Matrix Perturbation Theory Research Articles

Dynamic response analysis for structures with interval parameters

In this paper, a new method to solve the dynamic response problem for structures with interval parameters is presented. It is difficult to obtain all possible solutions with sharp bounds even an optimum scheme is adopted when there are many interval structural parameters. With the interval algorithm, the expressions of the interval stiffness matrix, damping matrix and mass matrices are developed. Based on the matrix perturbation theory and interval extension of function, the upper and lower bounds of dynamic response are obtained, while the sharp bounds are guaranteed by the interval operations. A numerical example, dynamic response analysis of a box cantilever beam, is given to illustrate the validity of the present method.

Microscopic interface asymmetry and spin-splitting of electron subbands in semiconductor quantum structures

The microscopic interface asymmetry of (001)-grown semiconductor heterostructures that gives rise to heavy-light hole coupling even at zero in-plane wave vector k ‖, modifies also the subband dispersion of confined electrons. Starting from a multiband envelope formulation we apply matrix perturbation theory to derive explicit expressions caused by this interface asymmetry, which in the 2×2 conduction band Hamiltonian appear as a warping and a spin-splitting term. The warping term results in an inequivalence of the dispersion along [110] and [1 1 0] as required by the microscopic C 2v symmetry, while the spin-splitting term has the same structure as the corresponding term derived from the zinc-blende bulk inversion asymmetry. Implications with respect to spin-relaxation will be discussed.

Parameter depending state space descriptions of index-2-matrix polynomials

To a quadratic matrix polynomial P with coefficients in R n×n , which originated from an electrical network and depending on a parameter vector q∈ R ϱ , a matrix A( q) and a parameter set Ω are assigned such that for all q∈Ω the eigenvalues of A( q) coincide with the zeros of det P(.; q). To find A( q) and the corresponding parameter set Ω two algorithms are proposed where the first one is similar to the Algorithm 3.6 of Van Dooren [Linear Algebra Appl. 27 (1979) 103]. The reason to compute A( q) parameter depending is given by the desire to apply the matrix perturbation theory of Stewart and Sun [Matrix Perturbation Theory (Academic Press, 1990)] to study the influence of q on the zeros of the determinant of P. The assumption that P is derived from the Laplace transform of a DAE system describing an electrical network implies that its coefficients admit representations containing sums of the form ∑ v k q k w k T, where v k and w k are the unit vectors or the nonvanishing difference of two such vectors. This circumstance is decisive for the efficiency of our two algorithms.

Specification of Varma Echelon form Models

Vector autoregressive moving average (VARMA) models have certain advantages over pure vector autoregressive (VAR) models because they may permit more parsimonious representations of data generation processes. Moreover for many economic data, for instance, temporal and contemporaneous aggregation leads directly to mixed VARMA models. The major problems in VARMA modelling are related to specification unique representation. To solve these problems, we consider a VARMA model in echelon form, which is automatically identified. To specify the orders of the model, we propose an algorithm which selects a maximal set of linearly independent rows of the Hankel matrix of the estimated covariances. This set is obtained by sequentially testing the smallest singular value of the Hankel matrix and yields estimates of Kronecker indices which characterise the echelon form. Using the matrix perturbation theory, the asymptotic distribution of the test statistic is seen to be chi-squared. An example based on economic data series illustrates the procedure and demonstrates its feasibility in practice. Its performance is compared with other existing methods.

Error Characterization of the Factorization Method

This paper studies error characterization of the factorization method for 3-D shape and motion recovery from image sequences using matrix perturbation theory and covariance propagation for linear models. Given the 2-D projections of a set of feature points across multiple image frames and small perturbations/covariances of the feature point coordinates, first-order perturbation and covariance matrices of 3-D shape and motion are derived and validated with the ground truth. This work provides quantitative analysis of error sensitivity of 3-D shape and motion estimation subject to small feature correspondence errors. It can be used to pinpoint system performance, such as which point/frame/coordinate has relatively higher uncertainty, and gain insight for further improvement. We show the 3-D shape uncertainty as ellipsoids on top of the 3-D reconstruction as an enhanced visualization, leading to better use of the factorization method in engineering applications. Experimental results are demonstrated to support the analysis and to show how the error analysis and error measures can be used.

SENSITIVITY ANALYSES OF SENSOR LOCATIONS FOR VIBRATION CONTROL AND DAMAGE DETECTION OF THIN-PLATE SYSTEMS

This paper addresses the sensitivity problem of sensor locations for vibration control and damage detection of thin-plate systems with parameter variation or noise. For vibration control, the technique for robust determination of sensor locations is presented. Based on the spectral condition number of the Hankel matrix, the optimal sensor locations (OSLs) can be determined, and the effect of noise on the OSLs is investigated using the matrix perturbation theory. For damage detection, the damage locations can be determined using the damage index β derived from the curvature modes. The sensitivity analysis of sensor locations on the detection result for systems with parameter variation is presented. Some experiments are carried out to verify the effectiveness of the proposed method.

