Truncation Of Basis Research Articles

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to l = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for l ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbo-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = −14.6673473 corrected with these two truncation energy errors yields E = −14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = −14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.

A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows

Herein a new approach to enhance the accuracy of a novel Proper Orthogonal Decomposition (POD) model applied to moderate Reynolds number flows (of the type typically encountered in ocean models) is presented. This approach develops the POD model of Fang et al. [Fang, F., Pain, C.C., Navon, I.M., Piggott, M.D., Gorman, G.J., Allison, P., Goddard, A.J.H., 2008. Reduced-order modelling of an adaptive mesh ocean model. International Journal for Numerical Methods in Fluids. doi:10.1002/fld.1841] used in conjunction with the Imperial College Ocean Model (ICOM), an adaptive, non-hydrostatic finite element model. Both the velocity and vorticity results of the POD reduced order model (ROM) exhibit an overall good agreement with those obtained from the full model. The accuracy of the POD-Galerkin model with the use of adaptive meshes is first evaluated using the Munk gyre flow test case with Reynolds numbers ranging between 400 and 2000. POD models using the L 2 norm become oscillatory when the Reynolds number exceeds Re = 400 . This is because the low-order truncation of the POD basis inhibits generally all the transfers between the large and the small (unresolved) scales of the fluid flow. Accuracy is improved by using the H 1 POD projector in preference to the L 2 POD projector. The POD bases are constructed by incorporating gradients as well as function values in the H 1 Sobolev norm. The accuracy of numerical results is further enhanced by increasing the number of snapshots and POD bases. Error estimation was used to assess the effect of truncation (involved in the POD-Galerkin approach) when adaptive meshes are used in conjunction with POD/ROM. The RMSE of velocity results between the full model and POD-Galerkin model is reduced by as much as 50% by using the H 1 norm and increasing the number of snapshots and POD bases.

Roth and Navrátil Reply:

In their comment on our recent Letter [1] Dean et al. [2] criticize the calculations for the ground-state energy of {sup 40}Ca within the importance truncated no-core shell model (NCSM). In particular they address the role of configurations beyond the 3p3h level, which have not been included in the {sup 40}Ca calculations for large N{sub max} {h_bar}{Omega} model spaces. Before responding to this point, the following general statements are in order. For the atomic nucleus as a self-bound system, translational invariance is an important symmetry. The only possibility to preserve translational invariance when working with a Slater determinant basis is to use the harmonic oscillator (HO) basis in conjunction with a basis truncation according to the total HO excitation energy, i.e. N{sub max} {h_bar}{Omega}, as done in the ab initio NCSM. This is important not only for obtaining proper binding or excitation energies, but also for a correct extraction of physical wavefunctions. The spurious center-of-mass components can be exactly removed only if the HO basis and the N{sub max} {h_bar}{Omega} truncation are employed. The minimal violation of the translational invariance was one of the main motivations for developing the importance-truncation scheme introduced in the Letter. In this scheme, we startmore » with the complete N{sub max} {h_bar}{Omega} HO basis space and select important configurations via perturbation theory. All symmetries are under control and our importance-truncated NCSM calculations are completely variational and provide an upper bound of the ground-state energy of the system. The restriction to the 3p3h level, made for computational reasons in the N{sub max} > 8 calculations for {sup 40}Ca, is not inherent to the importance truncation scheme. The explicit inclusion of 4p4h configurations--though computationally more demanding--is straight-forward, even for the largest N{sub max} {h_bar}{Omega} model spaces discussed. To demonstrate this fact we have performed full 4p4h calculations for {sup 40}Ca in a 14{h_bar}{Omega} no-core model space at {h_bar}{Omega} = 24 MeV using the V{sub low k} interaction employed in the Letter. The resulting ground state energy of E{sub 4p4h} = -471.0 MeV can be compared with E{sub 3p3h} = -461.2 MeV for the 3p3h calculation reported in Fig. 5(b) with an uncertainty of typically 1 MeV due to the extrapolation {kappa}{sub min} {yields} 0. Thus the 4p4h configurations change the resulting ground-state energy of {sup 40}Ca by approximately 2%. In addition to the explicit inclusion of 4p4h or higher-order configurations in the importance truncated space, there are various other means to account for their effect on the energy. We are presently developing a perturbative method for the inclusion of up to 6p6h configurations using the ground state obtained from the NCSM diagonalization as unperturbed state. Furthermore, there are even simpler methods, such as multireference Davidson corrections, which are employed successfully in quantum chemistry [3], for estimating the contribution of excluded configurations to the energy. They even restore size extensivity in an approximate way. The coupled-cluster method (CCM) used by the authors of the Comment lacks the above discussed features important for the nuclear many-body problem, in particular it violates translational invariance from the very beginning and it does not fulfill the variational principle. The problem of a spurious center-of-mass contamination of the many-body states in CCM is not resolved and often not even mentioned (see e.g. Ref. [4]). Moreover, non-iterative triples corrections like CCSD(T), as referred to in the comment, tend to overestimate the correlation energy or even collapse. It is claimed that the CCM is very accurate. A closer inspection of recently published nuclear many-body results does not support this statement. In Ref. [5], the {sup 4}He binding energy with the chiral N{sup 3}LO NN potential was determined with an uncertainty of several MeV. The same is obtained in the ab initio NCSM with accuracy of 10 keV [6, 7]. The CCM binding energy results for {sup 16}O with the identical chiral N{sup 3}LO NN potential obtained in Refs. [8] and [9] differ by more than 5 MeV. According to Ref. [9], the CD-Bonn NN potential overbinds {sup 16}O. This is contrary to results obtained in Ref. [10] and also contrary to expectations as the CD-Bonn underbinds lighter nuclei. The CCSD(T) ground state energy for {sup 40}Ca with V{sub low k} reported in Ref. [4] is about 30 MeV lower than our 4p4h-result (CCSD is about 20MeV lower). Taking into account the small 2% difference of our 3p3h and 4p4h results and the violation of the variational principle and the translational invariance in the CCM method, we believe that the CCM result overestimates the exact ground-state energy in this case.« less

