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Partial boundary value problems on finite networks Cristina Ara uz 1 , Angeles Carmona 1 and Andr es M. Encinas 1 The 6th de Br un Workshop Linear Algebra and Matrix Theory: connections, applications and computations NUI Galway

  1. Partial Dirichlet-Neumann BVPs A F R B Definition ( Homogeneous partial Dirichlet-Neumann BVP)  L q ( u h ) = 0  on F its solutions are a vector     ∂u h subspace of C ( F ∪ B ) that we = 0 on A   ∂ n F    denote by V B on A ∪ R u h = 0 Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 13 / 47 Partial BVPs on finite networks

  2. Partial Dirichlet-Neumann BVPs A F R B Definition ( Adjoint partial Dirichlet-Neumann BVP)  L q ( u a ) = 0  on F its solutions are a vector     ∂u a subspace of C ( F ∪ A ) that we = 0 on B   ∂ n F    denote by V A on B ∪ R u a = 0 Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 13 / 47 Partial BVPs on finite networks

  3. Partial Dirichlet-Neumann BVPs  L q ( u ) = h  on F     ∂u Remember our partial BVP = g on A   ∂ n F    on A ∪ R u = f Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  4. Partial Dirichlet-Neumann BVPs  L q ( u ) = h  on F     ∂u Remember our partial BVP = g on A   ∂ n F    on A ∪ R u = f Theorem | A | − | B | = dim V A − dim V B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  5. Partial Dirichlet-Neumann BVPs  L q ( u ) = h  on F     ∂u Remember our partial BVP = g on A   ∂ n F    on A ∪ R u = f Theorem | A | − | B | = dim V A − dim V B � Existence of solution for any data h , g , f ⇔ V A = { 0 } Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  6. Partial Dirichlet-Neumann BVPs  L q ( u ) = h  on F     ∂u Remember our partial BVP = g on A   ∂ n F    on A ∪ R u = f Theorem | A | − | B | = dim V A − dim V B � Existence of solution for any data h , g , f ⇔ V A = { 0 } � Uniqueness of solution for any data h , g , f ⇔ V B = { 0 } Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  7. Partial Dirichlet-Neumann BVPs  L q ( u ) = h  on F     ∂u Remember our partial BVP = g on A   ∂ n F    on A ∪ R u = f Theorem | A | − | B | = dim V A − dim V B � Existence of solution for any data h , g , f ⇔ V A = { 0 } � Uniqueness of solution for any data h , g , f ⇔ V B = { 0 } � In particular, if | A | = | B | then existence ⇔ uniqueness ⇔ the homogeneous problem has u = 0 as its unique solution Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  8. Partial Dirichlet-Neumann BVPs  L q ( u ) = h on F      ∂u Remember our partial BVP = g on A   ∂ n F    u = f on A ∪ R � We work with boundaries where | A | = | B | and assume there exists a unique solution u ∈ C ( ¯ F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  9. Partial Dirichlet-Neumann BVPs  L q ( u ) = h on F      ∂u Remember our partial BVP = g on A   ∂ n F    u = f on A ∪ R � We work with boundaries where | A | = | B | and assume there exists a unique solution u ∈ C ( ¯ F ) Question Can we find the solution? Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  10. Partial Dirichlet-Neumann BVPs  L q ( u ) = h on F      ∂u Remember our partial BVP = g on A   ∂ n F    u = f on A ∪ R � We work with boundaries where | A | = | B | and assume there exists a unique solution u ∈ C ( ¯ F ) Question Can we find the solution? Remark We need Green and Poisson operators! Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 14 / 47 Partial BVPs on finite networks

  11. Classical Green and Poisson operators Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 15 / 47 Partial BVPs on finite networks

  12. Classical Green and Poisson operators � The classical Green operator G q solves the problem  � �  L q G q ( h ) = h on F  G q ( h ) = 0 on δ ( F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 16 / 47 Partial BVPs on finite networks

  13. Classical Green and Poisson operators � The classical Green operator G q solves the problem  � �  L q G q ( h ) = h on F  G q ( h ) = 0 on δ ( F ) � The classical Poisson operator P q solves the problem  � �  L q P q ( f ) = 0 on F  P q ( f ) = f on δ ( F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 16 / 47 Partial BVPs on finite networks