Modified Cholesky Factorizations in Interior-Point Algorithms for Linear Programming

We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach.

Identification of physiological systems: a robust method for non-parametric impulse response estimation.

The identification of non-parametric impulse response functions (IRFs) from noisy finite-length data records is analysed using the techniques of matrix perturbation theory. Based on these findings, a method for IRF estimation is developed that is more robust than existing techniques, particularly when the input is non-white. Furthermore, methods are developed for computing confidence bounds on the resulting IRF estimates. Monte Carlo simulations are used to assess the capabilities of this new method and to demonstrate its superiority over classical techniques. An application to the identification of dynamic ankle stiffness in humans is presented.

Physiologically based pharmacokinetic modeling of a homologous series of barbiturates in the rat: a sensitivity analysis.

Sensitivity analysis studies the effects of the inherent variability and uncertainty in model parameters on the model outputs and may be a useful tool at all stages of the pharmacokinetic modeling process. The present study examined the sensitivity of a whole-body physiologically based pharmacokinetic (PBPK) model for the distribution kinetics of nine 5-n-alkyl-5-ethyl barbituric acids in arterial blood and 14 tissues (lung, liver, kidney, stomach, pancreas, spleen, gut, muscle, adipose, skin, bone, heart, brain, testes) after i.v. bolus administration to rats. The aims were to obtain new insights into the model used, to rank the model parameters involved according to their impact on the model outputs and to study the changes in the sensitivity induced by the increase in the lipophilicity of the homologues on ascending the series. Two approaches for sensitivity analysis have been implemented. The first, based on the Matrix Perturbation Theory, uses a sensitivity index defined as the normalized sensitivity of the 2-norm of the model compartmental matrix to perturbations in its entries. The second approach uses the traditional definition of the normalized sensitivity function as the relative change in a model state (a tissue concentration) corresponding to a relative change in a model parameter. Autosensitivity has been defined as sensitivity of a state to any of its parameters; cross-sensitivity as the sensitivity of a state to any other states' parameters. Using the two approaches, the sensitivity of representative tissue concentrations (lung, liver, kidney, stomach, gut, adipose, heart, and brain) to the following model parameters: tissue-to-unbound plasma partition coefficients, tissue blood flows, unbound renal and intrinsic hepatic clearance, permeability surface area product of the brain, have been analyzed. Both the tissues and the parameters were ranked according to their sensitivity and impact. The following general conclusions were drawn: (i) the overall sensitivity of the system to all parameters involved is small due to the weak connectivity of the system structure; (ii) the time course of both the auto- and cross-sensitivity functions for all tissues depends on the dynamics of the tissues themselves, e.g., the higher the perfusion of a tissue, the higher are both its cross-sensitivity to other tissues' parameters and the cross-sensitivities of other tissues to its parameters; and (iii) with a few exceptions, there is not a marked influence of the lipophilicity of the homologues on either the pattern or the values of the sensitivity functions. The estimates of the sensitivity and the subsequent tissue and parameter rankings may be extended to other drugs, sharing the same common structure of the whole body PBPK model, and having similar model parameters. Results show also that the computationally simple Matrix Perturbation Analysis should be used only when an initial idea about the sensitivity of a system is required. If comprehensive information regarding the sensitivity is needed, the numerically expensive Direct Sensitivity Analysis should be used.

Stochastic perturbation finite elements

This paper extends the stochastic perturbation method to vector-valued and matrix-valued functions. The numerical method for the response and reliability of uncertain structures is formulated using matrix calculus, Kronecker algebra and perturbation theory. Random variables and system derivatives are conveniently arranged into two-dimensional matrices and generalized mathematical formulae are obtained. The results derived are easily amenable to computational procedures.

Generalized probabilistic perturbation method for static analysis

This paper studies the static response and reliability of uncertain structures with vector-valued and matrix-valued functions. The finite element analysis method of uncertain structures is based on matrix calculus, Kronecker algebra and perturbation theory. Random variables and system derivatives are conveniently arranged into 2D matrices and generalized mathematical formulae for probabilistic perturbation are obtained.

Rejecting outliers and estimating errors in an orthogonal-regression framework

Least squares minimization is by nature global and, hence, vulnerable to distortion by outliers. We present a novel technique to reject outliers from an m -dimensional data set when the underlying model is a hyperplane (a line in two dimensions, a plane in three dimensions). The technique has a sound statistical basis and assumes that Gaussian noise corrupts the otherwise valid data. The majority of alternative techniques available in the literature focus on ordinary least squares , where a single variable is designated to be dependent on all others - a model that is often unsuitable in practice. The method presented here operates in the more general framework of orthogonal regression , and uses a new regression diagnostic based on eigendecomposition. It subsumes the traditional residuals scheme and, using matrix perturbation theory, provides an error model for the solution once the contaminants have been removed.

