Input recovery from noisy output data, using regularized inversion of the Laplace transform

In a dynamical system the input is to be recovered from finitely many measurements, blurred by random error, of the output of the system. As usual, the differential equation describing the system is reduced to multiplication with a polynomial after applying the Laplace transform. It appears that there exists a natural, unbiased, estimator for the Laplace transform of the output, from which an estimator of the input can be obtained by multiplication with the polynomial and subsequent application of a regularized inverse of the Laplace transform. It is possible, moreover, to balance the effect of this inverse so that ill-posedness remains restricted to its actual source: differentiation. The rate of convergence of the integrated mean-square error is a positive power of the number of data. The order of the differential equation has an adverse effect on the rate which, on the other hand, increases with the smoothness of the input as usual.