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A new direct constrained optimization algorithm for minimizing the Kohn-Sham (KS) total energy functional is presented in this paper. The key ingredients of this algorithm involve projecting the total energy functional into a sequences of subspaces of small dimensions and seeking the minimizer of total energy functional within each subspace. The minimizer of a subspace energy functional not only provides a search direction along which the KS total energy functional decreases but also gives an optimal "step-length" to move along this search direction. A numerical example is provided to demonstrate that this new direct constrained optimization algorithm can be more efficient than the self-consistent field (SCF) iteration.
Antibiotic resistance continues to be a major challenge we face today. Scientists and medical professionals are competing against a microbial evolutionary time bomb. The alarming increase in the number of deaths caused by multi-drug resistant infections (1) and the decrease in development of reliable treatment regimens is disturbing. Historically, most studies focus on the effects of fatal concentrations of antibiotics on the evolution of antibiotic resistance (2). Focusing on high antibiotic concentrations limits our understanding of antibiotic resistance and how its evolution is established. Especially since it has been shown that there is a greater selection of resistant bacteria at sub-lethal concentrations of antibiotics (3). Our main goal is to further investigate the impacts of sub-inhibitory concentrations of antibiotics on antibiotic resistance evolution. To do this, we studied two genes that confer resistance to β-lactam antibiotics,blaTEM-50 and blaTEM-85. We created adaptive landscapes from each of the 16 alleles of every combination of the four amino acid substitutions in each gene using bacterial growth rates as a measurement of fitness. We have shown that the topography of these adaptive landscapes depend on the type, and concentration, of the β-lactam antibiotic treatment (4). We also developed a rational design of antibiotic treatment plans based on mathematical models of the adaptive landscape data. We found that by cycling between structurally similar antibiotics, there is a 60%-100% probability of returning to a more-susceptible state. This is a favorable result for laying a foundation to use antibiotic cycling to help alleviate the effects of antibiotic resistance, which has recently shown promising (5) (6). Furthermore, we investigated the evolution of resistance within a local hospital by studying the trends in resistant phenotypes of patient isolates (7). We found there was no significant trend in antibiotic resistance occurring in the hospital, and suspect that the community contributes the majority of the selective pressures leading to multidrug resistant pathogens. Using our novel mathematical models, we were able to successfully predict the resistance genes that were present in the hospital and, by using genomic sequencing data; we confirmed the presence of these resistance genes. These studies show that sub-inhibitory concentrations of antibiotics, present in the environment, accelerate the diversity of antibiotic resistance genes. Also, we found that antibiotics used within the hospital do not impact the evolution of antibiotic resistance within the hospital. Altogether, we have 1) shown that sub-lethal concentrations of β-lactam antibiotics have an effect on the evolution of β-lactamase resistance genes, 2) developed mathematical models that can be used to lay a foundation for antibiotic cycling, and 3) developed a tool for hospitals to assess the transmission of antibiotic resistance trends using phenotypic data
The Self Consistent Field (SCF) iteration, widely used for computing the ground state energy and the corresponding single particle wave functions associated with a many-electronatomistic system, is viewed in this paper as an optimization procedure that minimizes the Kohn-Sham total energy indirectly by minimizing a sequence of quadratic surrogate functions. We point out the similarity and difference between the total energy and the surrogate, and show how the SCF iteration can fail when the minimizer of the surrogate produces an increase in the KS total energy. A trust region technique is introduced as a way to restrict the update of the wave functions within a small neighborhood of an approximate solution at which the gradient of the total energy agrees with that of the surrogate. The use of trust region in SCF is not new. However, it has been observed that directly applying a trust region based SCF(TRSCF) to the Kohn-Sham total energy often leads to slow convergence.We propose to use TRSCF within a direct constrained minimization(DCM) algorithm we developed in \cite dcm. The key ingredients of theDCM algorithm involve projecting the total energy function into a sequence of subspaces of small dimensions and seeking the minimizer of the total energy function within each subspace. The minimizer of a subspace energy function, which is computed by TRSCF, not only provides a search direction along which the KS total energy function decreases but also gives an optimal "step-length" that yields a sufficient decrease in total energy. A numerical example is provided to demonstrate that the combination of TRSCF and DCM is more efficient than SCF.
Many properties of nanostructures depend on the atomic configuration at the surface. One common technique used for determining this surface structure is based on the low energy electron diffraction (LEED) method, which uses a high-fidelity physics model to compare experimental results with spectra computed via a computer simulation. While this approach is highly effective, the computational cost of the simulations can be prohibitive for large systems. In this work, we propose the use of a direct search method in conjunction with an additive surrogate. This surrogate is constructed from a combination of a simplified physics model and an interpolation that is based on the differences between the simplified physics model and the full fidelity model.
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