  We consider the problem of computing the initial condition for a general
parabolic equation from the Cauchy lateral data. The stability of this problem
is well-known to be logarithmic. In this paper, we introduce an approximate
model, as a coupled linear system of elliptic partial differential equations.
The solution to this model is the vector of Fourier coefficients of the
solutions to the parabolic equation above. This approximate model is solved by
the quasi-reversibility method. We will prove the convergence for the
quasi-reversibility method as the measurement noise tends to 0. The convergent
rate is Lipschitz. We present the implementation of our algorithm in details
and verify our method by showing some numerical examples.
