Inverse Iteration for the Monge-Amp\`ere Eigenvalue Problem

Abstract

We present an iterative method based on repeatedly inverting the Monge-Amp\`ere operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain ⊂ Rn. We prove that the iterates uk generated by this method converge as k ∞ to a solution of the Monge-Amp\`ere eigenvalue problem cases det D2u = λMA (-u)n & in ,\\ u = 0 & on ∂ . cases Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates uk := uk||uk||L∞() converge to the eigenfunction of unit height. In addition, we show that k ∞ R(uk) = k ∞ R(uk) = λMA, where the Rayleigh quotient R(u) is defined as R(u) := ∫ (-u) \ det D2u∫ (-u)n+1. Our method converges for a wide class of initial choices u0 that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Amp\`ere eigenvalue λMA.