  14. Classical Green and Poisson operators � The classical Green operator G q solves the problem  � �  L q G q ( h ) = h on F  G q ( h ) = 0 on δ ( F ) � The classical Poisson operator P q solves the problem  � �  L q P q ( f ) = 0 on F  P q ( f ) = f on δ ( F ) However, our problem is different on the boundary Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 16 / 47 Partial BVPs on finite networks

  15. Classical Green and Poisson operators � The classical Green operator G q solves the problem  � �  L q G q ( h ) = h on F  G q ( h ) = 0 on δ ( F ) � The classical Poisson operator P q solves the problem  � �  L q P q ( f ) = 0 on F  P q ( f ) = f on δ ( F ) However, our problem is different on the boundary ⇒ We need to modify these operators (we will see it later) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 16 / 47 Partial BVPs on finite networks

  16. Dirichlet-to-Neumann map Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 17 / 47 Partial BVPs on finite networks

  17. Dirichlet-to-Neumann map Before modifying Green and Poisson operators, we need to define the Dirichlet-to-Neumann map as Λ q ( g ) = ∂ P q ( g ) for all g ∈ C ( δ ( F )) χ δ ( F ) ∂ n F Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 18 / 47 Partial BVPs on finite networks

  18. Dirichlet-to-Neumann map Before modifying Green and Poisson operators, we need to define the Dirichlet-to-Neumann map as Λ q ( g ) = ∂ P q ( g ) for all g ∈ C ( δ ( F )) χ δ ( F ) ∂ n F given by with kernel DN q : δ ( F ) × δ ( F ) − → R ( x, y ) �− → DN q ( x, y ) � Λ q ( g )( x ) = DN q ( x, y ) g ( y ) dy δ ( F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 18 / 47 Partial BVPs on finite networks

  19. Dirichlet-to-Neumann map Before modifying Green and Poisson operators, we need to define the Dirichlet-to-Neumann map as Λ q ( g ) = ∂ P q ( g ) for all g ∈ C ( δ ( F )) χ δ ( F ) ∂ n F given by with kernel DN q : δ ( F ) × δ ( F ) − → R ( x, y ) �− → DN q ( x, y ) � Λ q ( g )( x ) = DN q ( x, y ) g ( y ) dy δ ( F ) � � that is, DN q ( x, y ) = Λ q ( x ) for all x, y ∈ δ ( F ) ε y Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 18 / 47 Partial BVPs on finite networks

  20. Dirichlet-to-Neumann map - a little remark Definition (Schur complement) P ∈ M k × k ( R ) , Q ∈ M k × l ( R ) , C ∈ M l × k ( R ) and D ∈ M l × l ( R ) with D non–singular � P � Q The Schur Complement of D on M , where M = , is C D M � D = P − QD − 1 C ∈ M k × k ( R ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 19 / 47 Partial BVPs on finite networks

  21. Dirichlet-to-Neumann map - a little remark Definition (Schur complement) � P � M � Q D = P − QD − 1 C ⇒ M = C D Theorem The Dirichlet-to-Neumann map kernel DN q can be expressed as a Schur complement: F ) � DN q ( δ ( F ); δ ( F )) = L q ( ¯ F ; ¯ L q ( F ; F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 20 / 47 Partial BVPs on finite networks

  22. Dirichlet-to-Neumann map - a little remark Definition (Schur complement) � P � M � Q D = P − QD − 1 C ⇒ M = C D Theorem The Dirichlet-to-Neumann map kernel DN q can be expressed as a Schur complement: F ) � DN q ( δ ( F ); δ ( F )) = L q ( ¯ F ; ¯ L q ( F ; F ) Corollary If A, B ⊆ δ ( F ) , then DN q ( A ; B ) = L q ( A ∪ F ; B ∪ F ) � L q ( F ; F ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 20 / 47 Partial BVPs on finite networks

  23. Modified Green and Poisson operators Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 21 / 47 Partial BVPs on finite networks

  24. Modified Green and Poisson operators   L q ( u ) = h on F    ∂u = g on A  ∂ n F    u = f on A ∪ R Using the Dirichlet-to-Neumann map, we can translate Theorem | A | − | B | = dim V A − dim V B � Existence of solution for any data h , g , f ⇔ V A = { 0 } � Uniqueness of solution for any data h , g , f ⇔ V B = { 0 } � In particular, if | A | = | B | then existence ⇔ uniqueness ⇔ the homogeneous problem has u = 0 as its unique solution Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 22 / 47 Partial BVPs on finite networks

