The Bayesian optimization is one of the machine learning methods to find the optimal value (maximum value or minimum value) of a function whose shape is unknown (black box function). From the function values at a finite number of points, the Bayesian optimization predicts the functional form by the Gaussian process from the function values at a limited number of points. Then, a point, which is expected to have the optimum value or that with large uncertainty is examined preferentially. By sequentially adding search points and gradually increasing the accuracy of prediction, this method can find the optimum point even in a relatively high-dimensional search space.
In the context of condensed matter physics, methods of Markov-chain Monte Carlo simulation for many-body systems obeying classical statistical mechanics are called classical Monte Carlo methods. The representative example is the one by the Metropolis-Hastings method. The Markov-chain Monte Carlo method for simulated annealing is also a classical Monte Carlo method. In particular, a local update method is the method in which a degree of freedom is selected at each step for update. A local update generally suffers from the slow convergence near the critical point (critical slowing-down). To solve this problem, global update methods are proposed, in which a group of elements are updated at each time. However, it is necessary to design the algorithm depending on each specific system, and the applicability of the algorithm is limited compared to the local update methods.
An approximation often used for disordered systems. In general, it is difficult to describe the disordered systems, including alloys, within the framework of the band theory that usually assumes periodic structures. CPA successfully describes realistic alloys quantitatively, by confining the effect of the disorder within each site in a self-consistent fashion. As a part of KKR method, CPA is used in computing the electronic state calculations of disordered systems.