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An iteration method in the problem of approximating functions from a finite number of observations ob odnom iteratsionnom metode v zadache approksimatsii funktsii po konechnomu chislu nablyudeniy
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A method of feasible directions using function approximations, with applications to min max problems.
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Function approximations
A development status evaluation is presented for applications of main approximation concepts in nonlinear programming for structural optimization requiring local, midrange, and global approximations. It is noted that there is not yet a sufficient basis for a comprehensive evaluation of relative effectiveness among approximation concepts. Attention is presently given to the careful selection of intermediate-variable and response quantity, and improvements in the traditional Taylor-series expansion by inclusion of either some higher-order terms or information at various points in the design space.
Impact of Density Functional Approximations on the Thermodynamics of Hydrogen-Helium Mixtures.
Abstract not provided.
Low-rank functional approximation for Gaussian %0Cfiltering.
Abstract not provided.
How Accurate Are Approximate Density Functionals for Noncovalent Interaction of Very Large Molecular Systems?
Noncovalent intermolecular interactions are very important in many research areas. Therefore, it is vital to understand the extent to which approximate density functionals give a proper description of noncovalent interactions. Previous research has demonstrated that some approximate density functionals can predict usefully accurate interaction energies for many noncovalent systems; however, most of that work is limited to small and moderate-sized molecules. Very recently though, accurate benchmarks have become available for some very large molecules. Here, the present work applies 21 approximate density functionals to compute the binding energies of seven large molecular systems that have a number of atoms ranging from 200 to 910. The results are judged by comparison to the recently published CIM-DLPNO-CCSD(T) results, which are assumed to provide a reliable benchmark. The five most accurate methods among those tested are found to be PW6B95-D4, PW6B95-D3(BJ), revM11, M06-L, and MN15.
Mapping the genome of meta-generalized gradient approximation density functionals: The search for B97M-V
A meta-generalized gradient approximation density functional paired with the VV10 nonlocal correlation functional is presented. The functional form is selected from more than 10{sup 10} choices carved out of a functional space of almost 10{sup 40} possibilities. Raw data come from training a vast number of candidate functional forms on a comprehensive training set of 1095 data points and testing the resulting fits on a comprehensive primary test set of 1153 data points. Functional forms are ranked based on their ability to reproduce the data in both the training and primary test sets with minimum empiricism, and filtered based on a set of physical constraints and an often-overlooked condition of satisfactory numerical precision with medium-sized integration grids. The resulting optimal functional form has 4 linear exchange parameters, 4 linear same-spin correlation parameters, and 4 linear opposite-spin correlation parameters, for a total of 12 fitted parameters. The final density functional, B97M-V, is further assessed on a secondary test set of 212 data points, applied to several large systems including the coronene dimer and water clusters, tested for the accurate prediction of intramolecular and intermolecular geometries, verified to have a readily attainable basis set limit, and checked for grid sensitivity. Compared to existing density functionals, B97M-V is remarkably accurate for non-bonded interactions and very satisfactory for thermochemical quantities such as atomization energies, but inherits the demonstrable limitations of existing local density functionals for barrier heights.
On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals
In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.
Legendre-tau approximation for functional differential equations. III - Eigenvalue approximations and uniform stability
The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A characteristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.
Practical auxiliary basis implementation of Rung 3.5 functionals
Approximate exchange-correlation functionals for Kohn-Sham density functional theory often benefit from incorporating exact exchange. Exact exchange is constructed from the noninteracting reference system's nonlocal one-particle density matrix γ(r{sup -vector},r{sup -vector}′). Rung 3.5 functionals attempt to balance the strengths and limitations of exact exchange using a new ingredient, a projection of γ(r{sup -vector},r{sup -vector} ′) onto a semilocal model density matrix γ{sub SL}(ρ(r{sup -vector}),∇ρ(r{sup -vector}),r{sup -vector}−r{sup -vector} ′). γ{sub SL} depends on the electron density ρ(r{sup -vector}) at reference point r{sup -vector}, and is closely related to semilocal model exchange holes. We present a practical implementation of Rung 3.5 functionals, expanding the r{sup -vector}−r{sup -vector} ′ dependence of γ{sub SL} in an auxiliary basis set. Energies and energy derivatives are obtained from 3D numerical integration as in standard semilocal functionals. We also present numerical tests of a range of properties, including molecular thermochemistry and kinetics, geometries and vibrational frequencies, and bandgaps and excitation energies. Rung 3.5 functionals typically provide accuracy intermediate between semilocal and hybrid approximations. Nonlocal potential contributions from γ{sub SL} yield interesting successes and failures for band structures and excitation energies. The results enable and motivate continued exploration of Rung 3.5 functional forms.
Rounded Approximate Step Functions For Interpolation
Rounded approximate step functions of form x(Sup m)/(x(Sup n) + 1) and 1/(x(Sup n) + 1) useful in interpolating between local steep slopes or abrupt changes in tabulated data varying more smoothly elsewhere. Used instead of polynomial curve fits. Interpolation formulas based on these functions implemented quickly and easily on computers. Used in real-time control computations to interpolate between tabulated data governing control responses.
Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability
The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.
Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations
The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional – the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.
Legendre-Tau approximations for functional differential equations
The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.
Recent advances in approximation concepts for optimum structural design
The basic approximation concepts used in structural optimization are reviewed. Some of the most recent developments in that area since the introduction of the concept in the mid-seventies are discussed. The paper distinguishes between local, medium-range, and global approximations; it covers functions approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It also appears that some new methodologies are emerging which could greatly benefit from the introduction of new computer architecture.