  25. Modified Green and Poisson operators   L q ( u ) = h on F    ∂u = g on A  ∂ n F    u = f on A ∪ R Into Theorem ⇔ � It has solution for any data DN q ( B ; A ) has maximum range � It has uniqueness of solution for any data ⇔ DN q ( A ; B ) has maximum range � In particular, if | A | = | B | then it has a unique solution for any data ⇔ ⇔ DN q ( A ; B ) non-singular DN q ( B ; A ) non-singular Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 22 / 47 Partial BVPs on finite networks

  26. Modified Green and Poisson operators   L q ( u ) = h on F    ∂u = g on A  ∂ n F    u = f on A ∪ R Into Theorem ⇔ � It has solution for any data DN q ( B ; A ) has maximum range � It has uniqueness of solution for any data ⇔ DN q ( A ; B ) has maximum range � In particular, if | A | = | B | then it has a unique solution for any data ⇔ ⇔ DN q ( A ; B ) non-singular DN q ( B ; A ) non-singular � From now on, we assume that DN q ( A ; B ) is invertible Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 22 / 47 Partial BVPs on finite networks

  27. Modified Green and Poisson operators   L q ( u ) = h on F     ∂u The unique solution of = g on A   ∂ n F    u = f on A ∪ R u = � G q ( h ) + � N q ( g ) + � on ¯ can be expressed as P q ( f ) F Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 23 / 47 Partial BVPs on finite networks

  28. Modified Green and Poisson operators   L q ( u ) = h on F     ∂u The unique solution of = g on A   ∂ n F    u = f on A ∪ R u = � G q ( h ) + � N q ( g ) + � on ¯ can be expressed as P q ( f ) F , where  � �  � �  � �  �  �  � L q G q ( h ) = h L q N q ( g ) = 0 L q P q ( h ) = 0 on F                ∂ � ∂ � ∂ � G q ( h ) N q ( g ) P q ( h ) = 0 = g = 0 on A       ∂ n F ∂ n F ∂ n F             � � � G q ( h ) = 0 N q ( g ) = 0 P q ( h ) = f on A ∪ R Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 23 / 47 Partial BVPs on finite networks

  29. Modified Green and Poisson operators   L q ( u ) = h on F     ∂u The unique solution of = g on A   ∂ n F    u = f on A ∪ R u = � G q ( h ) + � N q ( g ) + � on ¯ can be expressed as P q ( f ) F , where  � �  � �  � �  �  �  � L q G q ( h ) = h L q N q ( g ) = 0 L q P q ( h ) = 0 on F                ∂ � ∂ � ∂ � G q ( h ) N q ( g ) P q ( h ) = 0 = g = 0 on A       ∂ n F ∂ n F ∂ n F             � � � G q ( h ) = 0 N q ( g ) = 0 P q ( h ) = f on A ∪ R Modified Green operator Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 23 / 47 Partial BVPs on finite networks

  30. Modified Green and Poisson operators   L q ( u ) = h on F     ∂u The unique solution of = g on A   ∂ n F    u = f on A ∪ R u = � G q ( h ) + � N q ( g ) + � on ¯ can be expressed as P q ( f ) F , where  � �  � �  � �  �  �  � L q G q ( h ) = h L q N q ( g ) = 0 L q P q ( h ) = 0 on F                ∂ � ∂ � ∂ � G q ( h ) N q ( g ) P q ( h ) = 0 = g = 0 on A       ∂ n F ∂ n F ∂ n F             � � � G q ( h ) = 0 N q ( g ) = 0 P q ( h ) = f on A ∪ R Modified Green Modified Neumann operator operator Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 23 / 47 Partial BVPs on finite networks

  31. Modified Green and Poisson operators   L q ( u ) = h on F     ∂u The unique solution of = g on A   ∂ n F    u = f on A ∪ R u = � G q ( h ) + � N q ( g ) + � on ¯ can be expressed as P q ( f ) F , where  � �  � �  � �  �  �  � L q G q ( h ) = h L q N q ( g ) = 0 L q P q ( h ) = 0 on F                ∂ � ∂ � ∂ � G q ( h ) N q ( g ) P q ( h ) = 0 = g = 0 on A       ∂ n F ∂ n F ∂ n F             � � � G q ( h ) = 0 N q ( g ) = 0 P q ( h ) = f on A ∪ R Modified Green Modified Neumann Modified Poisson operator operator operator Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 23 / 47 Partial BVPs on finite networks

