Showing papers on "Basis function published in 2015"

A Multi-resolution Gaussian process model for the analysis of large spatial data sets

Abstract: We develop a multiresolution model to predict two-dimensional spatial fields based on irregularly spaced observations. The radial basis functions at each level of resolution are constructed using a Wendland compactly supported correlation function with the nodes arranged on a rectangular grid. The grid at each finer level increases by a factor of two and the basis functions are scaled to have a constant overlap. The coefficients associated with the basis functions at each level of resolution are distributed according to a Gaussian Markov random field (GMRF) and take advantage of the fact that the basis is organized as a lattice. Several numerical examples and analytical results establish that this scheme gives a good approximation to standard covariance functions such as the Matern and also has flexibility to fit more complicated shapes. The other important feature of this model is that it can be applied to statistical inference for large spatial datasets because key matrices in the computations are spars...

Generalized Jacobi functions and their applications to fractional differential equations

Abstract: In this paper, we consider spectral approximation of fractional dif- ferential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly de- signed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value prob- lems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well- conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.

Sparse maps—A systematic infrastructure for reduced-scaling electronic structure methods. I. An efficient and simple linear scaling local MP2 method that uses an intermediate basis of pair natural orbitals

Abstract: In this work, a systematic infrastructure is described that formalizes concepts implicit in previous work and greatly simplifies computer implementation of reduced-scaling electronic structure methods. The key concept is sparse representation of tensors using chains of sparse maps between two index sets. Sparse map representation can be viewed as a generalization of compressed sparse row, a common representation of a sparse matrix, to tensor data. By combining few elementary operations on sparse maps (inversion, chaining, intersection, etc.), complex algorithms can be developed, illustrated here by a linear-scaling transformation of three-center Coulomb integrals based on our compact code library that implements sparse maps and operations on them. The sparsity of the three-center integrals arises from spatial locality of the basis functions and domain density fitting approximation. A novel feature of our approach is the use of differential overlap integrals computed in linear-scaling fashion for screening products of basis functions. Finally, a robust linear scaling domain based local pair natural orbital second-order Moller-Plesset (DLPNO-MP2) method is described based on the sparse map infrastructure that only depends on a minimal number of cutoff parameters that can be systematically tightened to approach 100% of the canonical MP2 correlation energy. With default truncation thresholds, DLPNO-MP2 recovers more than 99.9% of the canonical resolution of the identity MP2 (RI-MP2) energy while still showing a very early crossover with respect to the computational effort. Based on extensive benchmark calculations, relative energies are reproduced with an error of typically <0.2 kcal/mol. The efficiency of the local MP2 (LMP2) method can be drastically improved by carrying out the LMP2 iterations in a basis of pair natural orbitals. While the present work focuses on local electron correlation, it is of much broader applicability to computation with sparse tensors in quantum chemistry and beyond.

A novel high order space-time spectral method for the time fractional fokker-planck equation ∗

Abstract: The fractional Fokker--Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker--Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker--Planck equation attains the same approximation order as the time fractional diffusion equation developed in [X. Li and C. Xu, SIAM J. Numer. Anal., 47 (2009), pp. 2108--2131] by using the present method. That indicates an e...

A multi-resolution approximation for massive spatial datasets

Abstract: Automated sensing instruments on satellites and aircraft have enabled the collection of massive amounts of high-resolution observations of spatial fields over large spatial regions. If these datasets can be efficiently exploited, they can provide new insights on a wide variety of issues. However, traditional spatial-statistical techniques such as kriging are not computationally feasible for big datasets. We propose a multi-resolution approximation (M-RA) of Gaussian processes observed at irregular locations in space. The M-RA process is specified as a linear combination of basis functions at multiple levels of spatial resolution, which can capture spatial structure from very fine to very large scales. The basis functions are automatically chosen to approximate a given covariance function, which can be nonstationary. All computations involving the M-RA, including parameter inference and prediction, are highly scalable for massive datasets. Crucially, the inference algorithms can also be parallelized to take full advantage of large distributed-memory computing environments. In comparisons using simulated data and a large satellite dataset, the M-RA outperforms a related state-of-the-art method.

Mixed Generalized Multiscale Finite Element Methods and Applications

Abstract: In this paper, we present a mixed generalized multiscale finite element method (GMsFEM) for solving flow in heterogeneous media Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface The snapshot vectors have zero Neumann boundary conditions on the outer boundaries, and we prescribe their values on the common interface We describe several spectral decompositions in the snapshot space motivated by the analysis In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM A main idea of oversampling techniques is to introduce a small dimensional snapshot space We present numerical results for two-phase flow and

Decomposed Vector Rotation-Based Behavioral Modeling for Digital Predistortion of RF Power Amplifiers

Abstract: A new behavioral model for digital predistortion of radio frequency (RF) power amplifiers (PAs) is proposed in this paper. It is derived from a modified form of the canonical piecewise-linear (CPWL) functions using a decomposed vector rotation (DVR) technique. In this model, the nonlinear basis function is constructed from piecewise vector decomposition, which is completely different from that used in the conventional Volterra series. Theoretical analysis has shown that this model is much more flexible in modeling RF PAs with non-Volterra-like behavior, and experimental results confirmed that the new model can produce excellent performance with a relatively small number of coefficients when compared to conventional models.

Interpolation and denoising of high-dimensional seismic data by learning a tight frame

Abstract: Sparse transforms play an important role in seismic signal processing steps, such as prestack noise attenuation and data reconstruction. Analytic sparse transforms (so-called implicit dictionaries), such as the Fourier, Radon, and curvelet transforms, are often used to represent seismic data. There are situations, however, in which the complexity of the data requires adaptive sparse transform methods, whose basis functions are determined via learning methods. We studied an application of the data-driven tight frame (DDTF) method to noise suppression and interpolation of high-dimensional seismic data. Rather than choosing a model beforehand (for example, a family of lines, parabolas, or curvelets) to fit the data, the DDTF derives the model from the data itself in an optimum manner. The process of estimating the basis function from the data can be summarized as follows: First, the input data are divided into small blocks to form training sets. Then, the DDTF algorithm is applied on the training set...

Rational Basis Functions in Iterative Learning Control—With Experimental Verification on a Motion System

Abstract: Iterative learning control (ILC) approaches often exhibit poor extrapolation properties with respect to exogenous signals, such as setpoint variations. This brief introduces rational basis functions in ILC. Such rational basis functions have the potential to both increase performance and enhance the extrapolation properties. The key difficulty that is associated with these rational basis functions lies in a significantly more complex optimization problem when compared with using preexisting polynomial basis functions. In this brief, a new iterative optimization algorithm is proposed that enables the use of rational basis functions in ILC for single-input single-output systems. An experimental case study confirms the advantages of rational basis functions compared with preexisting results, as well as the effectiveness of the proposed iterative algorithm.

Residual a posteriori error estimation for the Virtual Element Method for elliptic problems

Abstract: A posteriori error estimation and adaptivity are very useful in the context of the virtual element and mimetic discretization methods due to the flexibility of the meshes to which these numerical schemes can be applied. Nevertheless, developing error estimators for virtual and mimetic methods is not a straightforward task due to the lack of knowledge of the basis functions. In the new virtual element setting, we develop a residual based a posteriori error estimator for the Poisson problem with (piecewise) constant coefficients, that is proven to be reliable and efficient. We moreover show the numerical performance of the proposed estimator when it is combined with an adaptive strategy for the mesh refinement.

Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory

Abstract: A key component in calculations of exchange and correlation energies is the Coulomb operator, which requires the evaluation of two-electron integrals. For localized basis sets, these four-center integrals are most efficiently evaluated with the resolution of identity (RI) technique, which expands basis-function products in an auxiliary basis. In this work we show the practical applicability of a localized RI-variant ('RI-LVL'), which expands products of basis functions only in the subset of those auxiliary basis functions which are located at the same atoms as the basis functions. We demonstrate the accuracy of RI-LVL for Hartree–Fock calculations, for the PBE0 hybrid density functional, as well as for RPA and MP2 perturbation theory. Molecular test sets used include the S22 set of weakly interacting molecules, the G3 test set, as well as the G2–1 and BH76 test sets, and heavy elements including titanium dioxide, copper and gold clusters. Our RI-LVL implementation paves the way for linear-scaling RI-based hybrid functional calculations for large systems and for all-electron many-body perturbation theory with significantly reduced computational and memory cost.

