Reduced basis method for quantum models of crystalline solids

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2007.Includes bibliographical references (p. 203-213).Electronic structure problems in solids usually involve repetitive determination of quantities of interest, evaluation of which requires the solution of an underlying partial differential equation. We present in this thesis the application of the reduced basis method in accurate and rapid evaluations of outputs associated with some nonlinear eigenvalue problems related to electronic structure calculations. The reduced basis method provides a systematic procedure by which efficient basis sets and computational strategies can be constructed. The essential ingredients are (i) rapidly convergent global reduced basis approximation spaces; (ii) an offline-online computational procedure to decouple the generation and projection stages of the approximation process; and (iii) inexpensive a posteriori error estimation procedure for outputs of interest. We first propose two strategies by which we can construct efficient reduced basis approximations for vectorial eigensolutions - solutions consisting of several eigenvectors. The first strategy exploits the optimality of the Galerkin procedure to find a solution in the span of all eigenvectors at N judiciously chosen samples in the parameter space.(cont.) The second strategy determines a solution in the span of N vectorial basis functions that are pre-processed to better represent the smoothness of the solution manifold induced by the parametric dependence of the solutions. We deduce from numerical results conditions in which these approximations are rapidly convergent. For linear eigenvalue problems, we construct a posteriori asymptotic error estimators for our reduced basis approximations - extensions on existing work in algebraic eigenvalue problems. We further construct efficient error estimation procedures that allow efficient construction of reduced basis spaces based on the "greedy" sampling procedure. We extend our methods to nonlinear eigenvalue problems, utilizing the empirical interpolation method. We also provide a more efficient construction procedure for the empirical interpolation method. Finally, we apply our methods to two problems in electronic structure calculations - band structure calculations and electronic ground state calculations. Band structure calculations involve approximations of linear eigenvalue problems; we demonstrate the applicability of our methods in the many query limit with several examples related to determination of spectral properties of crystalline solids.(cont.) Electronic ground state energy calculations based on Density Functional Theory involve approximations of nonlinear eigenvalue problems; we demonstrate the potential of our methods within the context of geometry optimization.by George Shu Heng Pau.Ph.D

On the efficacy of anthracene isomers for triplet transmission from CdSe nanocrystals

The effect of isomeric substitutions on the transmitter for triplet energy transfer (TET) between nanocrystal (NC) donor and molecular acceptor is investigated. Each isomeric acceptor is expected to bind in a unique orientation with respect to the NC donor. We see that this orbital overlap drastically affects the transmission of triplets. Here, two functional groups, the carboxylic acid and dithiocarbamate, were varied between the 1-, 2- and 9-positions of the anthracene ring to give three ACA and three ADTC isomers. These six anthracene isomers served as transmitters for triplets between CdSe NC sensitizers and 9,10-diphenylanthracene annihilators for photon upconversion. The photon upconversion quantum yield (QY) is the highest for 9-ACA (12%), lowest for 9-ADTC (0.1%), around 3% for both 1-ACA and 1-ADTC, and about 1% for the 2-isomers. These trends in QYs are reflected in the rates of TET given by ultrafast transient absorption spectroscopy where a maximum of 3.8 × 107 s−1 for 9-ACA was measured. Molecular excited state energy levels were measured both in solution and polymer hosts to correlate structure to TET. This work confirms that anthracene excited states levels are very sensitive to molecular substitution, which in combination with orbital overlap, critically affect Dexter-based TET.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicada

Reduced Basis Method for Nanodevices Simulation

Ballistic transport simulation in nanodevices, which involves self-consistently solving a coupled Schrodinger-Poisson system of equations, is usually computationally intensive. Here, we propose coupling the reduced basis method with the subband decomposition method to improve the overall efficiency of the simulation. By exploiting a posteriori error estimation procedure and greedy sampling algorithm, we are able to design an algorithm where the computational cost is reduced significantly. In addition, the computational cost only grows marginally with the number of grid points in the confined direction

Reduced Basis Method for Nanodevices Simulation

Ballistic transport simulation in nanodevices, which involves self-consistently solving a coupled Schrodinger-Poisson system of equations, is usually computationally intensive. Here, we propose coupling the reduced basis method with the subband decomposition method to improve the overall efficiency of the simulation. By exploiting a posteriori error estimation procedure and greedy sampling algorithm, we are able to design an algorithm where the computational cost is reduced significantly. In addition, the computational cost only grows marginally with the number of grid points in the confined direction

A Parallel Second-Order Adaptive Mesh Algorithm for Incompressible Flow in Porous Media

In this paper we present a second-order accurate adaptive algorithm for solving multiphase, incompressible flows in porous media. We assume a multiphase form of Darcy's law with relative permeabilities given as a function of the phase saturation. The remaining equations express conservation of mass for the fluid constituents. In this setting the total velocity, defined to be the sum of the phase velocities, is divergence-free. The basic integration method is based on a total-velocity splitting approach in which we solve a second-order elliptic pressure equation to obtain a total velocity. This total velocity is then used to recast component conservation equations as nonlinear hyperbolic equations. Our approach to adaptive refinement uses a nested hierarchy of logically rectangular grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithm's accuracy and convergence properties and to illustrate the behavior of the method

Iterative Importance Sampling Algorithms for Parameter Estimation

In parameter estimation problems one computes a posterior distribution over uncertain parameters defined jointly by a prior distribution, a model, and noisy data. Markov chain Monte Carlo (MCMC) is often used for the numerical solution of such problems. An alternative to MCMC is importance sampling, which can exhibit near perfect scaling with the number of cores on high performance computing systems because samples are drawn independently. However, finding a suitable proposal distribution is a challenging task. Several sampling algorithms have been proposed over the past years that take an iterative approach to constructing a proposal distribution. We investigate the applicability of such algorithms by applying them to two realistic and challenging test problems, one in subsurface flow, and one in combustion modeling. More specifically, we implement importance sampling algorithms that iterate over the mean and covariance matrix of Gaussian or multivariate t-proposal distributions. Our implementation leverages massively parallel computers, and we present strategies to initialize the iterations using "coarse" MCMC runs or Gaussian mixture models.U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program [DE-AC02-05CH11231]; National Science Foundation [DMS-1619630]; Alfred P. Sloan Foundation through a Sloan Research Fellowship; Office of Science of the U.S. Department of Energy [DE-AC02-05CH11231]This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Iterative Importance Sampling Algorithms for Parameter Estimation

In parameter estimation problems one computes a posterior distribution over uncertain parameters defined jointly by a prior distribution, a model, and noisy data. Markov chain Monte Carlo (MCMC) is often used for the numerical solution of such problems. An alternative to MCMC is importance sampling, which can exhibit near perfect scaling with the number of cores on high performance computing systems because samples are drawn independently. However, finding a suitable proposal distribution is a challenging task. Several sampling algorithms have been proposed over the past years that take an iterative approach to constructing a proposal distribution. We investigate the applicability of such algorithms by applying them to two realistic and challenging test problems, one in subsurface flow, and one in combustion modeling. More specifically, we implement importance sampling algorithms that iterate over the mean and covariance matrix of Gaussian or multivariate t-proposal distributions. Our implementation leverages massively parallel computers, and we present strategies to initialize the iterations using "coarse" MCMC runs or Gaussian mixture models.U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program [DE-AC02-05CH11231]; National Science Foundation [DMS-1619630]; Alfred P. Sloan Foundation through a Sloan Research Fellowship; Office of Science of the U.S. Department of Energy [DE-AC02-05CH11231]This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]