Robust Methods for an Inverse Heat Conduction Problem

Abstract

Estimation methods based on the ℓ₁ norm, the ℓ_∞ norm, or Huber’s criterion function are proposed to solve the linear inverse heat conduction problem of determining the temperature or the heat flux on the inner surface of a tube using temperature measurements by thermocouples imbedded in the tube. Mathematically, the problem is to estimate the unknown boundary parameters of an one dimensional heat equation on an unit interval using temperature observations at some interior point. By approximating the solution of the heat equation using spectral method, the ℓ₁ norm or the ℓ_∞ norm estimates can then be found by solving some linear programming problems; the estimates based on Huber’s criterion function can be obtained by solving a non-linear least squares problem. Numerical examples are given to illustrate the effectiveness of the methods. For the non-linear inverse heat conduction problem, a quasilinearization method is proposed.