Michael Minkoff

Interpolating Moving Least-squares Methods for Fitting Potential Energy Surfaces: A Strategy for Efficient Automatic Data Point Placement in High Dimensions

An accurate and efficient method for automated molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating moving least-squares (IMLS) method is developed with the flexibility to fit various ab initio data: (1) energies, (2) energies and gradients, or (3) energies, gradients, and Hessian data. The method is automated and flexible so that a PES can be optimally generated for trajectories, spectroscopy, or other applications. High efficiency is achieved by employing local IMLS in which fitting coefficients are stored at a limited number of expansion points, thus eliminating the need to perform weighted least-squares fits each time the potential is evaluated. An automatic point selection scheme based on the difference in two successive orders of IMLS fits is used to determine where new ab initio data need to be calculated for the most efficient fitting of the PES. A simple scan of the coordinate is shown to work well to identify these maxima in one dimension, but this search strategy scales poorly with dimension. We demonstrate the efficacy of using conjugate gradient minimizations on the difference surface to locate optimal data point placement in high dimensions. Results that are indicative of the accuracy, efficiency, and scalability are presented for a one-dimensional model potential (Morse) as well as for three-dimensional (HCN), six-dimensional (HOOH), and nine-dimensional (CH4) molecular PESs.

An adaptive pseudo-spectral method for reaction diffusion problems

The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.

Interpolating moving least-squares methods for fitting potential energy surfaces: Detailed analysis of one-dimensional applications

We present the basic formal and numerical aspects of higher degree interpolated moving least-squares (IMLS) methods. For simplicity, applications of these methods are restricted to two one-dimensional (1D) test cases: a Morse oscillator and a 1D slice of the HN2→H+N2 potential energy surface. For these two test cases, we systematically examine the effect of parameters in the weight function (intrinsic to IMLS methods), the degree of the IMLS fit, and the number and placement of potential energy points. From this systematic study, we discover compact and accurate representations of potentials and their derivatives for first-degree and higher-degree (up to nine degree) IMLS fits. We show how the number of ab initio points needed to achieve a given accuracy declines with the degree of the IMLS. We outline automatic procedures for ab initio point selection that can optimize this decline.

Interpolating moving least-squares methods for fitting potential energy surfaces: computing high-density potential energy surface data from low-density ab initio data points.

A highly accurate and efficient method for molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating-moving-least-squares (IMLS)-based method is developed using low-density ab initio Hessian values to compute high-density PES parameters suitable for accurate and efficient PES representation. The method is automated and flexible so that a PES can be optimally generated for classical trajectories, spectroscopy, or other applications. Two important bottlenecks for fitting PESs are addressed. First, high accuracy is obtained using a minimal density of ab initio points, thus overcoming the bottleneck of ab initio point generation faced in applications of modified-Shepard-based methods. Second, high efficiency is also possible (suitable when a huge number of potential energy and gradient evaluations are required during a trajectory calculation). This overcomes the bottleneck in high-order IMLS-based methods, i.e., the high cost/accuracy ratio for potential energy evaluations. The result is a set of hybrid IMLS methods in which high-order IMLS is used with low-density ab initio Hessian data to compute a dense grid of points at which the energy, Hessian, or even high-order IMLS fitting parameters are stored. A series of hybrid methods is then possible as these data can be used for neural network fitting, modified-Shepard interpolation, or approximate IMLS. Results that are indicative of the accuracy, efficiency, and scalability are presented for one-dimensional model potentials as well as for three-dimensional (HCN) and six-dimensional (HOOH) molecular PESs.

A collaborative informatics infrastructure for multi-scale science

The Collaboratory for Multi-scale Chemical Science (CMCS) is developing a powerful informatics-based approach to synthesizing multi-scale information in support of systems-based research and is applying it within combustion science. An open source multi-scale informatics toolkit is being developed that addresses a number of issues core to the emerging concept of knowledge grids including provenance tracking and lightweight federation of data and application resources into cross-scale information flows. The CMCS portal is currently in use by a number of high-profile pilot groups and is playing a significant role in enabling their efforts to improve and extend community maintained chemical reference information.

Interpolating moving least-squares methods for fitting potential energy surfaces: applications to classical dynamics calculations.

