  We propose a robust numerical method to find the coefficient of the creation
or depletion term of parabolic equations from the measurement of the lateral
Cauchy information of their solutions. Most papers in the field study this
nonlinear and severely ill-posed problem using optimal control. The main
drawback of this widely used approach is the need of some advanced knowledge of
the true solution. In this paper, we propose a new method that opens a door to
solve nonlinear inverse problems for parabolic equations without any initial
guess of the true coefficient. This claim is confirmed numerically. The key
point of the method is to derive a system of nonlinear elliptic equations for
the Fourier coefficients of the solution to the governing equation with respect
to a special basis of L^2. We then solve this system by a predictor-corrector
process, in which our computation to obtain the first and second predictors is
effective. The desired solution to the inverse problem under consideration
follows.
