The least Square problem is an very old problem in mathematics. Recently, not only new ingredients in mathematics have enriched it's content, but also applications in fields such as chemistry, physics, neural network, finance, economic, mechanical system, communication, electronic engineering, medical imaging and so on, which always use least squares to measure the discrepancies between their model and observations, broaden it's context. These new applications, which usually are large scale problems, or highly nonlinear, or ill-posed, make the traditional numerical methods difficult to figure out a useful solution. Therefore, new stable and robust methods are of highly interests.

Parameter identification usually means the estimation of coefficients in a differential equation from observations of the solution to that equation. These coefficients are called system parameters, and the solution and its derivatives constitute the state variables. The forward problem is to compute the state variables given the system parameters and appropriate boundary conditions. The forward problem is typically well-posed. Parameter identification, the inverse problem of interest, is typically ill-posed. Moreover, even when the forward problem is linear in the state variable, the parameter identification problem is generally nonlinear. Ill-posed means the parameter to be reconstructed does not depend on the observation in a stable way, So regularization methods have to be used in order to compute a stable approximation of the parameter in the presence of data noise. Due to the ill-posedness of the identification problem, the numerical approximation of such problems is not a simple task.

Total Variation: A good choice of regularization term would lead to a good approximation of the solution. However, choice of regularization term depend on the definition of the admissible parameter set. If the coefficient is continuous, we can use H^2 or H^1 norm as the regularization term. However if the coefficient suffers large jumps, the use of these terms is not appropriate due to the discontinuous of the coefficient. Fortunately, the total variation will overcome such difficulties.