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

TL;DR

Addressing the ill-posed problem of recovering the projected mass density $κ$ from weak-lensing measurements, the paper reframes the lens equation as a quasi-conformal mapping with the Beltrami coefficient $μ = -g$ and develops QCLens, a finite-element solver that reduces the inversion to two elliptic PDEs for the real and imaginary parts of the lens mapping driven by the reduced shear field. The method is validated against analytic Schwarzschild and singular isothermal lens solutions, demonstrating consistency and convergence as the mesh is refined. This qc-based formulation offers a principled alternative to Kaiser–Squires and provides a path toward spherical generalization for wide-area surveys like Euclid, the Rubin Observatory, and the Roman Space Telescope. By linking geometric function theory with gravitational lensing, the work lays a foundation for more robust, boundary-aware mass-mapping in future cosmological analyses.

Abstract

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.

Figures (5)

• Figure 1: Geometric interpretation of q.c. mappings (figure from Ming2013).
• Figure 2: Schwarzschild lens: Comparison between actual lens mapping, $f = u+\mathrm{i}v$, and calculated lens mapping, $f^7 = u^7+\mathrm{i}v^7,$ with QCLens for a resolution of $n=7$ and Dirichlet boundary conditions.
• Figure 3: Schwarzschild lens error: $L_2$ and $H_1$ errors for different refinement orders with Dirichlet boundary conditions. The orange and red lines overlap, as do the green and blue lines.
• Figure 4: Singular isothermal lens: Comparison between actual lens mapping, $f = u+\mathrm{i}v$, and calculated lens mapping, $f^7 = u^7+\mathrm{i}v^7$, with QCLens for a resolution of $n=7$ and Dirichlet boundary conditions.
• Figure 5: Singular isothermal lens error: $L_2$ and $H_1$ errors for different refinement orders with Dirichlet boundary conditions. The orange and red lines overlap, as do the green and blue lines.