Showing papers on "Basis function published in 2010"

Property-optimized gaussian basis sets for molecular response calculations.

Abstract: With recent advances in electronic structure methods, first-principles calculations of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for molecules with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calculations since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H–Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small number of moderately diffuse basis functions. The exponents are determined variationally by maximization of atomic Hartree–Fock polarizabilities using analytical derivative methods. The performance of the resulting basis sets is assessed using a set of 313 molecular static Hartree–Fock polarizabilities. The mean...

Isogeometric collocation methods

Abstract: We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

Imposing Dirichlet boundary conditions with Nitsche's method and spline-based finite elements

Abstract: A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.

A generalized finite element formulation for arbitrary basis functions: From isogeometric analysis to XFEM

Abstract: Many of the formulations of cm-rent research interest, including iosogeometric methods and the extended finite element method, use nontraditional basis functions. Some, such as subdivision surfaces, may not have convenient analytical representations. The concept of an element, if appropriate at all, no longer coincides with the traditional definition. Developing a new software for each new class of basis functions is a large research burden, especially, if the problems involve large deformations, non-linear materials, and contact. The objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis, resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file. Elements ranging from a traditional linear four-node tetrahedron through a higher-order element combining XFEM and isogeometric analysis may be specified entirely through an input file without any additional programming. Examples of this framework to applications with Lagrange elements, isogeometric elements, and XFEM basis functions for fracture are presented.

Advanced applications of the holonomic systems approach

Abstract: The holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frederic Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion.Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Grobner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators.

Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces

Abstract: In this paper, robust preconditioners for multiscale flow problems are investigated. We consider elliptic equations with highly varying coefficients. We design and analyze two-level domain decomposition preconditioners that converge independent of the contrast in the media properties. The coarse spaces are constructed using selected eigenvectors of a local spectral problem. Our new construction enriches any given initial coarse space to make it suitable for high-contrast problems. Using the initial coarse space we construct local mass matrices for the local eigenvalue problems. We show that there is a gap in the spectrum of the eigenvalue problem when high-conductivity regions are disconnected. The eigenvectors corresponding to small, asymptotically vanishing eigenvalues are chosen to construct an enrichment of the initial coarse space. Only via a judicious choice of the initial space do we reduce the dimension of the resulting coarse space. Classical coarse basis functions such as multiscale or energy mi...

Hamiltonian Light-Front Field Theory in a Basis Function Approach

Abstract: Hamiltonian light-front quantum field theory constitutes a framework for the nonperturbative solution of invariant masses and correlated parton amplitudes of self-bound systems. By choosing the light-front gauge and adopting a basis function representation, a large, sparse, Hamiltonian matrix for mass eigenstates of gauge theories is obtained that is solvable by adapting the ab initio no-core methods of nuclear many-body theory. Full covariance is recovered in the continuum limit, the infinite matrix limit. There is considerable freedom in the choice of the orthonormal and complete set of basis functions with convenience and convergence rates providing key considerations. Here we use a two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall anti-de Sitter/quantum chromodynamics (AdS/QCD) model obtained from light-front holography. We outline our approach and present illustrative features of some noninteracting systems in a cavity. We illustrate the first steps toward solving quantum electrodynamics (QED) by obtaining the mass eigenstates of an electron in a cavity in small basis spaces and discuss the computational challenges.

A High-Order Internal Model Based Iterative Learning Control Scheme for Nonlinear Systems With Time-Iteration-Varying Parameters

Abstract: In this technical note, we propose a new iterative learning control (ILC) scheme for nonlinear systems with parametric uncertainties that are temporally and iteratively varying. The time-varying characteristics of the parameters are described by a set of unknown basis functions that can be any continuous functions. The iteratively varying characteristics of the parameters are described by a high-order internal model (HOIM) that is essentially an auto-regression model in the iteration domain. The new parametric learning law with HOIM is designed to effectively handle the unknown basis functions. The method of composite energy function is used to derive convergence properties of the HOIM-based ILC, namely the pointwise convergence along the time axis and asymptotic convergence along the iteration axis. Comparing with existing ILC schemes, the HOIM-based ILC can deal with nonlinear systems with more generic parametric uncertainties that may not be repeatable along the iteration axis. The validity of the HOIM-based ILC under identical initialization condition (i.i.c.) and the alignment condition is also explored.

Crossline wavefield reconstruction from multicomponent streamer data: Part 2 — Joint interpolation and 3D up/down separation by generalized matching pursuit

Abstract: Computation of the 3D upgoing/downgoing separated wavefield at any desired position within a marine streamer spread is enabled by multicomponent streamers that can measure the crossline and vertical components of water-particle motion in addition to the pressure. We introduce the concept of simultaneous interpolation and deghosting and describe a new technique, generalized matching pursuit (GMP), to achieve this. This method is based on the matching-pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions. In the GMP method, the basis functions describing the unknown 3D upgoing wavefield are filtered by appropriate forward ghost operators before being matched to the multicomponent measurements. As a data-dependent method, GMP can operate on data samples that are highly aliased in the crossline direction without relying on assumptions about seismic events such as linearity. The technique is naturally suitable for data with only a small number of samples that may be irregularly spaced. We demonstrate the efficacy and robustness of the GMP method on several synthetic data sets of increasing complexity and in the presence of noise.

