Any Tsunami Early Warning System (TEWS) should provide reliable prediction of the wave parameters as early as possible. Numerical calculation of the tsunami wave propagation over the particular part of the ocean can be implemented rather efficiently using modern high-performance system. It is possible when the initial wave shape at tsunami source is given. In this paper we discuss and show the fast algorithm for reconstructing the initial sea surface displacement at tsunami source. Based on the Fourier approximation theory, the presented algorithm treats the measured wave profile as a linear combination of synthetic (calculated) waves from the so-called unit sources. Using an artificial bathymetry, we evaluate the algorithm parameters focusing on its precision and performance. Moreover, a large tsunami sources (several hundred kilometers long) are also simulated.