4 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Forward Problems Absorption coefficient estimation: biological tissues Two dimensional transport equation: 2D Explicit Nodal Formulation [3] 1 Alternative quadrature schemes × angular directions representation [4] 2 Radiative transfer equation: anisotropy effects 3 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
This Talk we report some studies and preliminary results we have carried out in this context we have considered parameters estimation: coefficients of a proposed expansion Isotropic sources Polynomial source 1 Piecewise funcions 2 Tikhonov’s Regularization Two-dimensional Radiative Transfer Forward Fomulation New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
This Talk we report some studies and preliminary results we have carried out in this context we have considered parameters estimation: coefficients of a proposed expansion Isotropic sources Polynomial source 1 Piecewise funcions 2 Tikhonov’s Regularization Two-dimensional Radiative Transfer Forward Fomulation New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
This Talk we report some studies and preliminary results we have carried out in this context we have considered parameters estimation: coefficients of a proposed expansion Isotropic sources Polynomial source 1 Piecewise funcions 2 Tikhonov’s Regularization Two-dimensional Radiative Transfer Forward Fomulation New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
This Talk we report some studies and preliminary results we have carried out in this context we have considered parameters estimation: coefficients of a proposed expansion Isotropic sources Polynomial source 1 Piecewise funcions 2 Tikhonov’s Regularization Two-dimensional Radiative Transfer Forward Fomulation New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
The Model We begin with the time-independent neutron transport equation which considers the distribution of the particles in non-multiplying homogeneous media, with one energy group, written as follows � σ s ( r , Ω ′ · Ω )Ψ( r , Ω ′ ) d Ω ′ Ω · ∇ Ψ( r , Ω ) + σ t Ψ( r , Ω ) = + Q ( r , Ω ) (1) � �� � � �� � S streaming term � �� � total scattering source term collision term σ t represents the total macroscopic cross section; σ s ( r , Ω ′ · Ω ) represents the differential scattering macroscopic cross section; Ω = ( µ, η, ξ ) represents the direction of the particle as a vector on the unit sphere S; Q ( r , Ω ) is the fixed neutron source term.; Ψ( r , Ω ) is the angular flux at r = ( x , y , z ) along direction Ω . New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Balance- Phase Space New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Directions New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Variables Angular variable: discrete directions New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Problem of interest σ d z k − 1 z k z 0 a 0 b Figure: Multilayer slab New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Forward Problem L ψ = S , L transport operator (2) � 1 L L ψ ( z , µ ) = µ ∂ ∂ z ψ ( z , µ ) + σψ ( z , µ ) − c � P l ( µ ′ ) ψ ( z , µ ′ ) d µ ′ β l P l ( µ ) 2 − 1 l =0 (3) ψ is the angular flux of particles, ; µ ∈ [ − 1 , 1] is the cosine of the polar angle measured from the positive z -axis, z ∈ (0 , z 0 ). σ is the total macroscopic cross-section, c is the mean number of neutral particles emerging from collisions, β l ’s are the coefficients of the expansion of the scattering in terms of Legendre’s polynomials P l ’s. ψ (0 , µ ) = g 1 ( µ ) + α 1 ψ ( z , − µ ) , (4a) ψ ( z 0 , − µ ) = g 2 ( µ ) + α 2 ψ ( z 0 , µ ) , (4b) µ ∈ [0 , 1], (known) incoming fluxes at the boundaries g 1 and g 2 , α 1 , α 2 ∈ [0 , 1], the reflection coefficients. New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
σ d is the absorption macroscopic cross-section of a neutral particles detector located within (0 , z 0 ), � z 0 � 1 r = � ψ, σ d � ≡ σ d ( z , µ ) ψ ( z , µ ) d µ dz (5) 0 − 1 is a measure of the absorption rate of neutral particles by the detector. In this formulation, σ d is defined as a positive constant in a given contiguous region of (0 , z 0 ) and zero outside the region. Thus, r measures the absorption rate of neutral particles within the detector’s region migrating from all possible directions. New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
Closely related to the transport operator L , the adjoint transport operator L † is defined by [6] L † ψ † ( z , µ ) = − µ ∂ ∂ z ψ † ( z , µ ) + σψ † ( z , µ ) � 1 L − c � P l ( µ ′ ) ψ † ( z , µ ′ ) d µ ′ β l P l ( µ ) (6) 2 − 1 l =0 where all physical parameters are the same as the ones in the transport operator L . The rate of absorption of neutral particles defined in Equation (5) might be alternatively computed as [6] � � � g 1 , g 2 , ψ † � ψ † , S r = − P (7) ψ † computed once New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
solving the adjoint transport problem L † ψ † = σ d (8) subjected to boundary conditions prescribed by ψ † (0 , − µ ) = α 1 ψ † ( z , µ ) , (9a) ψ † ( z 0 , µ ) = α 2 ψ † ( z 0 , − µ ) , (9b) � g 1 , g 2 , ψ † � for µ ∈ [0 , 1]. The term P represents a contribution of particles migrating on both inward and outward directions at z = 0 and z = z 0 and is given by � 1 � g 1 , g 2 , ψ † � � � g 1 ( µ ) ψ † (0 , µ ) + g 2 ( µ ) ψ † ( z 0 , − µ ) P = − µ d µ. (10) 0 New Trends in Parameter Identification for Mathematical Models, RJ, Brasil
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