TY - JOUR

T1 - Fréchet derivatives for some bilinear inverse problems

AU - Dierkes, Thomas

AU - Dorn, Oliver

AU - Natterer, Frank

AU - Palamodov, Victor

AU - Sielschott, Helmut

PY - 2002/7

Y1 - 2002/7

N2 - In many inverse problems a functional of u is given by measurements, where u solves a partial differential equation of the type L(p)u+Su = q. Here q is a known source term, and L(p), S are operators, with p as an unknown parameter of the inverse problem. For the numerical reconstruction of p, the heuristically derived Fréchet derivative R' of the mapping R: p → "measurement functional of u" is often used. We show for three problems a transport problem in optical tomography, an elliptic equation governing near-infrared tomography, and the wave equation in moving media-that R' is the derivative in the strict sense. Our method is applicable to more general problems than are established methods for similar inverse problems.

AB - In many inverse problems a functional of u is given by measurements, where u solves a partial differential equation of the type L(p)u+Su = q. Here q is a known source term, and L(p), S are operators, with p as an unknown parameter of the inverse problem. For the numerical reconstruction of p, the heuristically derived Fréchet derivative R' of the mapping R: p → "measurement functional of u" is often used. We show for three problems a transport problem in optical tomography, an elliptic equation governing near-infrared tomography, and the wave equation in moving media-that R' is the derivative in the strict sense. Our method is applicable to more general problems than are established methods for similar inverse problems.

KW - Adjoint field methods

KW - Mathematical imaging

KW - Numerical inverse scattering

KW - Tomography

UR - http://www.scopus.com/inward/record.url?scp=0036665081&partnerID=8YFLogxK

U2 - 10.1137/S0036139901386375

DO - 10.1137/S0036139901386375

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AN - SCOPUS:0036665081

SN - 0036-1399

VL - 62

SP - 2092

EP - 2113

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 6

ER -