Variance minimization

It may seem that minimization of the energy should be a valid approach to trial function optimization. As it turns out, minimization of the energy using a fixed sample is problematic because it is easy for a few very low energies to dominate the sample. In this case, the optimization is not reliable and a good trial function is not obtained. This is not a concern in variance minimization because the variance is bounded from below by zero. However, even for variance minimization, care... 

In conventional variance minimization calculations (14) (i.e. the unconstrained A = 1 case), the above property is used to find an overall fit to IP (we drop the A superscript for simplicity). The procedure is to determine the parameters a in the trial function vPt(R-5 a ) by minimizing the variance of local energy a2... 

Following (14) we set the reweighting factors u>, in equations (27) and (30) equal to unity in order to avoid a numerical instability in the variance minimization procedure which occurs for systems with a large number of electrons (these factors, however, are included in calculating the expansion coefficients of the electron density). The above fixed-density variance minimization is then repeated several times until the procedure converges. 

We performed adiabatic connection calculations for cosine-wave jellium using six values of A 0,0.2,0.4,0.6,0.8,1. The many-body wavefunctions for A > 0 were optimized by fixed-density variance minimization using 10000 independent N—electron configurations at each A. These configurations were regenerated several times. The weight factor in expression (27) was set equal... 

Figure 1. Cost profiles during ground state energy optimization of O based on (a) energy and (b) variance minimization approach.

This structure has been in place for decades. Even when the manufacturing activity takes place in minutes, the final delivery of a product may take days, weeks, or months, depending on the efficiency of the supply chain. The operating objectives of a supply chain are to maximize response, minimize variance, minimize inventory, maximize consolidation, maintain high levels of quality, and provide life-cycle support. 

The summands for Eq. 9.42 must be positive (or zero) everywhere, so the variance of an approximate wave function is bounded from below by zero, even for a finite sample. The coincidence between the lower bound of the variance and the variance of the exact wave function enables the trial wave function to be optimized by variance minimization. The existence of an absolute lower bound makes variance minimization more robust than energy minimization [49]. 

Snajdr and Rothstein compared a number of properties including the average interelectronic distances and multipole moments of wave functions optimized by variance minimization to those optimized by energy minimization [155]. They... 

In the optimization of the wave function, we use the modification of the variance minimization technique [35]. In the original scheme, the sample points are generated according to the initial wave function and are fixed during optimization. In our study, we select the sample points used in the optimization as follows First, we generate... 

Wave function optimization Variance minimization, energy minimization, and their combination with fixed sample points. 

While the Newton method converges quickly even for many parameters, provided good starting values are available for the parameters, the stochastic implementation suffers from stochastic errors in the Hessian. Nonetheless, Lin et al. were able to optimize Jastrow parameters of the first-row atoms and demonstrated that energy minimization leads to significantly lower energy than variance minimization. On the other hand, the sample size for the optimization is considerably larger than in the case of variance minimization. ... 

Similarly, Riley and Anderson optimized Cl coefficients directly in the presence of a preoptimized Jastrow factor by solving the generalized eigenvalue problem based on matrix elements of the H and S matrices that were calculated by Monte Carlo integration. The method was applied to the Be atom and required a considerably larger sample size than a variance minimization method. 

---