The Eikonal Approximation for the Bilocal Vertex Function and NuW(2) in a Fermion-Neutral Vector Cluon Model
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Eichten, E.↗
SEARCH · Engineering Papers
Results for “Function approximation”
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Pseudo-spectral method to control three-degree-of-freedom wave energy converters
The invention provides optimal control of a three-degree-of-freedom wave energy converter using a pseudo-spectral control method. The three modes are the heave, pitch and surge. A dynamic model is characterized by a coupling between the pitch and surge modes, while the heave is decoupled. The heave, however, excites the pitch motion through nonlinear parametric excitation in the pitch mode. The invention can use a Fourier series as basis functions to approximate the states and the control. For the parametric excited case, a sequential quadratic programming approach can be implemented to numerically solve for the optimal control. The numerical results show that the harvested energy from three modes is greater than three times the harvested energy from the heave mode alone. Moreover, the harvested energy using a control that accounts for the parametric excitation is significantly higher than the energy harvested when neglecting this nonlinear parametric excitation term.
Abdelkhalik, Ossama↗
Localized Radial Basis Functions for Radiation Transport in Slab Geometry
Meshing the spatial domain for the neutron transport equation can be computationally expensive and labor-intensive. Mesh-free methods, such as radial basis function (RBF) approximations, may alleviate these concerns by using a pointwise solution of the transport equation instead of the standard volume-integrated solution using cells or elements. This allows for the use of any geometric description that can provide the material at solution points and the boundary normal directions for boundary points. RBFs give spectral convergence when used to solve differential equations, but only if they are infinitely smooth. This occurs when the RBFs are effectively flat, creating an ill-conditioned, dense system of equations. One simple way to alleviate this issue is to use localized RBFs. Global RBFs have been applied previously to the monoenergetic radiative transport equation in X-Y geometry using source iteration. In this summary, localized RBFs are applied to the multigroup neutron transport equation in slab geometry using Krylov iteration. The results show agreement with benchmark solutions, provided the coefficients for the RBF basis are properly chosen. (authors)
97 MATHEMATICS AND COMPUTING↗
Artificial intelligence “sees” split electrons
Chemical bonds between atoms are stabilized by the exchange-correlation (xc) energy, a quantum-mechanical effect in which “social distancing” by electrons lowers their electrostatic repulsion energy. Kohn-Sham density functional theory (DFT) states that the electron density determines this xc energy, but the density functional must be approximated. Furthermore, this is usually done by satisfying exact constraints of the exact functional (making the approximation predictive), by fitting to data (making it interpolative), or both. Two exact constraints—the ensemble-based piecewise linear variation of the total energy with respect to fractional electron number and fractional electron z-component of spin —require hard-to-control nonlocality. On page 1385 of this issue, Kirkpatrick et al. have taken a big step toward more accurate predictions for chemistry through the machine learning of molecular data plus the fractional charge and spin constraints, expressed as data that a machine can learn.
