Inverse Iteration for the Monge-Ampère Eigenvalue Problem

Abstract

We present an iterative method based on repeatedly inverting the Monge-Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $Ω\subset \mathbb{R}^n$. We prove that the iterates $u_k$ generated by this method converge as $k \to \infty$ to a solution of the Monge-Ampère eigenvalue problem $$\begin{cases} \text{det} D^2u = λ_{MA} (-u)^n & \quad \text{in } Ω,\\ u = 0 & \quad \text{on } \partial Ω. \end{cases}$$ Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat{u}_k := \frac{u_k}{||u_k||_{L^{\infty}(Ω)}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim\limits_{k \to \infty} R(u_k) = \lim\limits_{k \to \infty} R(\hat{u}_k) = λ_{MA}$, where the Rayleigh quotient $R(u)$ is defined as $$R(u) := \frac{\int_Ω (-u) \ \text{det} D^2u}{\int_Ω (-u)^{n+1}}.$$ Our method converges for a wide class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue $λ_{MA}$.