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

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. 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. 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. Experimental results for both the Schwarzschild and the singular isothermal lens demonstrate the agreement of our proposed method with the analytically computable solutions.