Truncation Of Basis Research Articles

Unbounded Domain Modeling in Axisymmetric Rotor Wake Analysis

Mitigating the effects of wind turbine wakes is a central part of wind farm design. This paper proposes a spectral solver for the axisymmetrical Ainslie wake model based on modified Laguerre basis functions over the semi-infinite radial domain and a marching scheme in the downstream direction. This orthogonal basis promises fast convergence and a low number of DOFs for discretizing the continuity and axial momentum equation, promising a computationally efficient method. The numerical implementation of the model could not be finished in time; preliminary results are presented but still show non-negligible conservation errors. The focus of this work lies, therefore, on a detailed derivation of the method and a discussion of the sources of numerical errors in the nonlinear terms, and due to the truncation of the spectral basis, a comparison of the solver outputs to other methods remains part of the ongoing work.

Evaluation of the excitation spectra with diffusion Monte Carlo on an auxiliary bosonic ground state.

We aim to improve upon the variational Monte Carlo (VMC) approach for excitations replacing the Jastrow factor by an auxiliary bosonic (AB) ground state and multiplying it by a fermionic component factor. The instantaneous change in imaginary time of an arbitrary excitation in the original interacting fermionic system is obtained by measuring observables via the ground-state distribution of walkers of an AB system that is subject to an auxiliary effective potential. The effective potential is used to (i) drive the AB system's ground-state configuration space toward the configuration space of the excitations of the original fermionic system and (ii) subtract from a diffusion Monte Carlo (DMC) calculation contributions that can be included in conventional approximations, such as mean-field and configuration interaction (CI) methods. In this novel approach, the AB ground state is treated statistically in DMC, whereas the fermionic component of the original system is expanded in a basis. The excitation energies of the fermionic eigenstates are obtained by sampling a fermion-boson coupling term on the AB ground state. We show that this approach can take advantage of and correct for approximate eigenstates obtained via mean-field calculations or truncated interactions. We demonstrate that the AB ground-state factor incorporates the correlations missed by standard Jastrow factors, further reducing basis truncation errors. Relevant parts of the theory have been tested in soluble model systems and exhibit excellent agreement with exact analytical data and CI and VMC approaches. In particular, for limited basis set expansions and sufficient statistics, AB approaches outperform CI and VMC in terms of basis size for the same systems. The implementation of this method in current codes, despite being demanding, will be facilitated by reusing procedures already developed for calculating ground-state properties with DMC and excitations with VMC.

An Iterative Implementation of the Signal Space Separation Method for Magnetoencephalography Systems with Low Channel Counts.

The signal space separation (SSS) method is routinely employed in the analysis of multichannel magnetic field recordings (such as magnetoencephalography (MEG) data). In the SSS method, signal vectors are posed as a multipole expansion of the magnetic field, allowing contributions from sources internal and external to a sensor array to be separated via computation of the pseudo-inverse of a matrix of the basis vectors. Although powerful, the standard implementation of the SSS method on MEG systems based on optically pumped magnetometers (OPMs) is unstable due to the approximate parity of the required number of dimensions of the SSS basis and the number of channels in the data. Here we exploit the hierarchical nature of the multipole expansion to perform a stable, iterative implementation of the SSS method. We describe the method and investigate its performance via a simulation study on a 192-channel OPM-MEG helmet. We assess performance for different levels of truncation of the SSS basis and a varying number of iterations. Results show that the iterative method provides stable performance, with a clear separation of internal and external sources.

Uncertainties in ab initio nuclear structure calculations with chiral interactions

We present theoretical ground state energies and their uncertainties for p-shell nuclei obtained from chiral effective field theory internucleon interactions as a function of chiral order, fitted to two- and three-body data only. We apply a Similary Renormalization Group transformation to improve the numerical convergence of the many-body calculations, and discuss both the numerical uncertainties arising from basis truncations and those from omitted induced many-body forces, as well as chiral truncation uncertainties. With complete Next-to-Next-to-Leading (N2LO) order two- and three-body interactions, we find significant overbinding for the ground states in the upper p-shell, but using higher-order two-body potentials, in combination with N2LO three-body forces, our predictions agree with experiment throughout the p-shell to within our combined estimated uncertainties. The uncertainties due to chiral order truncation are noticeably larger than the numerical uncertainties, but they are expected to become comparable to the numerical uncertainties at complete N3LO.

