A new approach of combination of the LBM and adjoint method based on the circular function idea is developed for optimization of unsteady incompressible/compressible inviscid flow in 1D and 2D problems. The circular function distribution is used for capturing the compressibility effect in the flowfield and the continuous adjoint method allows designers to implement large number of design variables in actual optimization problems. The adjoint approach is successfully derived for the first time based on the compressible LBM with terminal conditions for estimation of the cost function gradient vector in 1D/2D flow inverse design problem. To validate the new derived flow solver with new lattice, firstly, accuracy of the flow simulation is shown for the inviscid compressible discontinuous flows with shock wave. Secondly, for validation of novel derived adjoint optimization algorithm, three optimization problems in form of the inverse design, including the smooth and shock wave inviscid compressible flowfields, are presented. Also, trend of optimization algorithm is studied in all cases. The results indicate that the presented numerical optimization approach gives desirable accuracy in 1D/2D inverse design of inviscid unsteady compressible flowfields.