Showing papers on "Basis (linear algebra) published in 2015"

Reduced Basis Methods for Partial Differential Equations: An Introduction

Abstract: 1 Introduction.- 2 Representative problems: analysis and (high-fidelity) approximation.- 3 Getting parameters into play.- 4 RB method: basic principle, basic properties.- 5 Construction of reduced basis spaces.- 6 Algebraic and geometrical structure.- 7 RB method in actions.- 8 Extension to nonaffine problems.- 9 Extension to nonlinear problems.- 10 Reduction and control: a natural interplay.- 11 Further extensions.- 12 Appendix A Elements of functional analysis.

Person Re-Identification by Iterative Re-Weighted Sparse Ranking

Abstract: In this paper we introduce a method for person re-identification based on discriminative, sparse basis expansions of targets in terms of a labeled gallery of known individuals. We propose an iterative extension to sparse discriminative classifiers capable of ranking many candidate targets. The approach makes use of soft- and hard- re-weighting to redistribute energy among the most relevant contributing elements and to ensure that the best candidates are ranked at each iteration. Our approach also leverages a novel visual descriptor which we show to be discriminative while remaining robust to pose and illumination variations. An extensive comparative evaluation is given demonstrating that our approach achieves state-of-the-art performance on single- and multi-shot person re-identification scenarios on the VIPeR, i-LIDS, ETHZ, and CAVIAR4REID datasets. The combination of our descriptor and iterative sparse basis expansion improves state-of-the-art rank-1 performance by six percentage points on VIPeR and by 20 on CAVIAR4REID compared to other methods with a single gallery image per person. With multiple gallery and probe images per person our approach improves by 17 percentage points the state-of-the-art on i-LIDS and by 72 on CAVIAR4REID at rank-1. The approach is also quite efficient, capable of single-shot person re-identification over galleries containing hundreds of individuals at about 30 re-identifications per second.

Segmented contracted basis sets optimized for nuclear magnetic shielding.

Abstract: A family of segmented contracted basis sets is proposed, denoted pcSseg-n, which are optimized for calculating nuclear magnetic shielding constants. For the elements H–Ar, these are computationally more efficient than the previously proposed general contracted pcS-n basis sets, and the new basis sets are extended to also include the elements K–Kr. The pcSseg-n basis sets are optimized at the density functional level of theory, but it has been shown previously that these property-optimized basis sets are also suitable for calculating shielding constants with correlated wave function methods. The pcSseg-n basis sets are available in qualities ranging from (unpolarized) double-ζ to pentuple-ζ quality and should be suitable for both routine and benchmark calculations of nuclear magnetic shielding constants. The ability to rigorously separate basis set and method errors should aid in developing more accurate methods.

Enhancing Differential Evolution Utilizing Eigenvector-Based Crossover Operator

Abstract: Differential evolution has been shown to be an effective methodology for solving optimization problems over continuous space. In this paper, we propose an eigenvector-based crossover operator. The proposed operator utilizes eigenvectors of covariance matrix of individual solutions, which makes the crossover rotationally invariant. More specifically, the donor vectors during crossover are modified, by projecting each donor vector onto the eigenvector basis that provides an alternative coordinate system. The proposed operator can be applied to any crossover strategy with minimal changes. The experimental results show that the proposed operator significantly improves DE performance on a set of 54 test functions in CEC 2011, BBOB 2012, and CEC 2013 benchmark sets.

Estimation and Testing

Abstract: In this chapter we turn to estimation, the step in the marketing model building process that follows model specification, where we consider methods and procedures for obtaining numerical values for the model parameters in the model Throughout this chapter we will mainly consider the case where the independent variables are linearly related to the dependent variable or where one or more variables can be transformed in such a way that the relation between the variables becomes linear In such cases, it is appropriate to estimate a linear model Linear models do not only provide reasonable specifications for many practical applications, they are also attractive for a careful treatment of model assumptions, and for a conceptual explanation of the basis for the assumptions Most of the principles that apply to the linear model remain relevant as long as nonlinear effects for the original variables can be accommodated by transforming variables (so that the transformed variables are linearly related)

Biased tracers and time evolution

Abstract: We study the effect of time evolution on galaxy bias. We argue that at any order in perturbations, the galaxy density contrast can be expressed in terms of a finite set of locally measurable operators made of spatial and temporal derivatives of the Newtonian potential. This is checked in an explicit third order calculation. There is a systematic way to derive a basis for these operators. This basis spans a larger space than the expansion in gravitational and velocity potentials usually employed, although new operators only appear at fourth order. The basis is argued to be closed under renormalization. Most of the arguments also apply to the structure of the counter-terms in the effective theory of large-scale structure.

Reachable set estimation for discrete-time linear systems with time delays

Abstract: SUMMARY This paper is concerned with the reachable set estimation problem for discrete-time linear systems with multiple constant delays and bounded peak inputs. The objective is to check whether there exists a bounded set that contains all the system states under zero initial conditions. First, delay-dependent conditions for the solvability of the addressed problem are derived by employing a novel Lyapunov–Krasovskii functional. The obtained conditions are expressed in terms of matrix inequalities, which are linear when only one scalar variable is fixed. On the basis of these conditions, an ellipsoid containing the reachable set of the considered system is obtained. An approach for determining the smallest ellipsoid is also provided. Second, the approach and results developed in the first stage are generalized to the case of systems with polytopic parameter uncertainties, and delay-dependent conditions are given in the form of relaxed matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.

Weakly Monotonic Averaging Functions

Abstract: Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important nonmonotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions, and proves that several families of important nonmonotonic means are actually weakly monotonic averaging functions. Specifically, we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalized mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty-based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial-tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which monotonic aggregation functions and nonmonotonic averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.

Convolutional Sparse Coding for Trajectory Reconstruction

Abstract: Trajectory basis Non-Rigid Structure from Motion (NRSfM) refers to the process of reconstructing the 3D trajectory of each point of a non-rigid object from just their 2D projected trajectories. Reconstruction relies on two factors: (i) the condition of the composed camera & trajectory basis matrix, and (ii) whether the trajectory basis has enough degrees of freedom to model the 3D point trajectory. These two factors are inherently conflicting. Employing a trajectory basis with small capacity has the positive characteristic of reducing the likelihood of an ill-conditioned system (when composed with the camera) during reconstruction. However, this has the negative characteristic of increasing the likelihood that the basis will not be able to fully model the object’s “true” 3D point trajectories. In this paper we draw upon a well known result centering around the Reduced Isometry Property (RIP) condition for sparse signal reconstruction. RIP allow us to relax the requirement that the full trajectory basis composed with the camera matrix must be well conditioned. Further, we propose a strategy for learning an over-complete basis using convolutional sparse coding from naturally occurring point trajectory corpora to increase the likelihood that the RIP condition holds for a broad class of point trajectories and camera motions. Finally, we propose an $\ell_{1}$ inspired objective for trajectory reconstruction that is able to “adaptively” select the smallest sub-matrix from an over-complete trajectory basis that balances (i) and (ii). We present more practical 3D reconstruction results compared to current state of the art in trajectory basis NRSfM.

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.

An Unbiased Hessian Representation for Monte Carlo PDFs

Abstract: We develop a methodology for the construction of a Hessian representation of Monte Carlo sets of parton distributions, based on the use of a subset of the Monte Carlo PDF replicas as an unbiased linear basis, and of a genetic algorithm for the determination of the optimal basis. We validate the methodology by first showing that it faithfully reproduces a native Monte Carlo PDF set (NNPDF3.0), and then, that if applied to Hessian PDF set (MMHT14) which was transformed into a Monte Carlo set, it gives back the starting PDFs with minimal information loss. We then show that, when applied to a large Monte Carlo PDF set obtained as combination of several underlying sets, the methodology leads to a Hessian representation in terms of a rather smaller set of parameters (CMC-H PDFs), thereby providing an alternative implementation of the recently suggested Meta-PDF idea and a Hessian version of the recently suggested PDF compression algorithm (CMC-PDFs). The mc2hessian conversion code is made publicly available together with (through LHAPDF6) a Hessian representations of the NNPDF3.0 set, and the CMC-H PDF set.

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.

Polarization contributions to intermolecular interactions revisited with fragment electric-field response functions

Abstract: The polarization energy in intermolecular interactions treated by self-consistent field electronic structure theory is often evaluated using a constraint that the atomic orbital (AO) to molecular orbital transformation is blocked by fragments. This approach is tied to AO basis sets, overestimates polarization energies in the overlapping regime, particularly in large AO basis sets, and lacks a useful complete basis set limit. These problems are addressed by the construction of polarization subspaces based on the responses of isolated fragments to weak electric fields. These subspaces are spanned by fragment electric-field response functions, which can capture effects up to the dipole (D), or quadrupole (DQ) level, or beyond. Schemes are presented for the creation of both non-orthogonal and orthogonal fragment subspaces, and the basis set convergence of the polarization energies computed using these spaces is assessed. Numerical calculations for the water dimer, water-Na(+), water-Mg(2+), water-F(-), and water-Cl(-) show that the non-orthogonal DQ model is very satisfactory, with small differences relative to the orthogonalized model. Additionally, we prove a fundamental difference between the polarization degrees of freedom in the fragment-blocked approaches and in constrained density schemes. Only the former are capable of properly prohibiting charge delocalization during polarization.

Adaptive h‐refinement for reduced‐order models

Abstract: Our work presents a method to adaptively refine reduced-order models a posteriori without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive h-refinement: it enriches the reduced-basis space online by ‘splitting’ a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive k-means clustering of the state variables using snapshot data. This method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Furthermore, it enables the reduced-order model to satisfy any prescribed error tolerance regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.

NMF-based Target Source Separation Using Deep Neural Network

Abstract: Non-negative matrix factorization (NMF) is one of the most well-known techniques that are applied to separate a desired source from mixture data. In the NMF framework, a collection of data is factorized into a basis matrix and an encoding matrix. The basis matrix for mixture data is usually constructed by augmenting the basis matrices for independent sources. However, target source separation with the concatenated basis matrix turns out to be problematic if there exists some overlap between the subspaces that the bases for the individual sources span. In this letter, we propose a novel approach to improve encoding vector estimation for target signal extraction. Estimating encoding vectors from the mixture data is viewed as a regression problem and a deep neural network (DNN) is used to learn the mapping between the mixture data and the corresponding encoding vectors. To demonstrate the performance of the proposed algorithm, experiments were conducted in the speech enhancement task. The experimental results show that the proposed algorithm outperforms the conventional encoding vector estimation scheme.

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.

Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction

Abstract: Projection-based model reduction techniques rely on the definition of a small dimensional subspace in which the solution is approximated. Using local subspaces reduces the dimensionality of each subspace and enables larger speedups. Transitions between local subspaces require special care and updating the reduced bases associated with each subspace increases the accuracy of the reduced-order model. In the present work, local reduced basis updates are considered in the case of hyper-reduction, for which only the components of state vectors and reduced bases defined at specific grid points are available. To enable local reduced basis updates, two comprehensive approaches are proposed. The first one is based on an offline/online decomposition. The second approach relies on an approximated metric acting only on those components where the state vector is defined. This metric is computed offline and used online to update the local bases. An analysis of the error associated with this approximated metric is then conducted and it is shown that the metric has a kernel interpretation. Finally, the application of the proposed approaches to the model reduction of two nonlinear physical systems illustrates their potential for achieving large speedups and good accuracy.

Semiexperimental Equilibrium Structures for Building Blocks of Organic and Biological Molecules: The B2PLYP Route

Abstract: The B2PLYP double hybrid functional, coupled with the correlation-consistent triple-ζ cc-pVTZ (VTZ) basis set, has been validated in the framework of the semiexperimental (SE) approach for deriving accurate equilibrium structures of molecules containing up to 15 atoms. A systematic comparison between new B2PLYP/VTZ results and several equilibrium SE structures previously determined at other levels, in particular B3LYP/SNSD and CCSD(T) with various basis sets, has put in evidence the accuracy and the remarkable stability of such model chemistry for both equilibrium structures and vibrational corrections. New SE equilibrium structures for phenylacetylene, pyruvic acid, peroxyformic acid, and phenyl radical are discussed and compared with literature data. Particular attention has been devoted to the discussion of systems for which lack of sufficient experimental data prevents a complete SE determination. In order to obtain an accurate equilibrium SE structure for these situations, the so-called templating mo...

A TGV-Based Framework for Variational Image Decompression, Zooming, and Reconstruction. Part I: Analytics ∗

Abstract: A variational model for image reconstruction is introduced and analyzed in function space. Specific to the model is the data fidelity, which is realized via a basis transformation with respect to a Riesz basis followed by interval constraints. This setting in particular covers the task of reconstructing images constrained to data obtained from JPEG or JPEG 2000 compressed files. As image prior, the total generalized variation (TGV) functional of arbitrary order is employed. The present paper, the first of two works that deal with both analytical and numerical aspects of the model, provides a comprehensive analysis in function space and defines concrete instances for particular applications. A new, noncoercive existence result and optimality conditions, including a characterization of the subdifferential of the TGV functional, are obtained in the general setting.

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

Abstract: We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

Hybrid functionals for large periodic systems in an all-electron, numeric atom-centered basis framework

Abstract: We describe a framework to evaluate the Hartree–Fock exchange operator for periodic electronic-structure calculations based on general, localized atom-centered basis functions. The functionality is demonstrated by hybrid-functional calculations of properties for several semiconductors. In our implementation of the Fock operator, the Coulomb potential is treated either in reciprocal space or in real space, where the sparsity of the density matrix can be exploited for computational efficiency. Computational aspects, such as the rigorous avoidance of on-the-fly disk storage, and a load-balanced parallel implementation, are also discussed. We demonstrate linear scaling of our implementation with system size by calculating the electronic structure of a bulk semiconductor (GaAs) with up to 1,024 atoms per unit cell without compromising the accuracy.

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

Abstract: A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-P k polynomial tensors and the displacement is approximated by C −1-P k−1 polynomial vectors, for all k ⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued P k−1 polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.

Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs

Abstract: In this work we discuss the dynamically orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the $S$-terms DO solution by the corresponding $S$-terms best approximation, i.e., the truncated $S$-terms Karhunen--Loeve expansion at each time instant. The bound is applicable on the largest time interval in which the best $S$-terms approximation is continuously time differentiable. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoret...

Riemannian Geometry and Statistical Machine Learning

Abstract: Statistical machine learning algorithms deal with the problem of selecting an appropriate statistical model from a model space Θ based on a training set xiN i=1 ⊂ X or xi,y iN i=1 ⊂ X × Y. In doing so they either implicitly or explicitly make assumptions on the geometries of the model space Θ and the data space X. Such assumptions are crucial to the success of the algorithms as different geometries are appropriate for different models and data spaces. By studying these assumptions we are able to develop new theoretical results that enhance our understanding of several popular learning algorithms. Furthermore, using geometrical reasoning we are able to adapt existing algorithms such as radial basis kernels and linear margin classifiers to non-Euclidean geometries. Such adaptation is shown to be useful when the data space does not exhibit Euclidean geometry. In particular, we focus in our experiments on the space of text documents that is naturally associated with the Fisher information metric on corresponding multinomial models.

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

Abstract: A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovasz (J. Comb. Theory Ser. A, 95:169–179, 2001), but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.

Methods to calculate linear combination pre-coders for mimo wireless communication systems

Abstract: In a wireless communication system having an antenna array selectively configured to transmit channel state information reference signals (CSI-RS) using a plurality of antenna ports and basis beam vectors selected from a master beam set or retrieved from memory, a codebook enables selection of a linear combination of at least a subset of the beams and co-phases and coefficients for each selected beam, where the co-phases determine the co-phasing weights for the selected beams for a cross-polarized antenna array, and the coefficients determine the linear combination of the selected beams. Feedback contains an indication of channel state information (CSI) for the set of selected or retrieved basis beam vectors, the selected beams, co-phases, and coefficients. The CSI includes at least precoding matrix information (PMI) corresponding to a precoding vector based on a set of the basis beam vectors for the selected beams, corresponding co-phases, and corresponding coefficients.

Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions

Abstract: We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements. We prove the spectral duality for SU(2) Nekrasov functions and discuss its meaning for conformal blocks. We also clarify the relation between topological strings and q-Liouville vertex operators.