Nonadiabatic dynamics with the help of multiconfigurational Ehrenfest method: Improved theory and fully quantum 24D simulation of pyrazine

Abstract: This article proposes an improved version of recently developed multiconfigurational Ehrenfest approach to quantum dynamics. The idea of the approach is to use frozen Gaussians (FG) guided by Ehrenfest trajectories as a basis set for fully quantum propagation. The method is applied to simulation of nonadiabatic dynamics of pyrazine and shows that nonadiabatic dynamics on two coupled electronic states S2 and S1, which determines pyrazine absorption spectrum, can be simulated with the help of a basis comprised of very small number of trajectory guided basis functions. For the 24 dimensional (24D) model, good results were obtained with the basis of only 250 trajectories guided FG per electronic state. The efficiency of the method makes it particularly suitable for future application together with direct dynamics, calculating potentials on the fly.

Finding Deterministic Solution From Underdetermined Equation: Large-Scale Performance Variability Modeling of Analog/RF Circuits

Abstract: The aggressive scaling of integrated circuit technology results in high-dimensional, strongly-nonlinear performance variability that cannot be efficiently captured by traditional modeling techniques. In this paper, we adapt a novel L0-norm regularization method to address this modeling challenge. Our goal is to solve a large number of (e.g., 104-106) model coefficients from a small set of (e.g., 102-103) sampling points without over-fitting. This is facilitated by exploiting the underlying sparsity of model coefficients. Namely, although numerous basis functions are needed to span the high-dimensional, strongly-nonlinear variation space, only a few of them play an important role for a given performance of interest. An efficient orthogonal matching pursuit (OMP) algorithm is applied to automatically select these important basis functions based on a limited number of simulation samples. Several circuit examples designed in a commercial 65 nm process demonstrate that OMP achieves up to 25× speedup compared to the traditional least-squares fitting method.

Using basis functions in iterative learning control : analysis and design theory

Abstract: In this article, we discuss iterative learning control (ILC) for systems with input/output (i/o) basis functions. First, we show that various different ILC formulations in the literature can be cap...

Electromagnetic Modeling of Through-Silicon Via (TSV) Interconnections Using Cylindrical Modal Basis Functions

Abstract: This paper proposes an efficient method to model through-silicon via (TSV) interconnections, an essential building block for the realization of silicon-based 3-D systems. The proposed method results in equivalent network parameters that include the combined effect of conductor, insulator, and silicon substrate. Although the modeling method is based on solving Maxwell's equation in integral form, the method uses a small number of global modal basis functions and can be much faster than discretization-based integral-equation methods. Through comparison with 3-D full-wave simulations, this paper validates the accuracy and the efficiency of the proposed modeling method.

A criterion for selecting relevant intrinsic mode functions in empirical mode decomposition

Abstract: Information extraction from time series has traditionally been done with Fourier analysis, which use stationary sines and cosines as basis functions. However, data that come from most natural phenomena are mostly nonstationary. A totally adaptive alternative method has been developed called the Hilbert–Huang transform (HHT), which involves generating basis functions called the intrinsic mode functions (IMFs) via the empirical mode decomposition (EMD). The EMD is a numerical procedure that is prone to numerical errors that may persist in the decomposition as extra IMFs. In this study, results of numerical experiments are presented, which would establish a stringent threshold by which relevant IMFs are distinguished from IMFs that may have been generated by numerical errors. The threshold is dependent on the correlation coefficient between the IMFs and the original signal. Finally, the threshold is applied to IMFs of earthquake signals from five accelerometers located in a building.

A Sparsity Regularization Approach to the Electromagnetic Inverse Scattering Problem

Abstract: We investigate solving the electromagnetic inverse scattering problem using the distorted Born iterative method (DBIM) in conjunction with a variable-selection approach known as the elastic net. The elastic net applies both l 1 and l 2 penalties to regularize the system of linear equations that result at each iteration of the DBIM. The elastic net thus incorporates both the stabilizing effect of the l 2 penalty with the sparsity encouraging effect of the l 1 penalty. The DBIM with the elastic net outperforms the commonly used l 2 regularizer when the unknown distribution of dielectric properties is sparse in a known set of basis functions. We consider two very different 3-D examples to demonstrate the efficacy and applicability of our approach. For both examples, we use a scalar approximation in the inverse solution. In the first example the actual distribution of dielectric properties is exactly sparse in a set of 3-D wavelets. The performances of the elastic net and l 2 approaches are compared to the ideal case where it is known a priori which wavelets are involved in the true solution. The second example comes from the area of microwave imaging for breast cancer detection. For a given set of 3-D Gaussian basis functions, we show that the elastic net approach can produce a more accurate estimate of the distribution of dielectric properties (in particular, the effective conductivity) within an anatomically realistic 3-D numerical breast phantom. In contrast, the DBIM with an l 2 penalty produces an estimate which suffers from multiple artifacts.

