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

Simultaneous Reconstruction of Coefficients and Source Parameters in Elliptic Systems Modelled with Many Boundary Values Problems Nilson C. Roberty Nuclear Engineering Program Federal University of Rio de Janeiro-Brazil

The practical expression of linear elliptic partial differential equations found in most of the engineering application is represented by the following system, in which the fields may be a vector and coefficients can be represented by matrices and vectors according: To find u ( x ) such that  ∇ . ( − c ∇ u − α u + γ ) + β. ∇ u + au = f if x ∈ Ω;  hu = g if x ∈ ∂ Ω D ; (1) ν. ( c ∇ u + α u − γ ) + qu = g ν − h ∗ µ if x ∈ ∂ Ω N ;  where ν is the outward unit normal vector on ∂ Ω := ∂ Ω D ∪ Π ∪ ∂ Ω N ,

◮ Boundary integral formulation for the inverse problem: Let G ξ be the fundamental solution for the strongly elliptic system (1). ◮ Then the Calder´ on projector gap is: � γ u ( ξ ) � � � � Ω γ ξ [ G ξ ]( y ) f ( y , u ) dy = + � B ν u ( ξ ) Ω B ν ξ [ G ξ ]( y ) f ( y ) dy � � γ u ( ξ ) � 1 � 2 ( I x → ξ − T x → ξ ) S x → ξ , ( x , ξ ) ∈ Γ × Γ 1 R x → ξ 2 ( I x → ξ + T x → ξ ) B ν u ( ξ )

◮ Boundary integral formulation for the inverse problem: Let v be the regular fundamental solution for the strongly elliptic system (1). ◮ Variational : for all v ∈ ( H L ∗ (Ω) m ) ∗ . d d � � � � v ( x ) f ( x ) dx = − ( ν j A jk ∂ k v ( x )+ A j v ( x )) u ( x ) d σ x Ω Γ j =1 k =1 d d � � � + v ( x ) A jk ∂ k u ( x ) d σ x ν j Γ j =1 k =1

◮ Direct problem (Closed operators): Variational formulation and stabilization via Babuska-Brezzi-Necas-Banach condition. ◮ Inverse problem (Compact operators): Variational or strong formulation and stabilization thorough Picard-Tikhonov-Landweber-Morozov-Banach regularization. ◮ Functional Analysis Framework: Closed Range Banach Theorem and Fredholm Operator Theory. ◮ Solution of Direct problems with minimization of Least square discrepancy between calculated and measured Neumann data. ◮ Many solutions of two equivalent direct problems and minimization of the discrepancy between these two solutions.(This work !)

◮ Given The Dirichlet to Newmann map 1 2 ( ∂ Ω) → H − 1 2 ( ∂ Ω) Λ c : H ◮ To find u ∈ H 1 (Ω) , c ( x ) ∈ L ∞ (Ω) such that � ∇ . ( − c ∇ u ) = 0 in Ω; (2) g ν = Λ c [ g ] on ∂ Ω; ◮ Q c = Ω c ( x ) ||∇ u ( x ) || 2 dx = � � ∂ Ω g ( x )Λ c [ g ]( x ) d σ ( x )

◮ In 1980, (Seminar on Numerical Analysis and its Applications to Continuum Physics, SBM, Rio de Janeiro), Calder´ on posed the following problem: ◮ Decide whether c is uniquely determined by Q c , and, if so, calculate c in terms of Q c . ◮ the uniqueness problem, for conductivity or any other coefficients, is an open problem that has been only partially solved. ◮ ”Intrinsic non uniqueness in the inverse source problem”; ◮ The Calderon Projector Gap is the same for all Cauchy datum used to estimate it.

◮ Some uniqueness class ◮ The regular affine class ”If the sources are restricted to the affine class of functions C ( D ; F ) = { f ∈ H 2 (Ω) : Df = F } then we have uniqueness of the associated inverse source problem (Alves, Martins, Roberty, Cola¸ co, Olander, 2007 ); ◮ The characteristic class f = F χ ω ( P. Novikov, 1938, Isakov, 1990); ◮ The mono and dipolar source class A = { f := � m 1 j =1 λ j δ x j + � m 1 j =1 p j · ∇ δ x j (El Badia e Ha Duong, 2000).

