Inversion methods in helioseismology and solar tomography

Basic methods by which the internal structure and dynamics of the Sun can be inferred from observed frequencies of solar oscillations and acoustic travel times are discussed. The methods for inverting the oscillation frequencies are based on a variational formulation of the adiabatic eigenvalue problem for a star. The inversion technique formulated in terms of linear integral constraints provides estimates of localized averages of properties of the solar structure, such as density and sound speed, helium abundance in the convection zone for a given equation of state, and, in addition, the estimates for the internal rotation rate. The method of inverting acoustic travel times employs a geometrical ray approximation and provides 3D images of solar convective cells, active regions and sunspots. The information about the global and local structures and flow velocities in the solar interior is important for understanding solar evolution and mechanisms of solar activity. The high-resolution helioseismology projects from space provide a tremendous amount of data, the interpretation of which is increasingly challenging and requires the development of efficient inversion methods and algorithms.