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Present geophysical/geodetic observations and geological constraints provide a clue to understanding the dynamics of the Earths mantle in the past.Data assimilation allows to constrain the past mantle flow and temperature using dynamic models and available present observations.Quantitative tools are required to solve retrospective inverse problems.The basic inversion techniques(data assimilation methods) to restore the Earth interior will be presented.I shall show how these tools and computational methods(Ismail-Zadeh and Tackley,2010) allow reconstructing the evolution of mantle upwellings and lithospheric slabs using the example of a sinking relic slab in the intra-continental Vrancea region(Ismail-Zadeh et al.,2008) and the example of active subduction beneath the Japanese islands(Ismail-Zadeh et al.,2013).