74 ATOMIC AND MOLECULAR PHYSICS↗
Specific Reaction Parameter Density Functional Based on the Meta-Generalized Gradient Approximation: Application to H 2 + Cu(111) and H 2 + Ag(111)
Specific reaction parameter density functionals (SRP-DFs), which can describe dissociative chemisorption reactions on metals to within chemical accuracy, have so far been based on exchange functionals within the generalized gradient approximation (GGA) and on GGA correlation functionals or van der Waals correlation functionals. These functionals are capable of describing the molecule–metal surface interaction accurately, but they suffer from the general GGA problem that this can be done only at the cost of a rather poor description of the metal. Here, we show that it is possible also to construct SRP-DFs for H 2 dissociation on Cu(111) based on meta-GGA functionals, introducing three new functionals based on the “made-simple” (MS) concept. The exchange parts of the three functionals (MS-PBEl, MS-B86bl, and MS-RPBEl) are based on the expressions for the PBE, B86b, and RPBE exchange functionals. Quasi-classical trajectory (QCT) calculations performed with potential energy surfaces (PESs) obtained with the three MS functionals reproduce molecular beam experiments on H 2 , D 2 + Cu(111) with chemical accuracy. Therefore, these three non-empirical functionals themselves are also capable of describing H2 dissociation on Cu(111) with chemical accuracy. Similarly, QCT calculations performed on the MS-PBEl and MS-B86bl PESs reproduced molecular beam and associative desorption experiments on D 2 , H 2 + Ag(111) more accurately than was possible with the SRP48 density functional for H 2 + Cu(111). Also, the three new MS functionals describe the Cu, Ag, Au, and Pt metals more accurately than the all-purpose Perdew–Burke–Ernzerhof (PBE) functional. The only disadvantage we noted of the new MS functionals is that, as found for the example of H 2 + Cu(111), the reaction barrier height obtained by taking weighted averages of the MS-PBEl and MS-RPBEl functionals is tunable over a smaller range (9 kJ/mol) than possible with the standard GGA PBE and RPBE functionals (33 kJ/mol). As a result of this restricted tunability, it is not possible to construct an SRP-DF for H 2 + Ag(111) on the basis of the three examined MS meta-GGA functionals.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A simple generalized gradient approximation for the noninteracting kinetic energy density functional
A simple, novel, non-empirical, constraint-based orbital-free generalized gradient approximation (GGA) non-interacting kinetic energy density functional is presented along with illustrative applications. The innovation is adaptation of constraint-based construction to the essential properties of pseudo-densities from the pseudo-potentials that are essential in plane-wave-basis ab initio molecular dynamics. This contrasts with constraining to the qualitatively different Kato-cusp-condition densities. The single parameter in the new functional is calibrated by satisfying Pauli potential positivity constraints for pseudo-atom densities. Finally, in static lattice tests on simple metals and semiconductors, the new LKT functional outperforms the previous best constraint-based GGA functional, VT84F (Phys. Rev. B 88, 161108(R) (2013)), is generally superior to a recently proposed meta-GGA, is reasonably competitive with parametrized two-point functionals, and is substantially faster.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Multireference diffusion Monte Carlo reaches 2D materials
Abstract Quantum confinement in 2D materials strongly enhances electronic correlation effects. Therefore, predicting the properties of these unique materials, with both a high level of accuracy and computational efficiency, without relying on adjustable parameters or functionals, remains an outstanding theoretical challenge. The majority of theoretical studies are based on the approximations of density functional theory (DFT). The reliability of DFT predictions are heavily dependent on the choice of an approximated exchange-correlation functional. Here, we estimate the magnitude of impact of correlation on the total energy for the quintessential 2D material, graphene, by performing and comparing state-of-the-art selected CI and quantum Monte Carlo extrapolated calculations for a single unit cell at the$$\Gamma$$point. We demonstrate that Self-Healing Diffusion Monte Carlo (SHDMC) obtains a very compact, but high-quality wavefunction for this system that lacks the strong basis set dependence displayed by state of the art quantum chemistry methods. The SHDMC wavefunction is of higher quality compared to that obtained from sCI, in the same orbital basis, while being$$\sim$$ 1000 times smaller in terms of determinant count compared to sCI. We also demonstrate that extrapolating SHDMC results to the infinite determinant limit compares extremely well with complete basis set extrapolated sCI. Our work paves the way for future validation of SHDMC applied to challenging 2D materials.