Novel methods for calculating the fine vibronic spectra of polyatomic molecules

Novel approximate methods for calculating the vibrational structure of the electronic spectra of polyatomic molecules—a method for the direct calculation of the overlap integrals of vibrational wave functions for the electronic states involved in a transition and a variational method for the solution of the vibrational problem for the excited states—are discussed. The methods are based on the consideration of the displacement and entanglement of normal coordinates, the quasiorthogonality of the Dushinsky transformation, and the classification of the states by total vibrational quantum numbers. Matrix perturbation theory is employed. It is shown that the accuracy of these methods compares well with the accuracy of the available “exact” techniques (the errors are ∼1 cm−1 for frequency and 10% for relative intensity). At the same time, calculations by the new methods are performed more than two orders of magnitude faster than by the previously known methods.

An examination of procedures for determining body segment attitude and position from noisy biomechanical data

For the kinematic analysis of movement it is necessary to determine the position and attitude of rigid bodies in some specified reference frame. In biomechanics, these rigid bodies are usually segments of the human body, and the position of a distal segment is normally defined relative to a proximal segment. It was the purpose of this study to compare the accuracy of three approaches for the determination of the position and attitude of rigid bodies under four different noise conditions which were designed to model the conditions found in biomechanics studies. One technique investigated assumes the data are error free 1; another uses matrix algebra and employs matrix perturbation theory 2; and the third is a least-squares procedure 3. The evaluation was performed using a computer simulation which attempted to model ‘typical’ experimental conditions found in biomechanical studies. The attitude of the distal rigid body was defined using helical angles, with these angles being generated using a random number generator. All three techniques were assessed by their ability to predict a set of known helical angles and position vectors under different noise conditions. The study demonstrated that the least-squares technique was the most accurate for determining the attitude matrix and position vector for the cases investigated. None of the techniques investigated could allow for anisotropic noise conditions, yet anisotropic noise conditions are often obtained when using measurement procedures common in biomechanics. The study also highlighted the need to low-pass filter data prior to the computation of the position and attitude of rigid bodies.

Selective training of feedforward artificial neural networks using matrix perturbation theory

Many training algorithms for feedforward neural networks suffer from slow convergence. A new training method is presented which exploits results from matrix perturbation theory for significant training time improvement. This theory is used to assess the effect of a particular training pattern on the weight estimates prior to its inclusion in any iteration. Data which do not significantly change the weights are not used in that iteration obviating the computational expense of updating.

Methods for calculation of electronic—vibrational spectra of polyatomic molecules Part II. Variational solution of the electronic-vibrational problem

A more refined method for variational solution of the vibrational problem in excited electronic states has been devised. Calculations of the vibrational structure of polyatomic molecule electronic spectra are performed generally in the adiabatic approximation without invoking additional restrictions. Consecutively taking into account the effects determined by the shift of the potential surface and mixing of normal coordinates by excitation underlies this method. Also the matrix perturbation theory is used. In this method it is possible to use the obtained simple analytic expressions and recurrent relations to compute the overlap integrals. Therefore there is no need to diagonalize a variational matix of enormous dimensions. The method is well suited for program realization and is highly efficient for calculations of complicated polyatomic molecules.

Partial inverse scattering method for three-dimensional heterogeneous biological bodies by using a matrix perturbation theory

A partial inverse scattering method for three-dimensional, arbitrarily shaped heterogeneous dielectrics or biological bodies is presented by using a matrix perturbation (MPT). When small perturbations in the dielectric characteristics of an a priori known biological body occur, a relation between the varied scattering fields and the dielectric characteristics is derived in matrix form. Furthermore, the perturbation solutions for inverse scattering problem are investigated. A reconstruction example is given to show the validity of the method. >

Matrix perturbation theory of the eigenequation [A][u] = Δ[u] and its application in electromagnetic fields

AbstractA novel matrix perturbation theory to the eigenequation [h][u] = Δ[u] is investigated, in which the perturbation solutions to eigenvalue Δ and eigenvector [u] are given. One of the applications of the above theory‐application in a microwave resonate cavity‐is researched. © 1993 John Wiley & Sons, Inc.

Matrix perturbation theory in field computation by moment methods

A novel matrix perturbation theory in field computation by the moment method is presented, which can be used to calculate the field distributions of a partially varied system rapidly based on its original form. Compared with the usual moment methods, the new theory not only has an equivalent precision, but can save 60 times the execution time. This theory is especially suitable for the inside field computation of a biological body which has various diseases.

Stochastic Perturbation Theory

In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a first-order perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higher-order terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem.