Reconstruction of Coefficients and Source Parameters in Elliptic - PowerPoint PPT Presentation

Introduction Complementary Problems Diffusion-absorption elliptic system Model Optimization Methodology Results for the Diffusive-Absorption Model Results based on Nelder Mead algorithm Reconstruction of Coefficients and Source Parameters in Elliptic Systems by Nilson Costa Roberty Federal university of Rio de Janeiro News Trends in Parameter Identification for Mathematical Models IMPA, Rio de Janeiro, Brazil October 30th to November 03th, 2017 Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Complementary Problems Diffusion-absorption elliptic system Model Optimization Methodology Results for the Diffusive-Absorption Model Results based on Nelder Mead algorithm PEN/DNC/COPPE-EP/UFRJ/BR Figura: Programa/Departamento de Engenharia Nuclear-COPPE-UFRJ Thanks to Carlos J.S. Alves from CEMAT-Instituto Superior Tecnico- Lisbon University for the help full discussion. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination Objectives The objective of this presentation is describes some important aspects related with the reconstruction of parameters in models described with elliptic partial differential equations. We will not discuss regularization strategies. Incomplete information about coefficients and source is compensated by an overprescription of Cauchy data at the boundary. The methodology we propose explores concepts as: Lipschitz Boundary Dissection; 1 Complementary Mixed Problems with trial parameters; 2 Internal Discrepancy Fields. 3 The main techniques are variational formulation, boundary integral equations and Calderon projector. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination The engineering problem Most of the stationary engineering models can be represented as elliptic system of partial differential equations. Those models are mathematically elaborated with continuous thermomechanics and the constitutive theories of materials. Constitutive equations removes ambiguity in the model and frequently presents incomplete information about parameters. To assure uniqueness of model solution we must combine boundary information with correct parameters values. Estimation of missing parameters in diffusion reaction convection like systems of equations are the main problem. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination Introduction In this work we study the problem of reconstruction of coefficients and source parameters in second order strongly elliptic systems [1], [2]. Let Ω be a Lipschitz domain. Its boundary can be locally as the graph of a Lipschitz function, that is, a Holder continuous C 0 , 1 function. Let F α = [ f α , ..., f α ] ∈ ( L 2 (Ω)) m × N p be the source and 1 2 ( ∂ Ω) × ( H − 1 2 ( ∂ Ω))) m × N p the Cauchy data for ( H , H ν ) ∈ ( H N p problems based on the m − fields model. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination The inverse problem The inverse boundary value problem for parameter determination investigated here is: To find ( U , α ) ∈ H 1 (Ω) m × N p × R N a such that  L α U = F α if x ∈ Ω;  P α γ [ U ] = H if x ∈ ∂ Ω; (1) F α , H , H ν B ν [ U ] = H ν if x ∈ ∂ Ω;  Here γ is the boundary trace and B ν is the conormal trace. The coefficients of the strongly elliptic operator L α , self-adjoint, and the source depend on the parameters α . Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination The Parameters to Cauchy Data Implicit Function The main question is how Cauchy data ( H , H ν ) are related with the constitutive parameters α and the source F α ? Some functional equation C ( α, H , H ν , F α ) = 0 which could solve our problem, or, at least, conduct to a good framework to analysis. What are the consequence of incorrect values on the parameters? What are the consequence of incorrect values of the Cauchy Data? By using properties the Calderon Projector we will develop a methodology for the parameters determination problem by solving only direct problems and optimization problems. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination Auxiliary mixed problem Let ∂ Ω = ∂ Ω D ∪ Π ∪ ∂ Ω N a Lipschitz dissection of the boundary. The auxiliary mixed boundary value problem for inverse problem (1) is given by the well posed problem P α ν : For f α , g D , g N given Dirichlet and Neumann data 1 2 ( ∂ Ω D ) × H − 1 ( g D , g N 2 ( ∂ Ω N )) m , find u ∈ H 1 (Ω) m ν ) ∈ ( H such that  L α u = f α if x ∈ Ω;  P α γ [ u ] = g D if x ∈ ∂ Ω D ; (2) f α , g D , g N ν B ν u = g N if x ∈ ∂ Ω N ;  ν Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Objectives Complementary Problems The engineering problem Diffusion-absorption elliptic system Model The inverse problem Optimization Methodology The Parameters to Cauchy Data Implicit Function Results for the Diffusive-Absorption Model Auxiliary mixed problem Results based on Nelder Mead algorithm Discrepancy in Parameters Determination Complementary Problems and Internal Discrepancy Field The over prescription of data at the boundary is used to introduce the concept of complementary problems with the same constitutive equations. Internal discrepancies between solutions of complementary problems are observed when the model is supplied with parameters values in the constitutive equations that are inconsistent with the Cauchy data prescribed at the boundary. This discrepancy field measures the deviation from uniformity in sets of points in the interior of Ω. It is due to the solution of the complementary problems with incorrect parameters values. Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic

Introduction Complementary Problems on Lipschitz Domains Complementary Problems The Calderon Projector Diffusion-absorption elliptic system Model Theorem on Complementary Solutions Optimization Methodology Existence of Discrepancy between Complementary Solutions Results for the Diffusive-Absorption Model Discrepancy Field dependence on Lipschitz Dissection Results based on Nelder Mead algorithm Complementary Problems on Lipschitz Domains Definition Let us consider two mixed boundary value problems P α and f (1) , g (1) , g (1) ν P α defined on the same Lipschitz domain Ω with boundary f (1) , g (2) , g (2) ν dissection ∂ Ω = Γ (1) D ∪ Π ∪ Γ (1) N . We say that these problems are complementary if f (1) = f (2) α , α Γ (2) D = Γ (1) , Γ (2) N = Γ (1) D and there exist a Cauchy data ( g α , g α ν ) such that N g (1) = g α χ Γ (1) D and g (2) = g χ Γ (2) D . g (1) N and g (2) = g α = g α ν χ Γ (1) ν χ Γ (2) N . ν ν where χ Γ is the characteristic function for set Γ . Nilson Costa Roberty Reconstruction of Coefficients and Source Parameters in Elliptic