# Algorithm Solves Key Energy-reduction Issue in Describing Complex System

September 28, 2007

The SIAM Journal on Scientific Computing recently published the work by three CRD researchers that incorporated an algorithm they developed in computing the ground state energy and wave functions for describing the mechanics of a complex physical system.

The paper, “A Trust Region Direct Constrained Minimization Algorithm for the Kohn-Sham Equation,” described the use of the direct constrained minimization (DCM) algorithm with the “trust region” technique to compute single electron wavefunctions associated with the ground state energy of molecules or solids. These wavefunctions are also solutions to the Kohn-Sham equations, a set of equations that form the first order necessary condition satisfied by the minimizer of the total energy functional.

The Kohn-Sham equations are traditionally solved by a technique called the self-consistent field (SCF) iteration. However, the simplest form of SCF often fails to converge. The convergence of SCF can be improved by the use of a technique called “charge mixing.” Although it can be quite effective in practice, a charge mixing-enabled SCF iteration may not reduce the total energy approximation monotonically, and there is no guarantee that charge mixing will always fix the convergence failure of the SCF iteration. The DCM algorithm, on the other hand, is designed to minimize the total energy of the atomistic system directly. Through the use of adaptive “trust regions” of appropriate sizes, the reduction of the total energy is guaranteed to be monotonic.

Chao Yang, a member of the Scientific Computing Group, was the lead author. Juan Meza, head of the High-Performance Computing Research Department, and Lin-Wang Wang in the Scientific Computing Group co-authored the paper.

The researchers viewed the SCF iteration as an indirect way of minimizing the total energy function through the minimization of a sequence of quadratic surrogate functions. They noticed that, at each SCF iteration, the surrogate function shares the same gradient with that of the total energy function at the current approximation. Hence moving along a descent direction associated with the surrogate function near the current approximation is likely to result in a reduction in the total energy. However, if one moves too far along that direction, the total energy may actually increase, resulting in convergence failure of the SCF iteration.

So Yang and his fellow researchers proposed to stabilize the convergence of the SCF iteration by imposing an additional constraint to the surrogate minimization problem. This additional constraint sets up a region in which the minimizer of the surrogate function can be “trusted,” in the sense that a reduction in the surrogate in that region will lead to a reduction in the total energy as well.

However, identifying the appropriate size of a trust region would require computing all eigenvalues of a fixed Kohn-Sham Hamiltonian, which is not practical due to the dimension of the Hamiltonian. If the size of the trust region is too large, the SCF iteration may fail to converge. If the trust region is too small, the SCF iteration may converge very slowly.

To overcome this difficulty, the researchers came up with an alternative approach that minimizes the total energy directly. In this direct constrained minimization (DCM) algorithm, the total energy functional is projected into a sequence of overlapping subspaces. The projected problem has a significantly fewer degrees of freedom, thus can be solved efficiently by the trust region-enabled SCF iteration.

The optimal solution of the projected problem yields an optimal search direction and step length for updating the approximation to the minimizer of the true total energy function. They demonstrated this method using two numerical examples, which showed that “such a scheme outperforms the SCF iteration combined with charge mixing in terms of both efficiency and reliability.”

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