Computationally efficient approach for the minimization of volume constrained vector-valued Ginzburg-Landau energy functional

The minimization of volume constrained vector-valued Ginzburg-Landau energy functional is considered in the present study. It has many applications in computational science and engineering, like the conservative phase separation in multiphase systems (such as the spinodal decomposition), phase coarsening in multiphase systems, color image segmentation and optimal space partitioning. A computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem. The algorithm is based on the constrained nonmonotone L2 gradient flow of Ginzburg-Landau functional followed by a regularization step, which is resulted from the Tikhonov regularization term added to the objective functional, that lifts the solution from the L2 function space into H1 space. The regularization step not only improves the convergence rate of the presented algorithm, but also increases its stability bound. The step-size selection based on the Barzilai-Borwein approach is adapted to improve the convergence rate of the introduced algorithm. The success and performance of the presented approach is demonstrated throughout several numerical experiments. To make it possible to reproduce the results presented in this work, the MATLAB implementation of the presented algorithm is provided as the supplementary material.