Anisotropy-Based Adaptive Polynomial Chaos Method for Hybrid Uncertainty Quantification and Reliability-Based Design Optimization of Structural-Acoustic System

The evaluation of objective functions and component reliability in the optimisation of structural-acoustic systems with random and interval variables is computationally expensive, especially when strong nonlinearity exhibits between the response and input variables. To reduce the computational cost and improve the computational efficiency, a novel anisotropy-based adaptive polynomial chaos (ABAPC) expansion method was developed in this study. In ABAPC, the anisotropy-based polynomial chaos expansion, namely the retained order of polynomial chaos expansion (PCE) differs from each variable, is used to construct the initial surrogate model instead of first-order polynomial chaos expansion in conventional methods. Then, an anisotropy-based adaptive basis growth strategy was developed to reduce the estimation of the coefficients of the polynomial chaos expansion method and increase its computational efficiency. Finally, to solve problems with probabilistic and interval parameters, an adaptive basis truncation strategy was introduced and implemented. Using the ABAPC method, the computational cost of reliability-based design optimisation for structural-acoustic systems can be efficiently reduced. The effectiveness of the proposed method were demonstrated by solving two numerical examples and optimisation problems of a structural-acoustic system.

Retrieval of initial condition for Burgers' equation using reduced-order EnKF via POD-based sparse observations

<p style='text-indent:20px;'>The data assimilation (DA) is a popular method to solve uncertainty quantification problem which has attracted more and more attention in disaster assessment and climate change research. However, the DA usually faces the computational problem associated with "the curse of dimensionality'' when the high-dimensional inverse problem is involved. For this, the usual strategy is to combine the dimensionality reduction method with the DA to mitigate the computational demand in practical application. In this paper, we develop a reduced-order model (ROM)-based EnKF used for retrieving the initial condition, where the prediction model and the observation model are all POD-dependent. In addition to the reduced-order model (ROM) being prediction model, the idea of compressed sensing motivates the acquisition of the observation model that allows an efficient combination of the EnKF with QD algorithm to obtain the optimal sparse observation locations. In this way, computational acceleration is gained in retrieving the initial condition. The effectiveness of this algorithm is demonstrated through retrieving the initial condition of an one-dimensional (1-D) Burgers' equation. Experimental results show that a satisfactory retrieval result is obtained using the current ROM-based EnKF with the computational (CPU) time, <inline-formula><tex-math id="M1">\begin{document}$ 42.64\rm{s} $\end{document}</tex-math></inline-formula>, nearly 25 times faster, and the error result <inline-formula><tex-math id="M2">\begin{document}$ 7.47\times 10^{-3} $\end{document}</tex-math></inline-formula>, two order of magnitude <inline-formula><tex-math id="M3">\begin{document}$ O(10^2) $\end{document}</tex-math></inline-formula> more accurate, than the implementation of the traditional full-order model (FOM)-based EnKF where the computational instability is observed at each iteration though in the same condition as set in the ROM-based EnKF, leading to a poor result (the CPU time <inline-formula><tex-math id="M4">\begin{document}$ 1057.97\rm{s} $\end{document}</tex-math></inline-formula>, and the error accuracy <inline-formula><tex-math id="M5">\begin{document}$ 4.06\times 10^{-1}) $\end{document}</tex-math></inline-formula>. The present study extends the application of ROM to efficiently dealing with ill-posed problem as a regularization strategy through replacing the FOM with the ROM such that the low-order truncation of the POD basis leads to the ROM with as few degrees of freedom as possible, which can efficiently inhibit the higher frequency errors, and hence achieve stability of the proposed approach.</p>

An Anisotropic Sparse Adaptive Polynomial Chaos Method for the Uncertainty Quantification of Resonant Gratings-Performance

The response of resonant grating structures is studied, for a non-trivial number of random geometric design variables. Computationally inexpensive solutions are found by a new Anisotropic Sparse Adaptive Polynomial Chaos Expansions (ASA-PC) method for the quantities of interest. ASA-PC is based on the Least Angle Regression (LAR) algorithm in conjunction with an adaptive anisotropic basis truncation scheme.

TRENDS OF ESTABLISHMENT AND DEVELOPMENT OF THE TERM TRUNCATION IN UKRAINIAN DERIVATOLOGY

The article is devoted to the formation of a system of terms to define the concept of «cutting off» the phonemic end or a certain part of the creative basis in the process of new words formation in the Ukrainian language, which contributes to the terminological order in Ukrainian derivatology. Formation of the terms truncation and trimming in Ukrainian derivatology, notes the influence of the corresponding terms of Russian derivatology on this process is studied in the article. The correlation of the term truncation with numerous terms based on the clipping (rejection) of the part of the creative basis, accompanied by phonetic changes and without them, in particular with such as befixless (suffixless) method of creation, rederivation (inverse word formation), zero apophysis telescopy, etc., in different periods of linguistics development. The characteristic features of two terminological units at the present stage of development of Ukrainian derivatology are singled out: truncation as a morphological phenomenon and truncation as an independent way of word formation. It was found that truncation of the creative basis is possible not only during affixation, but also in the absence of any affixes in derived words. This phenomenon is observed in the creation of complex words of different types. It is called an abbreviation truncation of bases or an abbreviation type of abbreviation, based on the abbreviation type of word formation is distinguished, or an abbreviation, the essence of which is the formation of an abbreviation word due to abbreviation of words of the corresponding nominative phrase. There is a tendency for the term Elysium to function, in particular, it was that this term has not become widely used in Ukrainian word formation, apparently because the phenomenon it calls is not very characteristic of word-forming acts of the Ukrainian language. It is found out that in Ukrainian theoretical and applied derivatology the term truncation appeared in connection with the need for morphological adaptation of the phonemic end of the creative basis with the phonemic beginning of the word-forming suffix in the process of their combination and formation of a new word. It was considered one of the types of mutual adaptation of morphemes, along with the alternation of phonemes and interfixation during affix word formation. With this meaning it is codified both in word formation and in word-forming morphology.

Truncation and trimming in ukrainian theoretical derivatology

The relevance of the proposed study is due to the need to fully trace the formation of a system of terms to define the concept of «cutting off» the phonemic end or a certain part of the creative basis in the process of forming new words in the Ukrainian language, which will contribute to the terminological ordering in Ukrainian derivatology. The article investigates the formation of the terms truncation and trimming in Ukrainian derivatology, the influence of the corresponding terms of Russian derivatology on this process was noted. The correlation of the term «truncation» to numerous terms in different periods of development of linguistics was characterized. We paid special attention to those based on the clipping (rejection) of the part of the creative basis, accompanied by phonetic changes and without them, in particular with such as the affix-free (suffix-free) method of creation, rederivation (inverse word formation), zero suffixation, elision, apocope, telescopy, etc. The characteristic features of two terminological units at the present stage of development of Ukrainian derivatology were singled out: truncation as a morphological phenomenon and trimming as an independent way of word formation. It was found that truncation of the creative basis is possible not only during affixation, but also in the absence of any affixes in derived words. We can observe this phenomenon in the creation of complex words of different types. It is called an abbreviation truncation of bases or an abbreviation type of abbreviation, on the basis of which the abbreviation type of word formation was distinguished. Alternatively, we can call it an abbreviation, the essence of which is the formation of the word-abbreviation due to the reduction of words of the corresponding nominative phrase. The tendency of the term elision to function has been traced, in particular, it has been found out that this term has not become widely used in Ukrainian word formation, obviously because the phenomenon it causes is not very characteristic of word-forming acts of the Ukrainian language.

Recent advances in polynomial chaos method for uncertainty propagation

Uncertainty exists widely in engineering design. As one of the key components of engineering design, uncertainty propagation and quantification has always been an important research topic. Polynomial chaos (PC) is a highly efficient uncertainty propagation method which has been widely studied and applied. Therefore, this paper reviews recent advances in the PC method. First, the fundamentals of PC are introduced, including the construction of an orthogonal polynomial basis and the calculation of PC coefficients. Second, strategies such as basis truncation, sparse reconstruction, sparse grid and multi-fidelity modeling are described to address the curse of dimensionality issue of PC. Local and global sensitivity analyses based on PC are then introduced. Finally, the research prospects of PC are given.

Scattering in strong electromagnetic fields: Transverse size effects in time-dependent basis light-front quantization

The framework of 'time-dependent basis light-front quantisation' (tBLFQ) offers a non-perturbative approach to scattering problems in external fields, based on Fock space truncation. Here we extend tBLFQ to include spatio-temporal field inhomogeneities in multiple spacetime directions. This extension is necessary for the proper modelling of e.g. intense laser fields. We focus on the example of nonlinear Compton scattering of an electron on an axicon-type laser, with an emphasis on the transverse structure of the beam. We analyse the impact of field intensity and particle energy, as well as basis truncation effects, on the radiation spectrum of the scattered electron.

Saturations of subalgebras, SAGBI bases, and U-invariants

Saturations of subalgebras, SAGBI bases, and U-invariants

Reduced-Order Quasilinear Model of Ocean Boundary-Layer Turbulence

AbstractThe combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto a subset of modes determined by proper orthogonal decomposition (POD). When applied to boundary layers that are horizontally homogeneous, POD and a horizontally averaged quasilinear approximation both assume flow features that are horizontally wavelike, making the pairing very efficient. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mean fields as well as certain energetically important second-order turbulent transport statistics and energies to within 30% error. Reduced-basis quasilinear statistics must approach the full-basis statistics as the basis size approaches completion; however, some quasilinear statistics resemble those found in the fully nonlinear simulations at smaller basis truncations. Thus, model reduction could possibly improve upon the accuracy of quasilinear dynamics.

Estimating linear response statistics using orthogonal polynomials: An rkhs formulation

We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. Based on the fluctuation-dissipation theory, this problem is reformulated as an unsupervised learning task of estimating a density function. We consider a nonparametric density estimator formulated by the kernel embedding of distributions with kernels, constructed based on the classical orthogonal polynomials defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion (PCE), the connection to the reproducing kernel Hilbert space (RKHS) theory allows one to establish the uniform convergence of the estimator and to systematically address a practical question of identifying the PCE basis for a consistent estimation. We also provide practical conditions for the well-posedness of not only the estimator but also of the underlying response statistics. Finally, we provide a statistical error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to a finite basis truncation. This error bound provides a means to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.

Statistical Inference on Panel Data Models: A Kernel Ridge Regression Method

We propose statistical inferential procedures for nonparametric panel data models with interactive fixed effects in a kernel ridge regression framework. Compared with the traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters. The model complexity is controlled by a continuous regularization parameter which can be automatically selected by the generalized cross-validation. Based on the empirical process theory and functional analysis tools, we derive the joint asymptotic distributions for the estimators in the heterogeneous setting. These joint asymptotic results are then used to construct the confidence intervals for the regression means and the prediction intervals for future observations, both being the first provably valid intervals in literature. The marginal asymptotic normality of the functional estimators in a homogeneous setting is also obtained. Our estimators can also be readily modified and applied to other widely used semiparametric models, such as partially linear models. Simulation and real data analyses demonstrate the advantages of our method. Supplementary materials for this article are available online.

Microscopic polarization and magnetization fields in extended systems

We introduce microscopic polarization and magnetization fields at each site of an extended system, as well as free charge and current density fields associated with charge movement from site to site, by employing a lattice gauge approach based on a set of orthogonal orbitals associated with each site. These microscopic fields are defined using a single-particle electron Green function, and the equations governing its evolution under excitation by an electromagnetic field at arbitrary frequency involve the electric and magnetic fields rather than the scalar and vector potentials. If the sites are taken to be far from each other, we recover the limit of isolated atoms. For an infinite crystal we choose the orbitals to be maximally-localized Wannier functions, and in the long wavelength limit we recover the expected linear response of an insulator, including the zero frequency transverse conductivity of a topologically nontrivial insulator. For a topologically trivial insulator we recover the expected expressions for the macroscopic polarization and magnetization in the ground state, and find that the linear response to excitation at arbitrary frequency is described solely by the microscopic polarization and magnetization fields. For very general optical response calculations the microscopic fields necessarily satisfy charge conservation, even under basis truncation, and do not suffer from the false divergences at zero frequency that can plague response calculations using other approaches.

A modal coordinate catenary model for the real-time simulation of the pantograph-catenary dynamic interaction

A modal coordinate catenary model for the real-time simulation of the pantograph-catenary dynamic interaction

Importance Basis Truncation in the Symmetry-adapted No-core Shell Model

Importance Basis Truncation in the Symmetry-adapted No-core Shell Model

Wavejets: A Local Frequency Framework for Shape Details Amplification

AbstractDetail enhancement is a well‐studied area of 3D rendering and image processing, which has few equivalents for 3D shape processing. To enhance details, one needs an efficient analysis tool to express the local surface dynamics. We introduce Wavejets, a new function basis for locally decomposing a shape expressed over the local tangent plane, by considering both angular oscillations of the surface around each point and a radial polynomial. We link the Wavejets coefficients to surface derivatives and give theoretical guarantees for their precision and stability with respect to an approximate tangent plane. The coefficients can be used for shape details amplification, to enhance, invert or distort them, by operating either on the surface point positions or on the normals. From a practical point of view, we derive an efficient way of estimating Wavejets on point sets and demonstrate experimentally the amplification results with respect to noise or basis truncation.

Quantifying entanglement of rotor chains using basis truncation: Application to dipolar endofullerene peapods.

We propose a variational approach for the calculation of the quantum entanglement entropy of assemblies of rotating dipolar molecules. A basis truncation scheme based on the total angular momentum quantum number is proposed. The method is tested on hydrogen fluoride (HF) molecules confined in C60 fullerene cages themselves trapped in a nanotube to form a carbon peapod. The rotational degrees of freedom of the HF molecules and dipolar interactions between neighboring molecules are considered in our model Hamiltonian. Both screened and unscreened dipoles are simulated and results are obtained for the ground state and one excited state that is expected to be accessible via a far-infrared collective excitation. The effect of basis truncation on energetic and entanglement properties is examined and discussed in terms of size extensivity. It is empirically found that for unscreened dipoles, a total angular momentum cutoff that increases linearly with the number of rotors is required in order to obtain proper system size scaling of the chemical potential and entanglement entropy. Recent experiments [A. Krachmalnicoff et al., Nat. Chem. 8, 953 (2016)] suggest substantial screening of the HF dipole moment, so much smaller basis sets are required to obtain converged results in this realistic case. Static correlation functions are also computed and are shown to decay much quicker in the case of screened dipoles. Our variational results are also used to test the accuracy of perturbative and pairwise ansatz treatments.