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A quadratically constrained minimization problem arising from PDE of Monge-Ampère type. (English) Zbl 1187.65073

A quadratically constrained eigenvalue minimization problem is considered. These problems arise in the numerical solution of fully nonlinear three dimensional Monge-Ampère type equations, known as the Dirichlet problem for the \(\sigma_2\)-operator. The nonlinear Dirichlet problem with the elliptic \(\sigma_2\)-operator is linearized in a neighborhood of the solution. The theory and a solution technique is developed for the least-squares minimization of the linearized \(\sigma_2\) problem. The minimization subproblem has to be solved many times during the numerical solution of the Monge-Ampère equation. Therefore an efficient numerical technique is required for the solution of minimization problem. It turns out that the proposed algorithm is finite, of complexity \(\mathcal{O}(n^3)\) and requires additionally solving a simple scalar secular equation. As a numerical example, two dimensional minimization to the solution of the Dirichlet problem for the two-dimensional Monge-Ampère equation, is considered. The numerical results indicate the excellent convergence behavior of the algorithm.

MSC:

65K10 Numerical optimization and variational techniques
35J96 Monge-Ampère equations
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control
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References:

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