Calculations of modes in circumferentially nonuniform lined ducts

This paper proposes a computationally efficient method of determining eigenfunctions and eigenvalues of acoustic modes propagating in circular lined ducts with zero or uniform flow. Linings with circumferentially nonuniform impedance, as found in nacelle acoustics, are the focus. The method of solution adapts in two important respects--the presence of flow and the imposition of impedance boundary conditions--the series expansion method first proposed by Pagneux et al. [J. Acoust. Soc. Am. 110, 1307-1314 (2001)] to calculate the eigenvalues of Lamb modes. The inclusion of flow, and a corresponding different method of solution leading to an improved convergence of eigenvalues (O(N(-5)), N is the truncation of radial basis of expansion), is the important new feature as compared to the previous adaptation by Bi et al. [J. Acoust. Soc. Am. 122, 280-291 (2007)]. As a result, it becomes possible to safely determine and in a simple manner the eigenvalues and eigenfunctions of circumferentially nonuniform lined ducts in the presence of uniform flow, up to relatively high frequencies (e.g., 30<kR<80). All required eigenvalues are calculated at a time and no initial values guesses are needed. Results are presented and the method is discussed.

UWB Mixed-Signal Transform-Domain Direct-Sequence Receiver

We consider a mixed-signal ultra wideband (UWB) direct-sequence spread-spectrum (DS-SS) receiver. The receiver expands the signal over a basis set, and then operates on the basis coefficients. An analog computation of the basis coefficients efficiently parallelizes the signal for digital processing, relaxing the sampling requirements and enabling parallel digital processing at a much lower rate. Bit error rate (BER) performance is evaluated, as well as distortion due to basis truncation. The parallel processing also facilitates efficient interference mitigation. Various models for interference including adjacent multicarrier interference are developed and analyzed. Several simulation results confirm the analysis and provide insight into quantization error, basis truncation distortion, and the impact of the UWB channel. We show that in the AWGN uncoded case for SNR < 10 dB, the receiver can be operated at almost half the Nyquist rate with only 1 dB of performance loss. Moreover, when a common convolutional code is employed, the aggregate sampling rate can be 37.5% lower than Nyquist rate with little loss in receiver performance.