Science & Technology - Other Topics↗
Density functional theory for d- and f-electron materials and compounds
Here, the fundamental requirements for a computationally tractable Density Functional Theory-based method for relativistic f- and (nonrelativistic) d-electron materials and compounds are presented. The need for basing the Kohn–Sham equations on the Dirac equation is discussed. The full Dirac scheme needs exchange-correlation functionals in terms of four-currents, but ordinary functionals, using charge density and spin-magnetization, can be used in an approximate Dirac treatment. The construction of a functional that includes the additional confinement physics needed for these materials is illustrated using the subsystem-functional scheme. If future studies show that a full Dirac, four-current based, exchange-correlation functional is needed, the subsystem functional scheme is one of the few schemes that can still be used for constructing functional approximations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
WE-DE-201-01: BEST IN PHYSICS (THERAPY): A Fast Multi-Target Inverse Treatment Planning Strategy Optimizing Dosimetric Measures for High-Dose-Rate (HDR) Brachytherapy
Purpose: Inverse treatment planning (ITP) for interstitial HDR brachytherapy of gynecologic cancers seeks to maximize coverage of the clinical target volumes (tumor and vagina) while respecting dose-volume-histogram related dosimetric measures (DMs) for organs at risk (OARs). Commercially available ITP tools do not support DM-based planning because it is computationally too expensive to solve. In this study we present a novel approach that allows fast ITP for gynecologic cancers based on DMs for the first time. Methods: This novel strategy is an optimization model based on a smooth DM-based objective function. The smooth approximation is achieved by utilizing a logistic function for the evaluation of DMs. The resulting nonconvex and constrained optimization problem is then optimized with a BFGS algorithm. The model was evaluated using the implant geometry extracted from 20 patient treatment plans under an IRB-approved retrospective study. For each plan, the final DMs were evaluated and compared to the original clinical plans. The CTVs were the contoured tumor volume and the contoured surface of the vagina. Statistical significance was evaluated with a one-sided paired Wilcoxon signed-rank test. Results: As did the clinical plans, all generated plans fulfilled the defined DMs for OARs. The proposed strategy showed a statistically significant improvement (p<0.001) in coverage of the tumor and vagina, with absolute improvements of related DMs of (6.9 +/− 7.9)% and (28.2 +/− 12.0)%, respectively. This was achieved with a statistically significant (p<0.01) decrease of the high-dose-related DM for the tumor. The runtime of the optimization was (2.3 +/− 2.0) seconds. Conclusion: We demonstrated using clinical data that our novel approach allows rapid DM-based optimization with improved coverage of CTVs with fewer hot spots. Being up to three orders of magnitude faster than the current clinical practice, the method dramatically shortens planning time.
60 APPLIED LIFE SCIENCES↗
Quantifying the hydration structure of sodium and potassium ions: taking additional steps on Jacob's Ladder
The ability to reproduce the experimental structure of water around the sodium and potassium ions is a key test of the quality of interaction potentials due to the central importance of these ions in a wide range of important phenomena. In this study, we simulate the Na + and K + ions in bulk water using three density functional theory functionals: (1) the generalized gradient approximation (GGA) based dispersion corrected revised Perdew, Burke, and Ernzerhof functional (revPBE-D3) (2) the recently developed strongly constrained and appropriately normed (SCAN) functional (3) the random phase approximation (RPA) functional for potassium. We compare with experimental X-ray diffraction (XRD) and X-ray absorption fine structure (EXAFS) measurements to demonstrate that SCAN accurately reproduces key structural details of the hydration structure around the sodium and potassium cations, whereas revPBE-D3 fails to do so. However, we show that SCAN provides a worse description of pure water in comparison with revPBE-D3. RPA also shows an improvement for K + , but slow convergence prevents rigorous comparison. Finally, we analyse cluster energetics to show SCAN and RPA have smaller fluctuations of the mean error of ion–water cluster binding energies compared with revPBE-D3.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Approximate two-body generating Hamiltonian for the particle-hole Pfaffian wave function
We present two two-body Hamiltonians that approximate the exact particle-hole Pfaffian wave function with their ground states for all the system sizes where this wave function has been numerically constructed to date. The approximate wave functions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimization procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a noninteracting model. They belong to a finite region in the variational space of Hamiltonians where each point approximately generates the particle-hole Pfaffian. Here we diagonalize the identified Hamiltonians for up to 20 electrons and find that for them the particle-hole Pfaffian shift appears energetically more favorable. The possibility to interpret the data in terms of composite fermions is discussed.