Lemma Let V , W Banach spaces and V ′ , W ′ its respective dual. Let A ∈ L ( V , W ) and A T ∈ L ( W ′ , V ′ ) its transposed. Let ker ( A ) , ker ( A T ) , im ( A ) and im ( A T ) denotes its respective kernel and range. For M ⊂ V and N ⊂ W ′ , let M ⊥ := { v ′ ∈ V ′ |∀ m ∈ M , � v ′ , m � V ′ × V = 0 } and N ⊥ := { v ∈ V |∀ n ∈ N , � n , v � V ′ × V = 0 } . Then, the following properties hold: ◮ ker ( A ) = ( im ( A T )) ⊥ and ker ( A T ) = ( im ( A )) ⊥ ; ◮ im ( A ) = ( ker ( A T )) ⊥ and im ( A T ) = ( ker ( A )) ⊥

We first apply this Fundamental Lemma to the operator 0 (Ω)) m → ( L 2 (Ω)) m L 0 : ( H 2 0 (Ω)) m := { v ∈ ( H 2 (Ω)) m | v | ∂ Ω = 0; B ν v | ∂ Ω = 0 } has where ( H 2 transpose 0 : ( L 2 (Ω)) m → ( H − 2 (Ω)) m L ∗ Note that in this case, by (ii), 0 (Ω) ) m = ker ( L ∗ 0 ) = ( im ( L 0 )) ⊥ = ( L 0 (( H 2 0 (Ω)) m )) ⊥ and, by ( H L ∗ (iii) L 2 (Ω) ( L 2 (Ω)) m = ( H L ∗ 0 (Ω)) m ⊕ L 0 (( H 2 0 (Ω) m )) (3)

Now we define the following space D , N (Ω)) m := { v ∈ ( H 2 (Ω)) m | v | Γ D = 0; B| Γ N v = 0 } ( H 2 where Γ D ⊂ ∂ Ω and Γ N ⊂ ∂ Ω are arbitrary. Note that 0 (Ω)) m ⊂ ( H 2 H 2 D , N (Ω)) m D , N (Ω)) m → ( L 2 (Ω)) m has transpose The operator L 0 , D , N : ( H 2 0 , D c , N c : ( L 2 (Ω)) m → (( H 2 D , N (Ω)) m ) ∗ = ( H − 2 D c , N c (Ω)) m ⊂ L ∗ ( H − 2 (Ω)) m where ( H − 2 D c , N c (Ω)) m is a set of distribution with trace support in ∂ Ω \ Γ D and conormal trace with support in ∂ Ω \ Γ N and whose kernel is 0 , Dc , Nc (Ω)) m = { v ∈ ( L 2 (Ω)) m |L ∗ ( H L ∗ 0 , D c , N c v = 0 }

0 , Dc , Nc (Ω) ) m = ker ( L ∗ Note that in this case, by (ii), ( H L ∗ 0 , D c , N c ) = ( im ( L 0 , D , N )) ⊥ = ( L 0 (( H 2 D , N (Ω))) m ) ⊥ and, by (iii) L 2 (Ω) ( L 2 (Ω)) m = ( H L ∗ 0 , Dc , Nc (Ω)) m ⊕ L 0 , D , N (( H 2 D , N (Ω)) m ) (4) Note that for Γ D = Γ N = Γ, decomposition (4) reduces to (3). Also that when Γ D and Γ N are a Lipschitz dissection of ∂ Ω, the fact that the unique solution v χ of L ∗  0 , D c , N c v = χ if x ∈ Ω;  P c ∗ γ [ v ] = 0 if x ∈ ∂ Ω N = ∂ Ω \ Γ D ; (5) χ, 0 , 0 ˜ B ν v = 0 if x ∈ ∂ Ω D = ∂ Ω \ Γ D ;  0 , Dc , Nc (Ω)) m = ( { 0 } ) m is the trivial when χ ( x ) = 0,and ( H L ∗

◮ Characterizes materials parameters and source is a central question in the engineering project; ◮ it is important adequate existing engineering and multiphysics software to handle uncertainties in these parameters; ◮ be used as a tool for process experimental data; ◮ but respecting the actual engineering project project status of art. ◮ Applications when we have incomplete information about these coefficient and sources.

