Enhancing structure relaxations for first-principles codes: an approximate Hessian approach

We present a method for improving the speed of geometry relaxation by using a harmonic approximation for the interaction potential between nearest neighbor atoms to construct an initial Hessian estimate. The model is quite robust, and yields approximately a 30% or better reduction in the number of calculations compared to an optimized diagonal initialization. Convergence with this initializer approaches the speed of a converged BFGS Hessian, therefore it is close to the best that can be achieved. Hessian preconditioning is discussed, and it is found that a compromise between an average condition number and a narrow distribution in eigenvalues produces the best optimization.Comment: 9 pages, 3 figures, added references, expanded optimization sectio

PDFO: A Cross-Platform Package for Powell's Derivative-Free Optimization Solvers

The late Professor M. J. D. Powell devised five trust-region derivative-free optimization methods, namely COBYLA, UOBYQA, NEWUOA, BOBYQA, and LINCOA. He also carefully implemented them into publicly available solvers, which are renowned for their robustness and efficiency. However, the solvers were implemented in Fortran 77 and hence may not be easily accessible to some users. We introduce the PDFO package, which provides user-friendly Python and MATLAB interfaces to Powell's code. With PDFO, users of such languages can call Powell's Fortran solvers easily without dealing with the Fortran code. Moreover, PDFO includes bug fixes and improvements, which are particularly important for handling problems that suffer from ill-conditioning or failures of function evaluations. In addition to the PDFO package, we provide an overview of Powell's methods, sketching them from a uniform perspective, summarizing their main features, and highlighting the similarities and interconnections among them. We also present experiments on PDFO to demonstrate its stability under noise, tolerance of failures in function evaluations, and potential in solving certain hyperparameter optimization problems

An Optimal Interpolation Set for Model-Based Derivative-Free Optimization Methods

This paper demonstrates the optimality of an interpolation set employed in derivative-free trust-region methods. This set is optimal in the sense that it minimizes the constant of well-poisedness in a ball centred at the starting point. It is chosen as the default initial interpolation set by many derivative-free trust-region methods based on underdetermined quadratic interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis provides a theoretical justification for this choice

Development Of New Algorithms For Exploring The Potential Energy Landscape Of Chemical Reactions

The research presented in this dissertation is divided into 5 chapters. In Chapter 2, a method for reducing the number of coordinates required to accurately reproduce a known chemical reaction pathway by applying principal component analysis to a number of geometries along the pathway (expressed in either Cartesian coordinates or redundant internal coordinates) is described and applied to 9 example reactions. Chapter 3 introduces new methods for estimating the structure of and optimizing transition states by utilizing information about the atomic bonding in the reactants and products. These methods are then benchmarked against a standard transition state optimization approach utilizing a test set of 20 reactions, with energies computed at both semiempirical and density functional theory levels of theory. Chapter 4 is a collection of 3 new, alternative approaches (Flowchart Hessian updating, Scaled Rational Function Optimization, Quasi-Rotation coordinate propagation), to handling aspects of a typical Quasi-Newton minimization. These new approaches are then compared to their standard counterparts by optimizing a set of 20 molecules using either ab initio or density functional theory potential energy surfaces. The final two chapters of this thesis focus on the development of a new path optimization framework, the Variational Reaction Coordinate (VRC) method. This framework seeks to improve upon the “chain of states” methods, which minimize the energy of a series of structures while using constraints, fictitious forces or reparameterization schemes to maintain the distribution of points along the path. In the VRC method, a functional representing the energy of the entire reaction is minimized by varying the expansion coefficients of a continuous function used to represent the reaction path. In Chapter 5, an algorithm is outlined along with the discussion and application of constraints and coupling terms that may be used to improve the efficiency and reliability of the method, with analytical test surfaces used to demonstrate the method’s performance. Chapter 6 focuses on the inclusion of redundant internal coordinates and methods for approximating the potential energy surface into the VRC framework, in order to reduce the per-iteration computational cost of the VRC method to something comparable to the “chain of states” approaches so that it may be practically applied to the study of reactions using high accuracy density functional theory and ab initio potential energy surfaces

PkANN - II. A non-linear matter power spectrum interpolator developed using artificial neural networks

In this paper we introduce PkANN, a freely available software package for interpolating the non-linear matter power spectrum, constructed using Artificial Neural Networks (ANNs). Previously, using Halofit to calculate matter power spectrum, we demonstrated that ANNs can make extremely quick and accurate predictions of the power spectrum. Now, using a suite of 6380 N-body simulations spanning 580 cosmologies, we train ANNs to predict the power spectrum over the cosmological parameter space spanning $3\sigma$ confidence level (CL) around the concordance cosmology. When presented with a set of cosmological parameters ($\Omega_{\rm m} h^2, \Omega_{\rm b} h^2, n_s, w, \sigma_8, \sum m_\nu$ and redshift $z$), the trained ANN interpolates the power spectrum for $z\leq2$ at sub-per cent accuracy for modes up to $k\leq0.9\,h\textrm{Mpc}^{-1}$. PkANN is faster than computationally expensive N-body simulations, yet provides a worst-case error $<1$ per cent fit to the non-linear matter power spectrum deduced through N-body simulations. The overall precision of PkANN is set by the accuracy of our N-body simulations, at 5 per cent level for cosmological models with $\sum m_\nu<0.5$ eV for all redshifts $z\leq2$. For models with $\sum m_\nu>0.5$ eV, predictions are expected to be at 5 (10) per cent level for redshifts $z>1$ ($z\leq1$). The PkANN interpolator may be freely downloaded from http://zuserver2.star.ucl.ac.uk/~fba/PkANNComment: 21 pages, 14 figures, 2 table