36 MATERIALS SCIENCE↗
Charge localization and ordering in A 2 Mn 8 O 16 hollandite group oxides: Impact of density functional theory approaches
The phases of A 2 Mn 8 O 16 hollandite group oxides emerge from the competition between ionic interactions, Jahn-Teller effects, charge ordering, and magnetic interactions. Their balanced treatment with feasible computational approaches can be challenging for commonly used approximations in density functional theory. Three examples (A = Ag, Li, and K) are studied with a sequence of different approximate exchange-correlation functionals. Starting from a generalized gradient approximation (GGA), an extension to include van der Waals interactions and a recently proposed meta-GGA are considered. Then local Coulomb interactions for the Mn 3d electrons are more explicitly considered with the DFT + U approach. Finally, selected results from a hybrid functional approach provide a reference. Results for the binding energy of the A species in the parent oxide highlight the role of van der Waals interactions. Relatively accurate results for insertion energies can be achieved with a low-U and a high-U approach. In the low-U case, the materials are described as band metals with a high-symmetry, tetragonal crystal structure. In the high-U case, the electrons donated by A result in formation of local Mn 3+ centers and corresponding Jahn-Teller distortions characterized by a local order parameter. The resulting degree of monoclinic distortion depends on charge ordering and magnetic interactions in the phase formed. The reference hybrid functional results show charge localization and ordering. Comparison to low-temperature experiments of related compounds suggests that charge localization is the physically correct result for the hollandite group oxides studied here. Lastly, while competing effects in the local magnetic coupling are subtle, the fully anisotropic implementation of DFT + U gives the best overall agreement with results from the hybrid functional.
meta-GGA↗
Optical response and band structure of LiCoO 2 including electron-hole interaction effects
The optical response functions and band structures of LiCoO 2 are studied at different levels of approximation, from density functional theory (DFT) in the generalized gradient approximation (GGA) to quasiparticle self-consistent QSGW (with G for Green's function and W for screened Coulomb interaction) without and with ladder diagrams ($QSG\hat{W}$) and the Bethe Salpeter Equation (BSE) approach. The QSGW method is found to strongly overestimate the band gap and electron-hole or excitonic effects are found to be important. They lower the quasiparticle gap by only about 11~\% but the lowest energy peaks in absorption are found to be excitonic in nature. The contributions from different band to band transitions and the relation of excitons to band-to-band transitions are analyzed. Finally, the excitons are found to be strongly localized. Finally, a comparison to experimental data is presented.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not connected to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of “continuous” matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities.
42 ENGINEERING↗
Erratum: “Harnessing the meta-generalized gradient approximation for time-dependent density functional theory” [J. Chem. Phys. 137, 164105 (2012)]
This erratum corrects a factor 2 error in previous work on current-dependent TDDFT.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Basis adaptive sample efficient polynomial chaos (BASE-PC)
For a large class of orthogonal basis functions, there has been a recent identification of expansion methods for computing accurate, stable approximations of a quantity of interest. Here this paper presents, within the context of uncertainty quantification, a practical implementation using basis adaptation, and coherence motivated sampling, which under assumptions has satisfying guarantees. This implementation is referred to as Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC). A key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner, leading to consistently stable approximation for a practical class of smooth functionals. This fully adaptive, non-intrusive method, requires no a priori information of the solution, and has satisfying theoretical guarantees of recovery. A key contribution to stability is the use of a presented correction sampling for coherence-optimal sampling in order to improve stability and accuracy within the adaptive basis scheme. Theoretically, the method may dramatically reduce the impact of dimensionality in function approximation, and numerically the method is demonstrated to perform well on problems with dimension up to 1000.
97 MATHEMATICS AND COMPUTING↗
Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity
Here, the damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $\kappa$ is analyzed, when the external force is periodic in space and given by $\mathcal{f(x) = r}$ cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K , and the soliton width $l_s$, which is a function of the nonlinearity $\kappa$ as well as the initial frequency ω 0 of the solitary wave, there is a transition (at fixed ω 0 ) from trapped to unbound behavior of the soliton, which depends on the parameters $\mathcal{r}$ and K of the external force and the nonlinearity parameter $\kappa$. We previously studied this phenomena when $\kappa$ = 1 where we showed that for λ $\gg l_s$ the soliton oscillates in an effective potential, while forλ $\ll l_s$ it moves uniformly as a free particle. In this paper we focus on the $\kappa$ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of $\mathcal{r}$ and K when $\kappa$ = 1/2,1,2 at fixed value of the frequency ω 0 . Since the solitary wave gets narrower for fixed ω 0 as a function of $\kappa$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $\kappa$.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
On Approximation of Coefficient Inverse Problems for Differential Equations in Functional Spaces
This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.
97 MATHEMATICS AND COMPUTING↗