A novel inversion algorithm for weak gravitational lensing using quasi-conformal geometry

Context . The challenge in weak gravitational lensing caused by galaxies and clusters is to infer the projected mass density distribution from gravitational lensing measurements, known as the inversion problem. Aims . We introduce a novel theoretical approach to solving the inversion problem. The cornerstone of the proposed method lies in a complex formalism that describes the lens mapping as a quasi-conformal mapping with the Beltrami coefficient given by the negative of the reduced shear, which can, in principle, be observed from the image ellipticities. Methods . We propose an algorithm called QCLens that is based on this complex formalism. QCLens computes the underlying quasi-conformal mapping using a finite element approach by reducing the problem to two elliptic partial differential equations that solely depend on the reduced shear field. Results . Experimental results for both the Schwarzschild and the singular isothermal lens demonstrate the agreement of our proposed method with the analytically computable solutions.