Problems that led me to Gunther Uhlmann David Isaacson RPI 1. - PowerPoint PPT Presentation

Nachman’s D -Bar method. • Convert inverse conductivity problem to an Unphysical Inverse Scattering Problem for the Schrodinger Equation. • Use the measured D-N map to solve a boundary integral equation for the boundary values of the exponentially growing Faddeev solutions . • Compute the unphysical Scattering transform in the complex k-plane from these boundary values. • Solve the D-Bar integral equation in the whole complex k-plane for the Faddeev solutions in the region of interest. • Take the limit as k goes to 0 of these solutions to recover and display the conductivity in the region of interest.

Problem: Find the Conductivity σ from the measured Dirichlet to  Neumann map  Assume :      0 inside B. u   u V on B.        V u/ on B.     1 in a neighborho od of B.

Let ;       1/2 (p, ) u,      - 1/2 1/2 q q(p) Then      - q 0 in B         / on B    and q 0 in a neighborho od of B.

  n Look for Solutions on all of R (n 2 ) with   q 0 outside B that satisfy           exp(i p) as | p | , where 0.     2 In R take k(1, i) where k k ik 1 2 Let          (p, ) exp(i p) (p, )     where 1 as | p | .

Observe that          (- 2 ) 0 i q     and 1 as | p | .   We may recover from by the property t hat; 1/2       ( ) ( , 0 ) lim ( , ). p p p   0 Reason :      - (p,0) q (p,0) 0      1/2 1/2 - q 0     1/2 Since both and 1 at they are identical.

  Main Problem : Given find ?     1. First find and hence on B by solving         [I S( - )] exp( ) on B. i p  1 Here S denotes the operator     (Sw)(p) G(p - t)w(t)ds t  B where G(p) is the Faddeev Greens function         - G , G exp(i p) as | p | .

2. Compute the " unphysical " scattering transform          t( k) exp(i p) ( ) ( ) ( ) p ds p  1 B    3 . Solve the equation for (p, ); 1          / ( ) exp( ( ) ) ( , ) k t k i p p k  4 k     1/2 4 . Take lim (p, ) ( ) p  k 0  5 . Display .