Showing papers on "Basis (linear algebra) published in 2002"

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Transition moments and excited-state first-order properties in the coupled-cluster model CC2 using the resolution-of-the-identity approximation

Abstract: An implementation of transition moments and excited-state first-order properties is reported for the approximate coupled-cluster singles-and-doubles model (CC2) using the resolution of the identity (RI) approximation. In parallel to the previously reported code for the ground- and excited-state amplitude equations, we utilize a partitioned form of the CC2 equations and thus eliminate the need to store any N 4 intermediates. This opens the perspective for applications on molecules with 30 and more atoms. The accuracy of the RI approximation is tested for a set of 29 molecules for the aug-cc -p V X Z (X=D,T,Q) basis sets in connection with the recently optimized auxiliary basis sets. These auxiliary basis sets are found to be sufficient even for the description of diffuse states. The RI error is compared to the usual basis set error and is demonstrated to be insignificant.

Explicitly correlated second-order Møller–Plesset methods with auxiliary basis sets

Abstract: In explicitly correlated Moller–Plesset (MP2-R12) methods, the first-order wave function is expanded not only in terms of products of one-electron functions—that is, orbitals—but also in terms of two-electron functions that depend linearly on the interelectronic coordinates rij. With these functions, three- and four-electron integrals occur, but these integrals can be avoided by inserting a resolution of the identity (RI) in terms of the one-electron basis. In previous work, only one single basis was used for both the electronic wave function and the RI approximation. In the present work, a new computational approach is developed that uses an auxiliary basis set to represent the RI. This auxiliary basis makes it possible to employ standard basis sets in explicitly correlated MP2-R12 calculations.

Polarization consistent basis sets. III. The importance of diffuse functions

Abstract: A sequence of diffuse functions to be used in connections with the previously defined polarization consistent basis sets are proposed based on energetic criteria and results for molecular properties. At the Hartree–Fock level the addition of a single set of diffuse s- and p-functions significantly improves the convergence of calculated electron affinities. A corresponding analysis at the density functional level indicates that only systems with high electron affinities have well-defined basis set limits with common exchange-correlation functionals that have electron self-interaction errors. The majority of reported density functional calculations of electron affinities appear to be artifacts of the limited basis set used. The good agreement with experiments for such calculations is most likely due to a reasonable modeling of the physics of the anionic species, rather than being a theoretically sound procedure. For molecular properties like dipole and quadrupole moments, and static polarizabilities, the addition of diffuse functions up to d-functions is required to reach the basis set limit in a consistent fashion, but higher order angular momentum functions are significantly less important.

Systematic generation of finite-range atomic basis sets for linear-scaling calculations

Abstract: Basis sets of atomic orbitals are very efficient for density functional calculations but lack a systematic variational convergence. We present a method to optimize numerical atomic orbitals variationally, using a single parameter to control their range. The efficiency of the basis generation scheme is tested and compared with other schemes for multiple $\ensuremath{\zeta}$ basis sets. The scheme is shown to be comparable in quality to other widely used schemes albeit offering better performance for linear-scaling computations.

Direct method for optimized effective potentials in density-functional theory.

Abstract: The conventional optimized effective potential method is based on a difficult-to-solve integral equation. In the new method, this potential is constructed as a sum of a fixed potential and a linear combination of basis functions. The energy derivatives with respect to the coefficients of the linear combination are obtained. This enables calculations by optimization methods. Accurate atomic and molecular calculations with Gaussian basis sets are presented for exact exchange functionals. This efficient and accurate method for the optimized effective potential should play an important role in the development and application of density functionals.

Using manifold structure for partially labelled classification

Abstract: We consider the general problem of utilizing both labeled and unlabeled data to improve classification accuracy. Under the assumption that the data lie on a submanifold in a high dimensional space, we develop an algorithmic framework to classify a partially labeled data set in a principled manner. The central idea of our approach is that classification functions are naturally defined only on the sub-manifold in question rather than the total ambient space. Using the Laplace Beltrami operator one produces a basis for a Hilbert space of square integrable functions on the submanifold. To recover such a basis, only unlabeled examples are required. Once a basis is obtained, training can be performed using the labeled data set. Our algorithm models the manifold using the adjacency graph for the data and approximates the Laplace Beltrami operator by the graph Laplacian. Practical applications to image and text classification are considered.

Linear scaling local correlation approach for solving the coupled cluster equations of large systems.

Abstract: A linear scaling local correlation approach is proposed for approximately solving the coupled cluster doubles (CCD) equations of large systems in a basis of orthogonal localized molecular orbitals (LMOs). By restricting double excitations from spatially close occupied LMOs into their associated virtual LMOs, the number of significant excitation amplitudes scales only linearly with molecular size in large molecules. Significant amplitudes are obtained to a very good approximation by solving the CCD equations of various subsystems, each of which is made up of a cluster associated with the orbital indices of a subset of significant amplitudes and the local environmental domain of the cluster. The combined effect of these two approximations leads to a linear scaling algorithm for large systems. By using typical thresholds, which are designed to target an energy accuracy, our numerical calculations for a wide range of molecules using the 6-31G or 6-31G* basis set demonstrate that the present local correlation approach recovers more than 98.5% of the conventional CCD correlation energy.

A hierarchical family of global analytic Born-Oppenheimer potential energy surfaces for the H+H2 reaction ranging in quality from double-zeta to the complete basis set limit

Abstract: A hierarchical family of analytical Born–Oppenheimer potential energy surfaces has been developed for the H+H2 system. Ab initio calculations of near full configuration interaction (FCI) quality (converged to within ≈1 μEh) were performed for a set of 4067 configurations with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets. The complete basis set (CBS) limit energies were obtained using a highly accurate many-body basis set extrapolation scheme. Surfaces were fitted for the estimated CBS limit, as well as for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets using a robust new functional form. The mean unsigned fitting error for the CBS surface is a mere 0.0023 kcal/mol, and deviations for data not included in the fitting process are of similarly small magnitudes. Highly accurate calculations of the saddle point and van der Waals minimum configurations were performed using basis sets as large as aug-mcc-pV7Z, and these data show excellent agreement with the results of the extrapolated pote...

New ideas for using contracted basis functions with a Lanczos eigensolver for computing vibrational spectra of molecules with four or more atoms

Abstract: We propose new methods for using contracted basis functions in conjunction with the Lanczos algorithm to calculate vibrational (or rovibrational) spectra. As basis functions we use products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well, yet are simple enough that matrix-vector products may be evaluated efficiently. The methods we suggest obviate the need to transform from the contracted to an original product basis each time a matrix-vector product is evaluated. For HOOH the most efficient of the methods we present is about an order of magnitude faster than a product basis Lanczos calculation.

All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams

Abstract: A detailed investigation is presented of a set of algorithms which form the basis for a fast and reliable numerical integration of one-loop multi-leg (up to six) Feynman diagrams, with special attention to the behavior around (possibly) singular points in phase space. No particular restriction is imposed on kinematics, and complex masses (poles) are allowed.

A general framework for discrete variable representation basis sets

Abstract: A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.

A Reduced-Basis Element Method

Abstract: Reduced basis methods are particularly attractive to use in order to diminish the number of degrees of freedom associated with the approximation of a set of partial differential equations. The main idea is to construct ad hoc basis functions with a large information content. In this note, we propose to develop and analyze reduced basis methods for simulating hierarchical flow systems, which is of relevance for studying flows in a network of pipes, an example being a set of arteries or veins. We propose to decompose the geometry into generic parts (e.g., pipes and bifurcations), and to contruct a reduced basis for these generic parts by considering representative geometric snapshots. The global system is constructed by gluing the individual basis solutions together via Lagrange multipliers.

Stochastic Reduced Basis Methods

Abstract: Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.

LMIs - a fundamental tool in analysis and controller design for discrete linear repetitive processes

Abstract: Discrete linear repetitive processes are a distinct class of two-dimensional (2-D) linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The feature which makes them distinct from other classes of 2-D linear systems is that information propagation in one of the two distinct directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper, an LMI approach is used to produce highly significant new results on the stability analysis of these processes and the design of control schemes for them. These results are, in the main, for processes with singular dynamics and for those with so-called dynamic boundary conditions. Unlike other classes of 2-D linear systems, these feedback control laws have a firm physical basis, and the LMI setting is also shown to provide a (potentially) very powerful setting in which to characterize the robustness properties of these processes.

Local non-negative matrix factorization as a visual representation

Abstract: Proposes a novel method, called local non-negative matrix factorization (LNMF), for learning a spatially localized, parts-based subspace representation of visual patterns. An objective function is defined to impose the localization constraint, in addition to the non-negativity constraint in the standard non-negative matrix factorization (NMF). This gives a set of bases which not only allows a non-subtractive (part-based) representation of images but also manifests localized features. An algorithm is presented for the learning of such basis components. Experimental results are presented to compare LNMF with the NMF and principal component analysis (PCA) methods for face representation and recognition, which demonstrates the advantages of LNMF. Based on our LNMF approach, a set of orthogonal, binary, localized basis components are learned from a well-aligned face image database. It leads to a Walsh function-based representation of the face images. These properties can be used to resolve the occlusion problem, improve the computing efficiency and compress the storage requirements of a face detection and recognition system.

Modeling inverse covariance matrices by basis expansion

Abstract: This paper proposes a new covariance modeling technique for Gaussian Mixture Models. Specifically the inverse covariance (precision) matrix of each Gaussian is expanded in a rank-1 basis i.e., Σ j −1 = P j = Σ k = 1 D λ k ja k a k T, λ k j ∈ ℝd. A generalized EM algorithm is proposed to obtain maximum likelihood parameter estimates for the basis set {a k a k T} and the expansion coefficients {λ k j}. This model, called the Extended Maximum Likelihood Linear Transform (EMLLT) model, is extremely flexible: by varying the number of basis elements from d to d(d + 1)/2 one gradually moves from a Maximum Likelihood Linear Transform (MLLT) model to a full-covariance model. Experimental results on two speech recognition tasks show that the EMLLT model can give relative gains of up to 35% in the word error rate over a standard diagonal covariance model.