As a continuation of our efforts to develop efficient and accurate interpolating moving least-squares (IMLS) methods for generating potential energy surfaces, we carry out classical trajectories and compute kinetics properties on higher degree IMLS surfaces. In this study, we have investigated the choice of coordinate system, the range of points (i.e., the cutoff radius) used in fitting, and strategies for selections of data points and basis elements. We illustrate and test the method by applying it to hydrogen peroxide (HOOH). In particular, reaction rates for the O-O bond breaking in HOOH are calculated on fitted surfaces using the classical trajectory approach to test the accuracy of the IMLS method for providing potentials for dynamics calculations.

Interpolating moving least-squares methods for fitting potential energy surfaces: Improving efficiency via local approximants

The local interpolating moving least-squares (IMLS) method for constructing potential energy surfaces is investigated. The method retains the advantageous features of the IMLS approach in that the ab initio derivatives are not required and high degree polynomials can be used to provide accurate fits, while at the same time it is much more efficient than the standard IMLS approach because the least-squares solutions need to be calculated only once at the data points. Issues related to the implementation of the local IMLS method are investigated and the accuracy is assessed using HOOH as a test case. It is shown that the local IMLS method is at the same level of accuracy as the standard IMLS method. In addition, the scaling of the method is found to be a power law as a function of number of data points N, N(-q). The results suggest that when fitting only to the energy values for a d-dimensional system by using a Qth degree polynomial the power law exponent q approximately Qd when the energy range fitted is large (e.g., E<100 kcalmol for HOOH), and q>Qd when the energy range fitted is smaller (E<30 kcalmol) and the density of data points is higher. This study demonstrates that the local IMLS method provides an efficient and accurate means for constructing potential energy surfaces.

Interpolating moving least-squares methods for fitting potential energy surfaces : an application to the H2CN unimolecular reaction.

Classical trajectories have been used to compute rates for the unimolecular reaction H2CN-->H+HCN on a fitted ab initio potential energy surface (PES). The ab initio energies were obtained from CCSD(T)/aug-cc-pvtz electronic structure calculations. The ab initio energies were fitted by the interpolating moving least-squares (IMLS) method. This work continues the development of the IMLS method for producing ab initio PESs for use in molecular dynamics simulations of many-atom systems. A dual-level scheme was used in which the preliminary selection of data points was done using a low-level theory and the points used for fitting the final PES were obtained at the desired higher level of theory. Classical trajectories were used on various low-level IMLS fits to tune the fit to the unimolecular reaction under study. Procedures for efficiently picking data points, selecting basis functions, and defining cutoff limits to exclude distant points were investigated. The accuracy of the fitted PES was assessed by comparing interpolated values of quantities to the corresponding ab initio values. With as little as 330 ab initio points classical trajectory rate constants were converged to 5%-10% and the rms error over the six-dimensional region sampled by the trajectories was a few tenths of a kcal/mol.

Randomly Generated Test Problems for Positive Definite Quadratic Programming

A procedure is described for randomly generating positive definite quadratic programming test problems. The test problems are constructed in the form of linear least squares problems subject to linear constraints. The probability measure for the problems so generated is invariant under orthogonal transformations. The procedure allows the user to specify the size of the least squares problem (number of unknown parameters, number of observations, and number of constraints); the relative magnitude of the residuals; the condition number of the Hessian matrix of the objective function; and the structure of the feasible region (number of equality constraints and the number of inequalities which will be active at the feasible starting point and at the optimal solution). An example is given illustrating how these problems can be used to evaluate the performance of a software package.

Interpolating moving least-squares methods for fitting potential-energy surfaces: further improvement of efficiency via cutoff strategies.

In standard applications of interpolating moving least squares (IMLS) for fitting a potential-energy surface (PES), all available ab initio points are used. Because remote ab initio points negligibly influence IMLS accuracy and increase IMLS time-to-solution, we present two methods to locally restrict the number of points included in a particular fit. The fixed radius cutoff (FRC) method includes ab initio points within a hypersphere of fixed radius. The density adaptive cutoff (DAC) method includes points within a hypersphere of variable radius depending on the point density. We test these methods by fitting a six-dimensional analytical PES for hydrogen peroxide. Both methods reduce the IMLS time-to-solution by about an order of magnitude relative to that when no cutoff method is used. The DAC method is more robust and efficient than the FRC method.