The author considers ill-posed Cauchy problems for elliptic partial differential equations with linear densely defined selfadjoint positive definite operators. It is assumed that the initial data for the solution and its time derivative are noisy with a prescribed level of accuracy. Under some assumptions about the smoothness of the solution the conditions for best possible accuracy for identifying the solution from the noisy data are obtained. It is shown that the best possible accuracy depends either in a Hölder continuous way or in a logarithmic form on the noise level. Next, the author considers a special generalization of Tikhonov regularization methods for the above problems which permit one to attain optimal error bounds.