  32. Modified Green and Poisson operators � We express these modified operators in terms of the classical ones and the matrix DN q ( A ; B ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 24 / 47 Partial BVPs on finite networks

  33. Modified Green and Poisson operators � We express these modified operators in terms of the classical ones and the matrix DN q ( A ; B ) Remark We can not express them in operator terms, as we need to invert a matrix. However, we can do it in matricial terms Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 24 / 47 Partial BVPs on finite networks

  34. Modified Green and Poisson operators Theorem G q ( F ; F ) = G q ( F ; F ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � G q ( A ∪ R ; F ) = 0 G q ( B ; F ) = − DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 25 / 47 Partial BVPs on finite networks

  35. Modified Green and Poisson operators Theorem G q ( F ; F ) = G q ( F ; F ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � G q ( A ∪ R ; F ) = 0 G q ( B ; F ) = − DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � N q ( F ; A ) = P q ( F ; B ) · DN q ( A ; B ) − 1 � N q ( A ∪ R ; A ) = 0 N q ( B ; A ) = DN q ( A ; B ) − 1 � Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 25 / 47 Partial BVPs on finite networks

  36. Modified Green and Poisson operators Theorem G q ( F ; F ) = G q ( F ; F ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � G q ( A ∪ R ; F ) = 0 G q ( B ; F ) = − DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � N q ( F ; A ) = P q ( F ; B ) · DN q ( A ; B ) − 1 � N q ( A ∪ R ; A ) = 0 N q ( B ; A ) = DN q ( A ; B ) − 1 � P q ( F ; A ∪ R ) = P q ( F ; A ∪ R ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) � � P q ( A ∪ R ; A ∪ R ) = I A ∪ R P q ( B ; A ∪ R ) = − DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) � Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 25 / 47 Partial BVPs on finite networks

  37. Modified Green and Poisson operators Theorem G q ( F ; F ) = G q ( F ; F ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � G q ( A ∪ R ; F ) = 0 G q ( B ; F ) = − DN q ( A ; B ) − 1 · L q ( A ; F ) · G q ( F ; F ) � � N q ( F ; A ) = P q ( F ; B ) · DN q ( A ; B ) − 1 � N q ( A ∪ R ; A ) = 0 N q ( B ; A ) = DN q ( A ; B ) − 1 � P q ( F ; A ∪ R ) = P q ( F ; A ∪ R ) − P q ( F ; B ) · DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) � � P q ( A ∪ R ; A ∪ R ) = I A ∪ R P q ( B ; A ∪ R ) = − DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) � � They can be expressed in terms of the classical Green and Poisson operators and of the Dirichlet-to-Neumann map Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 25 / 47 Partial BVPs on finite networks

  38. Partial inverse boundary value problems on finite networks Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 26 / 47 Partial BVPs on finite networks

  39. Partial inverse BVPs on finite networks � We want to obtain the conductances by solving partial BVPs Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 27 / 47 Partial BVPs on finite networks

  40. Partial inverse BVPs on finite networks � We want to obtain the conductances by solving partial BVPs � We assume the network is in an equilibrium state   L q ( u ) = 0 on F     ∂u = g on A   ∂ n F    u = f on A ∪ R Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 27 / 47 Partial BVPs on finite networks

  41. Partial inverse BVPs on finite networks � We want to obtain the conductances by solving partial BVPs � We assume the network is in an equilibrium state   L q ( u ) = 0 on F     ∂u = g on A   ∂ n F    u = f on A ∪ R � We also assume the Dirichlet-to-Neumann map Λ q to be known, as it can be physically obtained from electrical boundary measurements Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 27 / 47 Partial BVPs on finite networks

  42. Partial inverse BVPs on finite networks � We want to obtain the conductances by solving partial BVPs � We assume the network is in an equilibrium state   L q ( u ) = 0 on F     ∂u = g on A   ∂ n F    u = f on A ∪ R � We also assume the Dirichlet-to-Neumann map Λ q to be known, as it can be physically obtained from electrical boundary measurements Remark (Alessandrini 1998, Mandache 2001) This problem is severelly ill-posed! Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 27 / 47 Partial BVPs on finite networks