◮ This work is addressed to investigate the class of problems in which we want determine unknown parameters in the functions that characterize these coefficients and sources. ◮ To compensate this incomplete information that ill-posed the problem, we suppose that both, Neumann and Dirichlet data, are prescribed for many boundary value problems. ◮ These problems are formulated for the same physical coefficients and source which depend on the same set of unknown parameters.

◮ L u = − � d j =1 ( � d k =1 ∂ j ( A jk ∂ k ) u + A j ∂ j u ) + Au ◮ ( A jk , A j , A ) : Ω → R m × m . ◮ u is a column vector with m scalar fields and L u : Ω → R m ◮ strongly elliptic system. d d � � L 0 u = − ∂ j B j u where B j = A jk ∂ k (6) j =1 k =1 ◮ Ω is a Lipschitz domain and γ is the trace operator ◮ the conormal derivative is d � B ν u = ν j γ [ B j u ] (7) j =1

Let Ω a domain with Lipschitz dissection boundary ∂ Ω = ∂ Ω N ∪ Π ∪ ∂ Ω N . The mixed boundary value problem for the physical model given by (1) is given by the well posed problem P f , g D , g N : To find u ∈ H 1 (Ω) m such that  L u = f if x ∈ Ω;  γ [ u ] = g D if x ∈ ∂ Ω D ; P f , g D , g N (8) B ν u = g N if x ∈ ∂ Ω N ;  we can show that (8) has the following weak formulation W f , g D , g N  ( L u , v ) Ω + ( B ν u , γ [ v ]) ∂ Ω = Φ( u , v ) =  if v ∈ H 1 D (Ω) m ; = ( f , v ) Ω + ( g N , γ [ v ]) ∂ Ω N (9) γ [ u ] = g D if x ∈ ∂ Ω D ; 

Definition When u = u + + u − ∈ L 2 ( R d ) m , with u ± ∈ H 1 (Ω ± ) m , has compact support in R d and f = f + + f − ∈ H − 1 ( R d ) m , we can enunciate the Third Green Identity u = G f + DL [ u ] Γ − SL [ B ν u ] Γ on R d . (10)

Definition When the mixed boundary value problem is posed with a non null source, P f , g D , g N ν , we have a gap in the Calder´ on projector: � γ u ( ξ ) � � � � � = Ω γ ξ [ G ξ ]( y ) f ( y ) dy Ω B ν ξ [ G ξ ]( y ) f ( y ) dy + B ν u ( ξ ) � � γ u ( ξ ) � 1 2 ( I x → ξ − T x → ξ ) S x → ξ � , ( x , ξ ) ∈ Γ × Γ 1 R x → ξ 2 ( I x → ξ + T x → ξ ) B ν u ( ξ )

Matrix equation for Calder´ on Projector Gap Lipschtiz Boundary Dissection: �     γ u ( ξ ) | Γ D Ω γ ξ G ξ | Γ D ( y ) f ( y ) dy � γ u ( ξ ) | Γ N Ω γ x iG ξ | Γ N ( y ) f ( y ) dy      =  +     � B ν u ( ξ ) | Γ D Ω B ν ξ G ξ | Γ D ( y ) f ( y ) dy   � B ν u ( ξ ) | Γ N Ω B ν ξ G ξ | Γ N ( y ) f ( y ) dy 1  2 ( I DD x → ξ − T DD − T ND S DD S ND x → ξ ) x → ξ x → ξ x → ξ 1 − T DN 2 ( I NN x → ξ − T NxN S DN S NN x → ξ )  x → ξ x → ξ x → ξ  1 x → ξ + ˜ ˜ R DD R ND 2 ( I DD T ∗ DD T ∗ ND x → ξ )  x → ξ x → ξ x → ξ  ˜ 1 x → ξ + ˜ R DN R NN T ∗ DN 2 ( I NN T ∗ NN x → ξ x → ξ x → ξ x → ξ