Bose–Fermi mixtures in 1D optical superlattices using full and truncated single-band bases

The zero temperature phase diagram of binary boson–fermion mixtures in two-colour superlattices is investigated. The eigenvalue problem associated with the Bose–Fermi–Hubbard Hamiltonian is solved using an exact numerical diagonalization technique, supplemented by an adaptive basis truncation scheme. The physically motivated basis truncation allows us to access larger systems in a fully controlled and very flexible framework. Several experimentally relevant observables, such as the matter–wave interference pattern and the condensate fraction, are investigated in order to explore the rich phase diagram. At symmetric half filling a phase similar to the Mott-insulating phase in a commensurate purely bosonic system is identified and an analogy to recent experiments is pointed out. Furthermore, a phase of complete localization of the bosonic species generated by the repulsive boson–fermion interaction is identified. These localized condensates are of a different nature than the genuine Bose–Einstein condensates in optical lattices.

Effective Hamiltonian Approach for Strongly Correlated Systems

We demonstrate the utility of effective Hamilonians for studying extended correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical contractor renormalization (CORE) method is applied in two steps (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut-off using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semi-quantitative numerical method. For ladder type geometries, we explicitly check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagome lattice as non-trivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagome lattice we find that the previously proposed twofold degenerate S=1/2 basis can account for a large number of phenomena of the spin 1/2 kagome system. In general, we are able to simulate system sizes which correspond to an 8× 8 lattice for the plaquette lattice or a 48-site kagome lattice, which are beyond the possibilities of a standard exact diagonalization approach.

Numerical contractor renormalization method for quantum spin models

We demonstrate the utility of the numerical contractor renormalization (CORE) method for quantum spin systems by studying one- and two-dimensional model cases. Our approach consists of two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cutoff using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semiquantitative numerical method. For ladder-type geometries, we explicitely check the accuracy of the effective models by increasing the range of the effective interactions until reaching convergence. Our results in the perturbative regime and also away from it are in good agreement with previously established results. In two dimensions we consider the plaquette lattice and the kagom\'e lattice as nontrivial test cases for the numerical CORE method. As it becomes more difficult to extend the range of the effective interactions in two dimensions, we propose diagnostic tools (such as the density matrix of the local building block) to ascertain the validity of the basis truncation. On the plaquette lattice we have an excellent description of the system in both the disordered and the ordered phases, thereby showing that the CORE method is able to resolve quantum phase transitions. On the kagom\'e lattice we find that the previously proposed twofold degenerate $S=1∕2$ basis can account for a large number of phenomena of the spin $1∕2$ kagom\'e system. For spin $3∕2$, however, this basis does not seem to be sufficient. In general we are able to simulate system sizes which correspond to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice for the plaquette lattice or a 48-site kagom\'e lattice, which are beyond the possibilities of a standard exact diagonalization approach.

Thin Plate Regression Splines

SummaryI discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother. By making use of Lanczos iteration the basis change and truncation are computationally efficient. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems that are associated with ‘knot placement’ that usually complicate modelling with regression splines or penalized regression splines, provide a sensible way of modelling interaction terms in generalized additive models, provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, provide a means for incorporating smooth functions of more than one variable into non-linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. Given that the approach produces spline-like models with a sparse basis, it also provides a natural way of incorporating unpenalized spline-like terms in linear and generalized linear models, and these can be treated just like any other model terms from the point of view of model selection, inference and diagnostics.

Optical absorption spectra of the Holstein molecular crystal for weak and intermediate electronic coupling

We investigate the optical absorption spectrum in a Holstein model for a molecular chain with Frenkel excitons and linear coupling to one internal vibration. The model is extended for nearest-neighbor charge-transfer excitons that mix with the Frenkel excitons. We represent the Hamiltonian in a displaced oscillator (Lang-Firsov) basis and employ a problem-adapted scheme for the truncation of the phonon basis. For weak and intermediate electronic coupling, the complete absorption spectrum and the structure of the relevant eigenstates become accessible by direct numerical diagonalization. We discuss the structure of the phonon clouds and the applicability of the molecular vibron model, in which only joint exciton-phonon configurations are included. As examples, we model absorption spectra of PTCDA (3,4,9,10-perylenetetracarboxylic dianhydride) and MePTCDI $(N\ensuremath{-}{N}^{\ensuremath{'}}$-dimethylperylene-3,4,9,10-dicarboximide).