Analysis and comparison of CVS-ADC approaches up to third order for the calculation of core-excited states

Abstract: The extended second order algebraic-diagrammatic construction (ADC(2)-x) scheme for the polarization operator in combination with core-valence separation (CVS) approximation is well known to be a powerful quantum chemical method for the calculation of core-excited states and the description of X-ray absorption spectra. For the first time, the implementation and results of the third order approach CVS-ADC(3) are reported. Therefore, the CVS approximation has been applied to the ADC(3) working equations and the resulting terms have been implemented efficiently in the adcman program. By treating the α and β spins separately from each other, the unrestricted variant CVS-UADC(3) for the treatment of open-shell systems has been implemented as well. The performance and accuracy of the CVS-ADC(3) method are demonstrated with respect to a set of small and middle-sized organic molecules. Therefore, the results obtained at the CVS-ADC(3) level are compared with CVS-ADC(2)-x values as well as experimental data by calculating complete basis set limits. The influence of basis sets is further investigated by employing a large set of different basis sets. Besides the accuracy of core-excitation energies and oscillator strengths, the importance of cartesian basis functions and the treatment of orbital relaxation effects are analyzed in this work as well as computational timings. It turns out that at the CVS-ADC(3) level, the results are not further improved compared to CVS-ADC(2)-x and experimental data, because the fortuitous error compensation inherent in the CVS-ADC(2)-x approach is broken. While CVS-ADC(3) overestimates the core excitation energies on average by 0.61% ± 0.31%, CVS-ADC(2)-x provides an averaged underestimation of −0.22% ± 0.12%. Eventually, the best agreement with experiments can be achieved using the CVS-ADC(2)-x method in combination with a diffuse cartesian basis set at least at the triple-ζ level.

Libcint: An efficient general integral library for Gaussian basis functions.

Abstract: An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian-type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one-electron overlap and nuclear attraction, two-electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4-component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth-order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi-CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20-core node, we are able to achieve an average output 8.3 GB/s for C60 molecule with cc-pVTZ basis. © 2015 Wiley Periodicals, Inc.

Linear-scaling implementation of the direct random-phase approximation.

Abstract: We report the linear-scaling implementation of the direct random-phase approximation (dRPA) for closed-shell molecular systems. As a bonus, linear-scaling algorithms are also presented for the second-order screened exchange extension of dRPA as well as for the second-order Moller–Plesset (MP2) method and its spin-scaled variants. Our approach is based on an incremental scheme which is an extension of our previous local correlation method [Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The approach extensively uses local natural orbitals to reduce the size of the molecular orbital basis of local correlation domains. In addition, we also demonstrate that using natural auxiliary functions [M. Kallay, J. Chem. Phys. 141, 244113 (2014)], the size of the auxiliary basis of the domains and thus that of the three-center Coulomb integral lists can be reduced by an order of magnitude, which results in significant savings in computation time. The new approach is validated by extensive test calculations for energies and energy differences. Our benchmark calculations also demonstrate that the new method enables dRPA calculations for molecules with more than 1000 atoms and 10 000 basis functions on a single processor.

A la carte | learning fast kernels

Abstract: Kernel methods have great promise for learning rich statistical representations of large modern datasets. However, compared to neural networks, kernel methods have been perceived as lacking in scalability and exibility. We introduce a family of fast, exible, lightly parametrized and general purpose kernel learning methods, derived from Fastfood basis function expansions. We provide mechanisms to learn the properties of groups of spectral frequencies in these expansions, which require only O ( m log d ) time and O ( m ) memory, for m basis functions and d input dimensions. We show that the proposed methods can learn a wide class of kernels, outperforming the alternatives in accuracy, speed, and memory consumption.

Compressed Representation of Kohn-Sham Orbitals via Selected Columns of the Density Matrix.

Abstract: Given a set of Kohn-Sham orbitals from an insulating system, we present a simple, robust, efficient, and highly parallelizable method to construct a set of optionally orthogonal, localized basis functions for the associated subspace. Our method explicitly uses the fact that density matrices associated with insulating systems decay exponentially along the off-diagonal direction in the real space representation. We avoid the usage of an optimization procedure, and the localized basis functions are constructed directly from a set of selected columns of the density matrix (SCDM). Consequently, the core portion of our localization procedure is not dependent on any adjustable parameters. The only adjustable parameters present pertain to the use of the SCDM after their computation (for example, at what value should the SCDM be truncated). Our method can be used in any electronic structure software package with an arbitrary basis set. We demonstrate the numerical accuracy and parallel scalability of the SCDM procedure using orbitals generated by the Quantum ESPRESSO software package. We also demonstrate a procedure for combining the orthogonalized SCDM with Hockney's algorithm to efficiently perform Hartree-Fock exchange energy calculations with near-linear scaling.

Gaussian Ring Basis Functions for the Analysis of Modulated Metasurface Antennas

Abstract: This paper presents a novel type of basis functions, whose spectral- and space-domain properties can be exploited for the efficient method of moments (MoM) analysis of planar metasurface (MTS) antennas. The effect of the homogenized MTS is introduced in the integral equation as an impedance boundary condition (IBC). The proposed basis functions are shaped as Gaussian-type rings with small width and linear azimuthal phase. The analytical form of the spectrum of the Gaussian ring basis allows for a closed-form evaluation of the MoM impedance matrix’s entries. Moreover, these basis functions account for the global evolution of the surface current density in an effective manner, reducing the size of the MoM system of equations with respect to the case of subdomain basis functions. These features allow one to carry out a direct solution for problems with a diameter of up to 15 wavelengths in less than 1 min using a conventional laptop. The applicability on practical antennas has been tested through the full-wave analysis of MTS antennas implemented with small printed elements.

A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

Abstract: In this article, a new spectral meshless radial point interpolation (SMRPI) technique is proposed and, as a test problem, applied to the two-dimensional diffusion equation with an integral (non-classical) condition. This innovative method is based on erudite combination of meshless methods and spectral collocation techniques but it is not traditional meshless collocation method. As the meshless method, the point interpolation method with the help of radial basis functions is used to construct shape functions as basis functions in the frame of spectral collocation methods. These basis functions (shape functions) have Kronecker delta function property. In the proposed method, high order derivatives (in high order partial differential equations) are evaluated by constructing and using operational matrices. In SMRPI method, it does not require any kind of integration locally over small quadrature domains as it is essential in Galerkin weak form meshless methods such as element free Galerkin (EFG) and meshless local Petrov–Galerkin (MLPG) methods. Also, it is not needed to determine shape parameter which is often played important role in these methods, for instance recall test (weight functions) for moving least squares (MLS) approximation in the frame of MLPG. Therefore, computational costs of SMRPI method is less expensive. A comparison study of the efficiency and accuracy of the present method and meshless local Petrov–Galerkin (MLPG) method is given by applying on mentioned diffusion equation. Convergence studies in the numerical examples show that SMRPI method possesses reasonable rate of convergence.

Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QVPPD) for RI-MP2 and RI-CC calculations

Abstract: We report optimized auxiliary basis sets for use with the Karlsruhe segmented contracted basis sets including moderately diffuse basis functions (Rappoport and Furche, J. Chem. Phys., 2010, 133, 134105) in resolution-of-the-identity (RI) post-self-consistent field (post-SCF) computations for the elements H-Rn (except lanthanides). The errors of the RI approximation using optimized auxiliary basis sets are analyzed on a comprehensive test set of molecules containing the most common oxidation states of each element and do not exceed those of the corresponding unaugmented basis sets. During these studies an unsatisfying performance of the def2-SVP and def2-QZVPP auxiliary basis sets for Barium was found and improved sets are provided. We establish the versatility of the def2-SVPD, def2-TZVPPD, and def2-QZVPPD basis sets for RI-MP2 and RI-CC (coupled-cluster) energy and property calculations. The influence of diffuse basis functions on correlation energy, basis set superposition error, atomic electron affinity, dipole moments, and computational timings is evaluated at different levels of theory using benchmark sets and showcase examples.