Calderón problem in inverse conductivity
E817947
The Calderón problem in inverse conductivity is a fundamental question in mathematical inverse problems that asks whether one can determine the electrical conductivity inside a medium uniquely from voltage and current measurements made at its boundary.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
PDE inverse problem
ⓘ
inverse problem ⓘ mathematical problem ⓘ |
| asks | whether one can determine the electrical conductivity inside a medium from boundary measurements ⓘ |
| centralObject | Dirichlet-to-Neumann map Λ_γ NERFINISHED ⓘ |
| concerns | uniqueness of determining conductivity from Dirichlet-to-Neumann map ⓘ |
| data | Dirichlet-to-Neumann map on the boundary ⓘ |
| difficulty |
ill-posedness
ⓘ
instability with respect to noise ⓘ |
| domain | bounded domain in Euclidean space ⓘ |
| equation | div(γ(x)∇u(x)) = 0 ⓘ |
| field |
applied analysis
ⓘ
electrical impedance tomography ⓘ inverse problems ⓘ mathematical physics ⓘ partial differential equations ⓘ |
| goal | recover the conductivity coefficient inside a domain ⓘ |
| hasApplication |
geophysical prospecting
ⓘ
medical imaging ⓘ non-destructive testing ⓘ |
| hasVariant |
Calderón problem on Riemannian manifolds
NERFINISHED
ⓘ
anisotropic conductivity Calderón problem NERFINISHED ⓘ partial data Calderón problem ⓘ |
| historicalOrigin | work of Alberto P. Calderón in the 1980s ⓘ |
| involves |
Dirichlet boundary data
ⓘ
Dirichlet-to-Neumann operator NERFINISHED ⓘ Neumann boundary data ⓘ boundary measurements of voltage and current ⓘ conductivity equation ⓘ elliptic partial differential equations ⓘ |
| mainQuestion | does Λ_γ determine γ uniquely? ⓘ |
| namedAfter | Alberto P. Calderón NERFINISHED ⓘ |
| questionType |
reconstruction
ⓘ
stability ⓘ uniqueness ⓘ |
| relatedTo |
Carleman estimates
NERFINISHED
ⓘ
Schrödinger equation inverse problem NERFINISHED ⓘ boundary control method ⓘ complex geometrical optics solutions ⓘ electrical impedance tomography NERFINISHED ⓘ unique continuation principle ⓘ |
| typicalAssumption |
conductivity has some regularity (e.g. Lipschitz or smoother)
ⓘ
conductivity is bounded and strictly positive ⓘ |
| typicalBoundaryCondition |
measured current flux on boundary
ⓘ
prescribed voltage on boundary ⓘ |
| unknown | conductivity γ(x) ⓘ |
| uses |
boundary value problems for elliptic operators
ⓘ
functional analysis ⓘ harmonic analysis ⓘ |
Referenced by (1)
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