Empirical Mode Decomposition - an introduction

Abstract: Due to external stimuli, biomedical signals are in general non-linear and non-stationary. Empirical Mode Decomposition in conjunction with a Hilbert spectral transform, together called Hilbert-Huang Transform, is ideally suited to extract essential components which are characteristic of the underlying biological or physiological processes. The method is fully adaptive and generates the basis to represent the data solely from these data and based on them. The basis functions, called Intrinsic Mode Functions (IMFs) represent a complete set of locally orthogonal basis functions whose amplitude and frequency may vary over time. The contribution reviews the technique of EMD and related algorithms and discusses illustrative applications.

Ghost transmission: How large basis sets can make electron transport calculations worse.

Abstract: The Landauer approach has proven to be an invaluable tool for calculating the electron transport properties of single molecules, especially when combined with a nonequilibrium Green’s function approach and Kohn–Sham density functional theory. However, when using large nonorthogonal atom-centered basis sets, such as those common in quantum chemistry, one can find erroneous results if the Landauer approach is applied blindly. In fact, basis sets of triple-zeta quality or higher sometimes result in an artificially high transmission and possibly even qualitatively wrong conclusions regarding chemical trends. In these cases, transport persists when molecular atoms are replaced by basis functions alone (“ghost atoms”). The occurrence of such ghost transmission is correlated with low-energy virtual molecular orbitals of the central subsystem and may be interpreted as a biased and thus inaccurate description of vacuum transmission. An approximate practical correction scheme is to calculate the ghost transmission and subtract it from the full transmission. As a further consequence of this study, it is recommended that sensitive molecules be used for parameter studies, in particular those whose transmission functions show antiresonance features such as benzene-based systems connected to the electrodes in meta positions and other low-conducting systems such as alkanes and silanes.

A matched filter for chaos.

Abstract: A novel chaotic oscillator is shown to admit an exact analytic solution and a simple matched filter. The oscillator is a hybrid dynamical system including both a differential equation and a discrete switching condition. The analytic solution is written as a linear convolution of a symbol sequence and a fixed basis function, similar to that of conventional communication waveforms. Waveform returns at switching times are shown to be conjugate to a chaotic shift map, effectively proving the existence of chaos in the system. A matched filter in the form of a delay differential equation is derived for the basis function. Applying the matched filter to a received waveform, the bit error rate for detecting symbols is derived, and explicit closed-form expressions are presented for special cases. The oscillator and matched filter are realized in a low-frequency electronic circuit. Remarkable agreement between the analytic solution and the measured chaotic waveform is observed.

Continuous Time-Varying Kriging for Spatial Prediction of Functional Data: An Environmental Application

Abstract: Spatially correlated functional data are present in a wide range of environmental disciplines and, in this context, efficient prediction of curves is a key issue. We present an approach for spatial prediction based on the functional linear pointwise model adapted to the case of spatially correlated curves. First, a smoothing process is applied to the curves by expanding the curves and the functional parameters in terms of a set of basis functions. The number of basis functions is chosen by cross-validation. Then, the spatial prediction of a curve is obtained as a pointwise linear combination of the smoothed data. The prediction problem is solved by estimating a linear model of coregionalization to set the spatial dependence among the fitted coefficients. We extend an optimization criterion used in multivariable geostatistics to the functional context. The method is illustrated by smoothing and predicting temperature curves measured at 35 Canadian weather stations.

Sparse coding for speech recognition

Abstract: This paper proposes a novel feature extraction technique for speech recognition based on the principles of sparse coding. The idea is to express a spectro-temporal pattern of speech as a linear combination of an overcomplete set of basis functions such that the weights of the linear combination are sparse. These weights (features) are subsequently used for acoustic modeling. We learn a set of overcomplete basis functions (dictionary) from the training set by adopting a previously proposed algorithm which iteratively minimizes the reconstruction error and maximizes the sparsity of weights. Furthermore, features are derived using the learned basis functions by applying the well established principles of compressive sensing. Phoneme recognition experiments show that the proposed features outperform the conventional features in both clean and noisy conditions.

Generalized Functional Linear Models with Semiparametric Single-Index Interactions

Abstract: We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single-index structure. We propose a two-step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are predetermined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance due to the estimation error of the basis functio...

Optimized basis sets for the calculation of indirect nuclear spin-spin coupling constants involving the atoms B, Al, Si, P, and Cl

Abstract: The aug-cc-pVTZ-J series of basis sets for indirect nuclear spin-spin coupling constants has been extended to the atoms B, Al, Si, P, and Cl. The basis sets were obtained according to the scheme previously described by Provasi et al. [J. Chem. Phys. 115, 1324 (2001)]. First, the completely uncontracted correlation consistent aug-cc-pVTZ basis sets were extended with four tight s and three tight d functions. Second, the s and p basis functions were contracted with the molecular orbital coefficients of self-consistent-field calculations performed with the uncontracted basis sets on the simplest hydrides of each atom. As a first illustration, we have calculated the one-bond indirect spin-spin coupling constants in BH4−, BF, AlH, AlF, SiH4, SiF4, PH3, PF3, H2S, SF6, HCl, and ClF at the level of density functional theory using the Becke three parameter Lee–Yang–Parr and the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes.