Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Multilevel domain decomposition for electronic structure calculations
Journal of Computational Physics
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We analyze a decomposition algorithm for minimizing a quadratic objective function, separable in $\mathbf{x}_1$ and $\mathbf{x}_2$, subject to the constraint that $\mathbf{x}_1$ and $\mathbf{x}_2$ are orthogonal vectors on the unit sphere. Our algorithm consists of a local step where we minimize the objective function in either variable separately, while enforcing the constraints, followed by a global step where we minimize over a subspace generated by solutions to the local subproblems. We establish a local convergence result when the global minimizers are nondegenerate. Our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint. The analysis is connected with a new domain decomposition algorithm for electronic structure calculations.