Inverse iteration for the Monge–Ampère eigenvalue problem,Proceedings of the American Mathematical Society

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Inverse iteration for the Monge–Ampère eigenvalue problem
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-08-11 , DOI: 10.1090/proc/15157
Farhan Abedin , Jun Kitagawa

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 $\Omega \subset \mathbb{R}^{n}$ . We prove that the iterates

generated by this method converge as $k \to \infty$ to a solution of the Monge-Ampère eigenvalue problem

$\begin{displaymath}\begin {cases}\mathrm {det} D^2u = \lambda _{MA} (-u)^n & \qu... ...ega ,\\ u = 0 & \quad \text {on } \partial \Omega . \end{cases}\end{displaymath}$

Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat {u}_k \colonequals \frac {u_k}{\vert\vert u_k\vert\vert _{L^{\infty }(\Omega )}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim _{k \to \infty } R(u_k) = \lim _{k \to \infty } R(\hat {u}_k) = \lambda _{MA}$ , where the Rayleigh quotient

is defined as

$\displaystyle R(u) \colonequals \frac {\int _{\Omega } (-u) \ \det D^2u}{\int _{\Omega } (-u)^{n+1}}.$

Our method converges for a wide class of initial choices

that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue $\lambda _{MA}$ .

中文翻译：

Monge–Ampère特征值问题的逆迭代

摘要：我们提出了一种基于Dirichlet边界条件并在有界凸域上规定了右手边的Monge-Ampère算子反复求逆的迭代方法。我们证明了该方法生成的迭代收敛到Monge-Ampère特征值问题的解 $\ Omega \ subset \ mathbb {R} ^ {n}$

$k \ to \ infty$

$\ begin {displaymath} \ begin {cases} \ mathrm {det} D ^ 2u = \ lambda _ {MA}（-u）^ n＆\ qu ... ... ega，\\ u = 0＆\ quad \ text {on} \ partial \ Omega。 \ end {cases} \ end {displaymath}$

由于此问题的解决方案在一个正的乘法常数之前都是唯一的，因此归一化的迭代收敛到单位高度的本征函数。此外，我们证明，其中瑞利商定义为 $\ hat {u} _k \ colonequals \ frac {u_k} {\ vert \ vert u_k \ vert \ vert _ {L ^ {\ infty}（\ Omega}}}$ $\ lim _ {k \ to \ infty} R（u_k）= \ lim _ {k \ to \ infty} R（\ hat {u} _k）= \ lambda _ {MA}$

$\ displaystyle R（u）\ colonequals \ frac {\ int _ {\ Omega}（-u）\ \ det D ^ 2u} {\ int _ {\ Omega}（-u）^ {n + 1}}。$

我们的方法适用于

可以明确构造的多种初始选择，并且不依赖于对Monge-Ampère特征值的先验知识。 $\ lambda _ {MA}$

更新日期：2020-09-02

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