Gram-Schmidt Orthonormalization Process Research Articles

Implementation of photon partial distinguishability in a quantum optical circuit simulation

We are concerned with numerical simulations of quantum optical circuits under certain realistic conditions, specifically that photon quantum states are not perfectly indistinguishable. The partial photon distinguishability presents a serious limitation in implementation of optical quantum information processing. In order to properly assess its effect on quantum information protocols, accurate numerical simulations, which closely emulate quantum circuit operations, are essential. Our specific objective is to provide a computer implementation of the partial photon distinguishability which is in principle applicable to existing simulation techniques used for ideal quantum circuits and which avoids a need for their significant modification. Our approach is based on the Gram-Schmidt orthonormalization process, which is well suited for our purpose. Photonic quantum states are represented by wavepackets which contain information on their time and frequency distributions. In order to account for the partial photon distinguishability, we expand the number of degrees of freedom associated with the circuit operation extending the definition of the photon channels to incorporate wavepacket degrees of freedom. This strategy allows to define delay operations in the same footing as the linear optical elements.

Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative

<abstract><p>In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.</p></abstract>

Stable computation of higher order Charlier moments for signal and image reconstruction

The computation of high-order discrete orthogonal moments is very complex and unstable due to fluctuations in numerical polynomial values. In this paper, we will present new algorithms for the stable computation of high-order discrete orthogonal Charlier polynomials and moments. These algorithms are based on the use of modified recurrence relations of Charlier polynomials with respect to the order n and with respect to the variable x, and on the use of the modified Gram-Schmidt orthonormalization process (GSOP). The algorithms in question allow the cancellation of terms that cause the numerical fluctuations of Charlier polynomial values during the recursive calculations. They also preserve the orthogonality property of high-order Charlier polynomials, which enables an efficient analysis of large-size bio-signals and 2D/3D images. Therefore, the simulation results prove the efficiency of the proposed algorithms when it comes to reconstructing large-size bio-signals and images.

Robust Kernel Principal Component Analysis With ℓ2,1-Regularized Loss Minimization

Principal component analysis (PCA) is a widely used unsupervised method for dimensionality reduction. The kernelized version is called kernel principal component analysis (KPCA), which can capture the nonlinear data structure. KPCA is derived from the Gram matrix, which is not robust when outliers exist in the data. This may yield the principal axis in the feature space deviated by outliers, leading to misinterpretation of the principal components. In this paper, we propose a robust method for KPCA with a reformulation in Euclidean space to construct a robust KPCA method, where an error measurement is introduced into the loss function, and l 2,1 -regularization is added to the loss function. The idea of l 2,1 -regularization of the proposed method is motivated by sparse PCA via variable projection. However, because orthogonality is not satisfied in the proposed method, orthonormal bases are obtained by using the Gram-Schmidt orthonormalization process. In the experiments, a toy example and real data are used for outlier detection to verify the method's performance and effectiveness. In the toy example, the proposed method reduces the influence of outliers and detects more outliers than KPCA. For the real data, the proposed method improves detection in comparison to other existing methods.

Performance Evaluation of DS-CDMA System based Orthogonal Chaotic Vectors over Rayleigh Fading Channel

codes with low cross-correlation values are needed to suppress the effect of multiple access interference (MAI). The chaotic system can be used to generate very large number of sequences due to its sensitive property to initial conditions but due to the quasi-orthogonal nature of chaotic sequences the MAI between active users is generated which leads to degrade the performance of the system. Another type of chaotic sequences with zero values of cross-correlation called orthogonal chaotic vectors (OCVs) are generated by using logistic map and the basic idea of Gram-Schmidt ortho- normalization process. In this paper, performance of OCVs in DS-CDMA system for transmission data for different active users over different channels (Rayleigh Flat fading channel and Rayleigh Frequency selective channel) is given. Simulation results show that the DS-CDMA based on OCVs gives better BER performance than other systems. The results also show that the system based on OCVs has the same BER performance regardless of the number of users under the effect of flat fading channel. While at BER=10 −3 , number of active users =8 and spread factor 160 there is about 6dB performance difference between the system based on OCVs and system based on Non-OCVs under the effect of time invariant frequency selective fading channel and this difference is decreased to 1dB when spread factor is 320. Keywordsrthogonal chaotic vectors, Non-Orthogonal chaotic vectors, Gram-Schmidt ortho-normalization process, Fading channel.

Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering.

Gram-Schmidt orthonormalization is a very fast and efficient method for the fringe pattern phase demodulation. It requires only two arbitrarily phase-shifted frames. Images are treated as vectors and upon orthogonal projection of one fringe vector onto another the quadrature fringe pattern pair is obtained. Orthonormalization process is very susceptible, however, to noise, uneven background and amplitude modulation fluctuations. The Hilbert-Huang transform based preprocessing is proposed to enhance fringe pattern phase demodulation by filtering out the spurious noise and background illumination and performing fringe normalization. The Gram-Schmidt orthonormalization process error analysis is provided and its filtering-expanded capabilities are corroborated analyzing DSPI fringes and performing amplitude demodulation of Bessel fringes. Synthetic and experimental fringe pattern analyses presented to validate the proposed technique show that it compares favorably with other pre-filtering schemes, i.e., Gaussian filtering and continuous wavelet transform.

Principles of linear algebra with Maple

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

Construction of Shape Features for the Representation of Full-Field Displacement/Strain Data

The achievement of high levels of confidence in finite element models involves their validation using measured responses such as static strains or vibration mode shapes. A huge amount of data with a high level of information redundancy is usually obtained in both the detailed finite element prediction and the full-field measurements so that achieving a meaningful validation becomes a challenging problem. In order to extract useful shape features from such data, image processing and pattern recognition techniques may be used. One of the most commonly adopted shape feature extraction procedures is the Fourier transform in which the original data may be expressed as a set of coefficients (coordinates) of the decomposition kernels (bases) in the feature space. Localised effects can be detected by the wavelet transform. The acquired shape features are succinct and therefore simplify the model validation, based on the full-field data, allowing it to be achieved in a more effective and efficient way. In this paper, full-field finite element strain patterns of a plate with a centred circular hole are considered. A special set of orthonormal shape decomposition kernels based on the circular Zernike polynomials are constructed by the Gram-Schmidt orthonormalization process. It is found that the strain patterns can suitably be represented by only a very small number of shape features from the derived kernels.

Local Convergence Analysis of FastICA and Related Algorithms

The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.

Design of UWB Pulses in Terms of B-Splines

The present paper discusses a new construction of UWB pulses within the framework of soft-spectrum adaptation. The employed basis functions are B-splines having the following properties: (i) The B-splines are time-limited piecewise polynomials. (ii) The first-order B-splines are rectangular pulses and they converge band-limited functions at the limit that their order tends to infinity. (iii) There are an analog circuit and a fast digital filter for the generation of B-splines. Simple application of Gram-Schmidt orthonormalization process to the shifted B-splines results in a few basic pulses, which are well time-limited and have a broad band width, but do not comply with the FCC spectral mask. A constrained approximation technique is proposed for adaptively designing pulses so that they approximate target frequency characteristics. At the cost of using eleven shifted B-splines, an example set of four pulses comforting the FCC spectral mask is obtained.

Real Time Local Approximation of Deformations using Rotations

Additional realism can be achieved in computer generated images using smooth and increasingly complex deformations. Though significant effort has been spent on improving these deformations, no general method has been proposed yet to deal with rigid pieces connected to soft objects. This paper proposes a general framework to solve this problem. We will present several types of applications, such as flowing small objects in a deformation field, animating rigid features connected to some deformed object, or smoothly attached limbs to a deforming body. All the calculations presented here can be applied to any type of deformation, provided that the deformation at each point only depends on the point itself. Even though we can directly compute the result for some analytical deformation fields, we will show that a good sampling of the deformation in the area of interest is generally enough. One intermediate result consists of a practical method to find the best rotation that approximates a linear transformation. The proposed method is a superset of the Gram-Schmidt orthonormalization process, and is much easier to compute than global methods based on Taylor series.

Continuous time algorithm for Lyapunov exponents. I.

The Lyapunov exponent algorithm of Benettin et al. [C.R. Acad. Sci. Paris 286A, 431 (1978); Meccanica 15, 9 (1980)] and of Shimada and Nagashima [Prog. Theor. Phys. 61, 1605 (1979)] is converted to a continuous time algorithm. The Gram-Schmidt orthonormalization process is incorporated into the differential equations, making orthogonalization continuous. Differential equations for the k-dimensional fiducial volumes are also derived. It is further shown that this algorithm is a factorization of the fundamental matrix \ensuremath{\Phi} in the form \ensuremath{\Phi}(t)=E(t)W(t)${\mathit{E}}^{\mathrm{\ensuremath{-}}1}$(${\mathit{t}}_{0}$), where E(t) is orthonormal and W(t) is upper triangular. Numerical stability of the algorithm is considered, and it is shown that the standard Gram-Schmidt process can be used to stabilize the (possibly) unstable equations of motion for the orthonormal basis vectors. A numerical example is presented.

On the nonlinear convexity theorem of Kostant

A classical result of Schur and Horn [Sc, Ho] states that the set of diagonal elements of all n x n Hermitian matrices with fixed eigenvalues is a convex set in IRn. Kostant [Kt] has generalized this result to the case of any semisimple Lie group. This is often referred to as the linear convexity theorem of Kostant: picking up the diagonal of a Hermitian matrix is a linear operation. This result was later put into the framework of symplectic geometry by Atiyah [Ay] , Guillemin-Sternberg [Gu-St] and Duistermaat [Du1], the key argument being that the map that picks up the diagonal of a Hermitian matrix is a moment map for a Hamiltonian torus action and that the image of such a moment map is always a convex polytope. There is also a nonlinear version of the convexity theorem of Kostant. It has not been connected so far with symplectic geometry. The purpose of this paper is to make this connection. We first describe the nonlinear convexity theorem for the case of SL(n, q (see §2 for more details). The Gram-Schmidt orthonormalization process in linear algebra asserts that every nonsingular n x n complex matrix g can be uniquely written as the product g = kan for some unitary matrix k, positive diagonal matrix a, and strictly upper-triangular matrix n. We use A to denote the group of all positive diagonal matrices, and we call the matrix a in the above decomposition the A-component of g. The nonlinear convexity theorem of Kostant now asserts that the set of A-components of g as g runs through all positive definite Hermitian matrices with fixed eigenvalues a = (aJ ' ••• , an) is the convex hull of the set of all points obtained by permuting the coordinates of a. Here we identify A with its Lie algebra a via the exponential map; thus convexity in A makes sense. For a general semisimple Lie group G, the Gram-Schmidt decomposition is replaced by an Iwasawa decomposition G = KAN of G, and the permutation group that appeared above is replaced by the Weyl group W of (K, A). If an element g EGis written as the product g = kan for k E K, a E A,

Vortex Methods for Free Interface Problems

Numerical methods for the solution of free interface problems are reviewed. For two-dimensional problems, an application of the random vortex method is proposed in which the rotational and irrotational flows are first calculated and then reconstituted into the time-dependent velocity field through the use of Hodge's decomposition theorem. The irrotational part is calculated by conformally mapping the flow, bounded on one side by the interface, into a strip at every time step, followed by use of the Gram-Schmidt orthonormalization process to solve Laplace's equation for the velocity potential. An alternative for the irrotational flow calculation, in which the free interface is represented by a vortex sheet and the boundary integral method is applied, is also discussed. The rotational field is calculated by generating vortex sheets to satisfy the no-slip boundary conditions, and by following the convective and diffusive motion of the sheets and vortex blobs. The technique is shown to yield accurate results for damping of solitary waves on shallow liquids.

Heat Transfer in a Capillary Flow Emerging from a Reservoir

An extended Graetz problem is analyzed, with a semi-infinite axial domain and the Robin boundary condition on the heat-transfer wall. The heat-transfer problem examines a viscous fluid entering a cylindrical capillary from a reservoir. The capillary fluid is exchanging heat with the surrounding environment of prescribed temperature and thus the Robin boundary condition is employed on the wall. Since axial heat conduction is included in the analysis, a generalized Danckwerts boundary condition is shown to be most appropriate for the tube entrance. The energy equation is decomposed into a system of first-order partial differential equations, as in [4, 8, 9], to obtain a selfadjoint formalism. The Gram-Schmidt orthonormalization process is finally used to obtain what is technically an analytical solution, which is computationally simple and efficient. Other entrance boundary conditions are also discussed and analyzed.

Conjugated Graetz problems—II: Fluid-fluid problems

Conjugated Graetz problems [1], involving two co- or counter-currently flowing phases are discussed following the general formalism of [1]. Although the difficulty, which arises from the changing sign of the velocity profile over the total cross-section of the two-fluid conduit of the counter-current problems, may be readily handled by the general formalism, the inclusion of axial heat conduction in the analysis presents special difficulties in obtaining analytical and/or computationally efficient solutions. The present analysis shows that analytical and computationally efficient solutions may be obtained only for these problems where the temperature profile at the entrance of the heating section is known for at least one of the fluids. The solution of a class of problems with long heating sections is obtained untilizing the general formalism together with the Gram-Schmidt orthonormalization process in the spirit of [5]. Problems with low Peclet numbers for both fluids and with an adiabatic section preceding the heating section or problems with very short heating sections are also briefly discussed.

Remarks on a matrix transformation for linear differential equations

The remarks of this note are concerned with a result on transformations stated below as Theorem A, and are two-fold in nature: firstly, there are comments on the relation of this theorem to results of Perron [3] and Diliberto [1; 2], in the hope of correcting a misunderstanding that has arisen in this regard; secondly, there are remarks stressing two general properties of admissible transformation matrices which together afford a very elementary matrix proof of Theorem A. Matrix notation will be used throughout, with a vector considered as a one-column matrix. If M is a matrix then the corresponding transpose and conjugate-transpose matrices are denoted by M and M*, respectively. The symbol YI will be employed for the norm (y*y)112 of a vector y. For M-IIMiAlI, (i, j=1, , n), the corresponding lower case bold-face letter mj will denote the jth column vector of M. In particular, if M is a nonsingular n Xn matrix, and N is the unitary matrix whose sequence of column vectors nli, * * *, nn is the set of vectors obtained by applying the Gram-Schmidt orthonormalization process to the sequence of column vectors mi, * , Mn of M, then we shall write simply N=gs [M]. nonsingular matrix Y(x) whose column vectors are solutions of (1) will be called a fundamental matrix for (1). matrix M(x) will be termed on A whenever its individual elements are bounded functions on this interval.