  43. Partial inverse BVPs on finite networks  L q ( u ) = 0 on F  Remember that under our     conditions ( | A | = | B | and ∂u = g on A DN q ( A ; B ) invertible) this  ∂ n F   problem has a unique solution   u = f on A ∪ R Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 28 / 47 Partial BVPs on finite networks

  44. Partial inverse BVPs on finite networks  L q ( u ) = 0 on F  Remember that under our     conditions ( | A | = | B | and ∂u = g on A DN q ( A ; B ) invertible) this  ∂ n F   problem has a unique solution   u = f on A ∪ R Corollary The unique solution is characterized by the equations u B = DN q ( A ; B ) − 1 · g − DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) · f on B u ( x ) = P q ( x ; A ∪ R ) · f + P q ( x ; B ) · u B for all x ∈ F Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 28 / 47 Partial BVPs on finite networks

  45. Partial inverse BVPs on finite networks  L q ( u ) = 0 on F  Remember that under our     conditions ( | A | = | B | and ∂u = g on A DN q ( A ; B ) invertible) this  ∂ n F   problem has a unique solution   u = f on A ∪ R Corollary The unique solution is characterized by the equations u B = DN q ( A ; B ) − 1 · g − DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) · f on B u ( x ) = P q ( x ; A ∪ R ) · f + P q ( x ; B ) · u B for all x ∈ F Remark Althogh u is not determined yet on F , Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 28 / 47 Partial BVPs on finite networks

  46. Partial inverse BVPs on finite networks  L q ( u ) = 0 on F  Remember that under our     conditions ( | A | = | B | and ∂u = g on A DN q ( A ; B ) invertible) this  ∂ n F   problem has a unique solution   u = f on A ∪ R Corollary The unique solution is characterized by the equations u B = DN q ( A ; B ) − 1 · g − DN q ( A ; B ) − 1 · DN q ( A ; A ∪ R ) · f on B u ( x ) = P q ( x ; A ∪ R ) · f + P q ( x ; B ) · u B for all x ∈ F Remark Althogh u is not determined yet on F , u B gives the values of the solution on B ! Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 28 / 47 Partial BVPs on finite networks

  47. Partial inverse BVPs on finite networks � However, this is not enough Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 29 / 47 Partial BVPs on finite networks

  48. Partial inverse BVPs on finite networks � However, this is not enough with all these last steps we only get to know u on δ ( F ) and no conductances Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 29 / 47 Partial BVPs on finite networks

  49. Partial inverse BVPs on finite networks � However, this is not enough with all these last steps we only get to know u on δ ( F ) and no conductances � We restrict to circular planar networks to obtain some conductances Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 29 / 47 Partial BVPs on finite networks

  50. Partial inverse BVPs on finite networks � However, this is not enough with all these last steps we only get to know u on δ ( F ) and no conductances � We restrict to circular planar networks to obtain some conductances it can be drawn on the plane planar network ⇔ without crossings between edges Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 29 / 47 Partial BVPs on finite networks

  51. Partial inverse BVPs on finite networks � However, this is not enough with all these last steps we only get to know u on δ ( F ) and no conductances � We restrict to circular planar networks to obtain some conductances it can be drawn on the plane planar network ⇔ without crossings between edges circular planar planar & all the boundary vertices ⇔ network can be found in the same (exterior) face Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 29 / 47 Partial BVPs on finite networks

  52. Partial inverse BVPs on finite networks boundary vertex boundary circle exterior face boundary spike boundary edge Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 30 / 47 Partial BVPs on finite networks

  53. Partial inverse BVPs on finite networks boundary vertex we will consider certain circular order on the boundary boundary spike boundary edge Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 30 / 47 Partial BVPs on finite networks

  54. Partial inverse BVPs on finite networks a circular pair is connected through the network if there exists a set of disjoint paths between them Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 30 / 47 Partial BVPs on finite networks

  55. Partial inverse BVPs on finite networks � Generalization of Curtis and Morrow’s results in 2000 Theorem ( P, Q ) circular pair -of size k - of δ ( F ) , where P and Q are disjoint arcs of the boundary circle ( P, Q ) not connected through Γ ⇔ det ( DN q ( P ; Q )) = 0 . ( P, Q ) connected through Γ ⇔ ( − 1) k det ( DN q ( P ; Q )) > 0 . Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 31 / 47 Partial BVPs on finite networks