Important correlation effects for the Er3+ 4f114S3/2→4I15/2 laser transition energy

AbstractMost of the important correlation effects transitions within the 4fn manifold of rare‐earth atomic ions are identified as coming from three sources: 4f2 pair correlations, single excitations from the valence shell and shallow core, and pair excitations where both of the electrons go into the incomplete 4f subshell (exclusion effects). A method (REDUCE) of managing the large basis sets encountered is discussed. Relativistic configuration interaction calculations are done to obtain the energy of the laser transition (in, e.g., GaN) Er3+ 4I15/2→4S3/2, to which correlation contributes ∼4000 cm−1. The residual error of 864 cm−1 may be due to missing second‐order effects and to truncation of the angular basis. © 2002 John Wiley &amp; Sons, Inc. Int J Quantum Chem, 2002

Methods of Synthesis of Efficient Orthogonal Transforms of High and Low Correlation and Their Fast Algorithms for Coding and Compressing Digital Images

Based on recurrent matrix methods of synthesis of orthogonal transforms, efficient fast algorithms are developed that need only output data rearrangement. These transforms are tested on non-separable and correlation models. A comparative analysis of these transforms and the well-known slant and Walsh transforms is given in respect to the following characteristics: transformation efficiency, maximum reducible bits, mean-square error of basis truncation, transform coding efficiency and computational complexity.

Quasi- SU(3) truncation scheme for odd–even and odd–odd sd-shell nuclei

The quasi- SU(3) symmetry, as found in shell-model calculations, refers to the dominance of the single-particle plus quadrupole–quadrupole terms in the Hamiltonian used to describe well-deformed nuclei, and to the subspace relevant in its diagonalization. It provides a very efficient basis truncation scheme. It is shown that a small number of SU(3) coupled irreps, those with the largest C 2 values within the direct product of the proton and neutron SU(3) irreps with spin 0 and 1 (for even number of particles), and spin 1/2 and 3/2 for (for odd number of nucleons), are enough to describe the low-energy spectra, B(E2) transition strengths and band structure of 21Ne, 23Na and 25Mg odd-mass and 22Na, 24Na, 26Al and 28Al odd–odd nuclei. A simple but realistic Hamiltonian is employed. Results compare favorably both with experimental data and with full shell-model calculations. Limitations and possible improvements of the schematic Hamiltonian are discussed.

Quasi- [formula omitted] truncation scheme for even–even sd-shell nuclei

The quasi- SU(3) symmetry was uncovered in full pf and sdg shell-model calculations for both even–even and odd–even nuclei. It manifests itself through a dominance of single-particle and quadrupole–quadrupole terms in a Hamiltonian used to describe well-deformed nuclei. A practical consequence of the quasi- SU(3) symmetry is an efficient basis truncation scheme. In [C.E. Vargas et al., Phys. Rev. C 58 (1998) 1488] it is shown that when this type of Hamiltonian is diagonalized in an SU(3) basis, only a few irreducible representations (irreps) of SU(3) are needed to describe the yrast band, the leading S=0 irrep augmented with the leading S=1 irreps in the proton and neutron subspaces. In the present article the quasi- SU(3) truncation scheme is used, in conjunction with a “realistic but schematic” Hamiltonian that includes the most important multipole terms, to describe the energy spectra and B( E2) transition strengths of 20,22Ne, 24Mg and 28Si. The effect of the size of the Hilbert space on both sets of observables is discussed, as well as the structure of the yrast band and the importance of the various terms in the Hamiltonian. The limitations of the model are explicitly discussed.

Wave chaos in quantum billiards with a small but finite-size scatterer.

We study the low energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through the appropriate truncation of basis. We show that the signature of wave chaos in low energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited.

