Using a magneto-optical pump-probe technique with micrometer spatial resolution we show that magnetization precession can be launched in individual magnetic domains imprinted in a Co$_{40}$Fe$_{40}$B$_{20}$ (CoFeB) layer by elastic coupling to ferroelectric domains in a BaTiO$_{3}$ substrate. The dependence of the precession parameters on external magnetic field strength and orientation reveal that by laser-induced ultrafast partial quenching of the magnetoelastic coupling parameter of CoFeB by $approx$27% along with 10% ultrafast demagnetization trigger the magnetization precession. The relation between the laser-induced reduction of the magnetoelastic coupling and the demagnetization is approximated by the $n(n+1)/2$-law with n$approx$2. This correspondence confirms the thermal origin of the laser-induced anisotropy change. Based on the analysis and modeling of the excited precession we find signatures of laser-induced precessional switching, which occurs when the magnetic field is applied along the hard magnetization axis and its value is close to the effective magnetoelastic anisotropy field. The precession excitation process in an individual magnetoelastic domain is found to be unaffected by neighboring domains. This makes laser-induced changes of magnetoelastic anisotropy a promising tool for driving magnetization dynamics and switching in composite multiferroics with spatial selectivity.
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