  56. Partial inverse BVPs on finite networks Corollary (Boundary Spike formula) If xy is a boundary spike with y ∈ δ ( F ) and contracting xy to a single boundary vertex means breaking the connection through Γ between a circular pair ( P, Q ) , then � � c ( x, y ) = ω ( y ) DN q ( y ; y ) − DN q ( y ; Q ) · DN q ( P ; Q ) − 1 · DN q ( P ; y ) − λ ω ( x ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 32 / 47 Partial BVPs on finite networks

  57. Partial inverse BVPs on finite networks Corollary (Boundary Spike formula) If xy is a boundary spike with y ∈ δ ( F ) and contracting xy to a single boundary vertex means breaking the connection through Γ between a circular pair ( P, Q ) , then � � c ( x, y ) = ω ( y ) DN q ( y ; y ) − DN q ( y ; Q ) · DN q ( P ; Q ) − 1 · DN q ( P ; y ) − λ ω ( x ) � We can recover certain conductances on planar networks! Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 32 / 47 Partial BVPs on finite networks

  58. Partial inverse BVPs on finite networks Corollary (Boundary Spike formula) If xy is a boundary spike with y ∈ δ ( F ) and contracting xy to a single boundary vertex means breaking the connection through Γ between a circular pair ( P, Q ) , then � � c ( x, y ) = ω ( y ) DN q ( y ; y ) − DN q ( y ; Q ) · DN q ( P ; Q ) − 1 · DN q ( P ; y ) − λ ω ( x ) � We can recover certain conductances on planar networks! � We can try to recover all the conductances in special cases: well-connected spider networks Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 32 / 47 Partial BVPs on finite networks

  59. Conductance reconstruction on well-connected spider networks Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 33 / 47 Partial BVPs on finite networks

  60. Conductance reconstruction on w-c spider networks � A well-connected spider network has n ≡ 3( mod 4) boundary nodes and m = n − 3 circles 4 v S 1 v S v S n 2 δ ( F ) radius j v S 3 x S ji F circle i x S 00 Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 34 / 47 Partial BVPs on finite networks

  61. Conductance reconstruction on w-c spider networks Remark Taking A = { v S 1 , . . . , v S 2 } , B = { v S 2 , . . . , v S n − 1 } and R = { v S n } (or n − 1 n +1 equivalent configurations), then A and B is a circular pair always connected through the network R A A R B B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 35 / 47 Partial BVPs on finite networks

  62. Reconstruction - Step 1 � Boundary spike formula R A B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 36 / 47 Partial BVPs on finite networks

  63. Reconstruction - Step 2 � We choose f = ε v S n and g = 0  L q S ( u ) = 0 on F S    ∂u = u = 0 on A � Considering problem then ∂ n FS    on R = { v S u = 1 n } , u B = − DN q S ( A ; B ) − 1 · DN q S ( A ; v S n ) Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 37 / 47 Partial BVPs on finite networks

  64. Reconstruction - Step 3 � Moreover, we obtain a zero zone of the solution of this BVP problem R A A R B B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 38 / 47 Partial BVPs on finite networks

  65. Reconstruction - Step 4 � We also get to know the values of u on the neighbours of B � � u N ( B ) = u B − L q S ( B ; N ( B )) − 1 · DN q S ( B ; v S n ) + DN q S ( B ; B ) · u B ↑ ↑ ↑ ↑ ↑ already known! Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 39 / 47 Partial BVPs on finite networks

  66. Reconstruction - Step 5 � With this information, we obtain two new conductances R A B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 40 / 47 Partial BVPs on finite networks

  67. Reconstruction - Step 6 � ...and rotating the BVP, we obtain more conductances R A B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 41 / 47 Partial BVPs on finite networks

  68. Reconstruction - Step 7 � Now we can even obtain two more conductances R A B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 42 / 47 Partial BVPs on finite networks

  69. Reconstruction - Step 8 � ...and rotating the BVP again, R A B Cristina Ara´ uz (UPC) 6th de Br´ un Workshop 43 / 47 Partial BVPs on finite networks

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