Riesz basis property and exponential stability of controlled euler-bernoulli beam equations with

Abstract: This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler-Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrum- determined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam. 1. Introduction. The Riesz basis property, meaning that the generalized eigen- vectors of the system form an unconditional basis for the state Hilbert space, is one of the fundamental properties of a linear vibrating system. The establishment of the basis property will naturally lead to solutions to such problems as the spectrum- determined growth condition and the exponential stability for infinite dimensional systems. Unfortunately, verification of the Riesz basis generation is challenging even for extensively studied systems such as Euler-Bernoulli beam equations. Recently, a new approach has been suggested (1) to obtain a complete solution to the basis prop- erty of the following uniform Euler-Bernoulli beam equation under linear boundary feedback control: ⎧ ⎪

OLAP dimension constraints

Abstract: In multidimensional data models intended for online analytic processing (OLAP), data are viewed as points in a multidimensional space. Each dimension has structure, described by a directed graph of categories, a set of members for each category, and a child/parent relation between members. An important application of this structure is to use it to infer summarizability, that is, whether an aggregate view defined for some category can be correctly derived from a set of precomputed views defined for other categories. A dimension is called heterogeneous if two members in a given category are allowed to have ancestors in different categories. In previous work, we studied the problem of inferring summarizability in a particular class of heterogeneous dimensions. In this paper, we propose a class of integrity constraints and schemas that allow us to reason about summarizability in general heterogeneous dimensions. We introduce the notion of frozen dimensions, which are minimal homogeneous dimension instances representing the different structures that are implicitly combined in a heterogeneous dimension. Frozen dimensions provide the basis for efficiently testing implication of dimension constraints, and are useful aid to understanding heterogeneous dimensions. We give a sound and complete algorithm for solving the implication of dimension constraints, that uses heuristics based on the structure of the dimension and the constraints to speed up its execution. We study the intrinsic complexity of the implication problem, and the running time of our algorithm.

Least squares fitting of an affine function and strength of association for interval-valued data

Abstract: The ultimate goal of this paper is to determine a measure of the degree of dependence between two interval-valued random sets, when the dependence is intended in the sense of an affine function relating these random elements. For this purpose, a general study on the least squares fitting of an affine function for interval-valued data is first carried out, where the least squares method we will present considers that squared residuals are based on a generalized metric on the space of nonempty compact intervals, and output and input random mechanisms are modelled by means of convex compact random sets. For the general case of nondegenerate convex compact random sets, solutions are presented in an algorithmic way, and the few cases leading to nonunique solutions are characterized. On the basis of this regression study we later introduce and analyze a well-defined determination coefficient of two interval-valued random sets, which will allow us to quantify the strength of association between them, and an algorithm for the computation of the coefficient has been also designed. Finally, a real-life example illustrates the study developed in the paper.

Learning Sparse Multiscale Image Representations

Abstract: We describe a method for learning sparse multiscale image representations using a sparse prior distribution over the basis function coefficients. The prior consists of a mixture of a Gaussian and a Dirac delta function, and thus encourages coefficients to have exact zero values. Coefficients for an image are computed by sampling from the resulting posterior distribution with a Gibbs sampler. The learned basis is similar to the Steerable Pyramid basis, and yields slightly higher SNR for the same number of active coefficients. De-noising using the learned image model is demonstrated for some standard test images, with results that compare favorably with other denoising methods.

Beyond stabilizer codes .I. Nice error bases

Abstract: Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.

Problems for Ordinary Differential Equations with Transmission Conditions

Abstract: We investigate a boundary-functional problem with transmission conditions for ordinary differential-operator equation in Sobolev spaces with a weight. We prove an isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis of a system of root functions of the problem. Obtained results in the article are new even in case of Sobolev spaces without the weight.

Electromagnetic field representations and computations in complex structures III: network representations of the connection and subdomain circuits

Abstract: When complex structures are divided into subdomains for electromagnetic field computations, it is necessary to specify which of the tangential electromagnetic field components at the boundaries may be regarded as independent and which have to be treated as dependent. In this third of our three-part study, we establish some rules for the choice of primary and secondary fields at the connection interface, i.e. at the interface between adjacent subdomains. The discretized field continuity equations at the connection interface provide the connection network whose canonical forms are illustrated, and it is shown that the normalized scattering matrix is symmetric, orthogonal and unitary. Tellegen's theorem is introduced in order to provide the basis for consistent choice of primary and secondary fields and for deriving canonical forms of the connection network. The field problem is systematically treated by partitioning and by specifying canonical Foster representations for the subcircuits. Connection between different subdomains is obtained by selecting the appropriate independent field quantities via Tellegen's theorem. For each subdomain, as well as for the entire circuit, an equivalent circuit extraction procedure is feasible, either in closed form for subdomains amenable to analytical treatment or via the relevant pole structure description when a numerical solutions is available. Moreover, circuit-based resonant expansions in the complex frequency plane are provided here for the analytic dyadic Green's functions in [1]. Copyright © 2002 John Wiley & Sons, Ltd.