Explicit calculation of smoothed sensitivity coefficients for linear problems

A technique of explicit calculation of sensitivity coe cients based on the approximation of the retrieved function by a linear combination of trial functions of compact support is presented. The method is applicable to steady state and transient linear inverse problems where unknown distributions of boundary uxes, temperatures, initial conditions or source terms are retrieved. The sensitivity coe cients are obtained by solving a sequence of boundary value problems with boundary conditions and source term being homogeneous except for one term. This inhomogeneous term is taken as subsequent trial functions. Depending on the type of the retrieved function, it may appear on boundary conditions (Dirichlet or Neumann), initial conditions or the source term. Commercial software and analytic techniques can be used to solve this sequence of boundary value problems producing the required sensitivity coe cients. The choice of the approximating functions guarantees a ltration of the high frequency errors. Several numerical examples are included where the sensitivity coe cients are used to retrieve the unknown values of boundary uxes in transient state and volumetric sources. Analytic, boundary-element and nite-element techniques are employed in the study