Sources of Error in Eigenvalue Spectra for Pipe-Poiseuille Flows

An operational Chebyshev Tau approach was used to establish temporal eigenvalue stability spectra for pipe-Poiseuille flows, using the Sexl equation which was transformed to remove the singularity at the origin. Substantial deviations were noted between the long-standing results of Davey and Drazin (1969) which are scattered and display branching patterns, and those of the present work which manifest a more orderly progression of the stable high-order eigenvalues. Formal distinctions are made between errors arising from round-off, basis truncation, and poor choice of basis function: the inaccuracies in Davey and Drazin’s data are thought to have arisen from the latter two effects.

Accuracy of the density-matrix renormalization-group method.

White's density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.

Exact one-loop vacuum polarization effects in (1+1)-dimensional quantum hydrodynamics.

We present a spectral method for computing one-loop vacuum polarization effects [ital exactly] in (1+1)-dimensional finite nuclei with hadronic degrees of freedom. The Dirac equation is solved exactly with periodic boundary conditions by diagonalizing the Dirac Hamiltonian matrix in a finite basis of free parity eigenstates, which in one spatial dimension amounts to a discrete Fourier decomposition of the exact fermion wave functions. We then carry out the renormalization in a manner consistent with the truncation of the finite basis which leads to well-behaved vacuum polarization densities for any basis size. To account for additional vacuum contributions arising from sea states outside our truncated basis, we employ the familiar derivative expansion (DE) in a form consistent with our finite basis. We emphasize the presence of a vector potential in these calculations, and show that, while a variety of methods which are adequate in the scalar-only case fail for the vector-scalar problem, our method treats both cases satisfactorily because consistency between the basis and the renormalization subtractions is scrupulously maintained. Finally, we examine the bulk properties of closed-shell nuclear solutions for a range of input parameters and make a careful comparison between our exact results and a variety of approximations. Wemore » find that the usual DE in general accurately approximates one-loop contributions.« less

Non‐linear seismic response analysis using vector superposition methods

AbstractComputational algorithms based on the pseudo‐force method (PFM) and the tangent spectrum method (TSM) for the seismic analysis of elasto‐plastic MDOF structures by mode superposition are presented. The emphasis is put on the effect of the truncation of higher modes on the convergence of the ductility demand and energy dissipated during the earthquake. Eigenvectors and load‐dependent vectors have been used in comparative analyses. Applications on a flexible 25‐storey building and a stiff, 5‐storey shear building indicate that, for a flexible structure, the ductility demand computed from the PFM is more sensitive to basis truncation than that computed from the TSM. However, for the stiff structure, the opposite behaviour is observed. The results indicate that the use of load‐dependent vectors in inelastic analyses maintains the computational advantages found for elastic analyses in previous investigations. Although the PFM is more stable and computationally more effective than the TSM, it does not provide any information on the evolution of tangent modal properties in time that reflects dynamic response modification as the structure becomes inelastic.

Potential-harmonic expansion for atomic wave functions.

One way to reduce the large degeneracy of the hyperspherical-harmonic basis for solving few- and many-body bound-state problems is to introduce an optimal basis truncation called the potential-harmonic (PH) basis. In this paper we introduce various potential-harmonic truncation schemes and assess their accuracies in predicting the energies of the helium and H{sup {minus}} ground states and the excited 2 {sup 1}{ital S} level of the helium atom. We first find that the part of the PH basis that accounts for one-body correlations gives a better ground-state energy for He than Hartree-Fock (2.8790 a.u. versus 2.8617 for Hartree-Fock and 2.9037 exact). When an orthogonal complement is introduced to the basis to account for {ital e}-{ital e} correlations, we find that the error in the binding energy is 0.000 25 a.u., and 0.000 15 a.u. for ground-state and excited helium, and 0.000 35 a.u. for {ital H}{sup {minus}}. Furthermore, the PH truncation is about 99.9% accurate in accounting for contributions coming from large values of the global angular momentum. This PH scheme is also much more accurate than previous versions based on the Faddeev equations. The present results indicate that the PH truncation can render the hyperspherical-harmonic method useful for systems withmore » {ital N}{gt}3.« less