Identification of the Source for Full Parabolic Equations

Abstract

In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

“…The general convergence framework states that the stability and approximation properties guarantee the convergence of discrete solutions [1,2]. The deep, broad, and constructive theories of approximation and stability are developed, and they cover various important topics dealing with non-smooth data [3][4][5], weak solutions [6][7][8], energy and maximum principle stability estimates [1,2,9,10], ill-posed and inverse problems [11,12], and nonlocal mathematical models including fractional derivatives [13][14][15][16].…”

Optimal Estimation of Thermal Diffusivity in an Energy Transfer Problem

“…See, as an example, [11], [38]. The parameters that are most frequently estimated, under different assumptions and with different techniques, are: thermal conductivity [21], [35], thermal diffusivity [8], [31], sources of heat generation [32], [34], [40] and the coefficient of heat transfer by convection [23], [33].…”