Quantum Monte Carlo is among the most precise simulation techniques to study realistic materials in physics and chemistry and provides a significant gain in precision compared with traditional density functional theory. One significant limitation of today?s QMC methods is the high computational demand. Since a substantial part of the QMC computation is spent in on forming and evaluating Slater determinants, the team plans to develop different localization transformations in order to obtain sparse determinants. The sparsity can be exploited in multilevel preconditioners, incomplete decomposition preconditioners, and iterative solvers to reach linear scaling with system size. The newly developed QMC methods will enable the team to obtain accurate equations of state, phase transitions, and elasticity of solid materials that are of high interest in geophysics. The spin state of iron in solid solutions magnesiowustite, perovskite and post-perovskite (Mg,Fe)SiO3 as well as the properties of water-carbon dioxide mixtures will be determined using QMC.
Our understanding of the interior of the Earth comes from seismic observations and from the characterization of geological materials at high pressure. This characterization is not only obtained with high-pressure laboratory experiments but also with computer simulations because the properties of materials depend on the interactions between the atoms and those can be determined with computer simulations from the fundamental laws of physics. This project focuses on making those simulation methods much more accurate by developing new mathematical techniques to improve the quantum Monte Carlo method. These newly developed methods will enable the team to characterize different metal oxides, silicates, and mixtures of fluid water and carbon dioxide at high pressure.
For applications of geophysics, such as earthquake simulations, and our fundamental understanding of the processes that take place deep inside the earth, it is important to know the properties of the materials deep inside the earth. However, there is no way to actually access such materials, and we must use computer simulations to obtain their properties. The quantum Monte Carlo method (QMC) for analyzing the properties of materials is among the most precise methods available, but unfortunately it is also very expensive. For this reason, we have developed new mathematical methods to make these calculations more efficient. The most important result is that, for many materials, we have reduced the cost of QMC from O(N3) to O(N2), where N is the number of valence electrons in the system. This leads to much more efficient simulations when N is large, that is, for more accurate solutions or more complex problems. Some of the methods that we have developed, iterative solvers for the solution of very large systems of linear equations, have much wider applicability than QMC alone; for example, they are also being used in large-scale engineering simulations of fluid flow. In addition to our research in new mathematical methods for QMC calculations, we have also taught several courses on these methods, their underlying theory, and their applications, to graduate students, researchers, and faculty in other disciplines. In particular, at the university of Illinois, we have taught a five day Summer School on Computational Materials Science: Quantum Monte Carlo: Theory and Fundamentals. The summer school was attended by 48 participants (roughly the maximum number possible) from North and South America, Europe, and Asia, and taught by six lecturers and two lab instructors. The lectures are available from YouTube (look for "MCC workshops"). We have also taught a short course on some of the methods developed in this project at the Ecole Normale Superieure de Lyon in France to graduate students and researchers in computational astrophysics. This nicely demonstrates the wide applicability of the mathematical methods developed; they are applicable from the very small scale of nuclei and electrons to the gigantic scale of galaxies. Finally, as part of this project, several graduate students from physics and mathematics have been educated and have collaborated on an important interdisciplinary project. This has prepared them much better for future interdisciplinary collaborations in academia, national laboratories, or industry than working just in their own field would have done.
