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This paper continues a series in which we intend to show how all observablesof galaxy clusters can be combined to recover the two-dimensional, projectedgravitational potential of individual clusters. Our goal is to develop anon-parametric algorithm for joint cluster reconstruction taking all clusterobservables into account. In this paper, we begin with the relation between theCompton-y parameter and the Newtonian gravitational potential, assuminghydrostatic equilibrium and a polytropic stratification of the intraclustergas. We show how Richardson-Lucy deconvolution can be used to convert theintensity change of the CMB due to the thermal Sunyaev-Zel'dovich effect intoan estimate for the two-dimensional gravitational potential. Synthetic datasimulated with characteristics of the ALMA telescope show that thetwo-dimensional potential of a cluster with mass 5*10^14 M_sun/h at redshift0.2 is possible with an error of < 5% between the cluster centre and a radius r< 0.9 Mpc/h.

Joint reconstruction of galaxy clusters from gravitational lensing and thermal gas. II. Inversion of the thermal Sunyaev-Zel'dovich effect