Orthogonal Approximation Research Articles

Legendre orthogonal polynomials approximation of system gramians and its application to balanced truncation

ABSTRACTThe paper presents a procedure for computation of system gramians by using Legendre orthogonal polynomials approximation of the system state impulse and output responses. The proposed approach is trajectory based and relies on the system state and output trajectories snapshots selected either by experiment or computer simulation. It is defined in deterministic settings as opposed to similar approaches defined in stochastic settings by using the data covariance matrix. The advantage of using orthogonal series approximation for the gramians is to avoid solving the usual Lyapunov equations. The proposed method can be equally well applied to linear time-invariant as well as time-varying systems, and even to unstable systems, since the gramians approximation is performed on a finite interval of time. When the observation interval contains the whole energy of the system state impulse and output responses, the proposed method gives similar results as the gramians computed by solving Lyapunov equations. Several experiments are performed showing the good approximation properties of the presented method.

Galerkin–Legendre spectral method for Neumann boundary value problems in three dimensions

ABSTRACTIn this paper, we investigate the Legendre spectral methods for problems with the essential imposition of Neumann boundary condition in three dimensions. A double diagonalization process has been employed, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of Neumann boundary condition. For analysing numerical errors, some results on Legendre orthogonal approximation in Jacobi weighted Sobolev space are established. As examples of applications, the spectral schemes are provided for two model problems. The convergences of the proposed schemes are proved, too. Numerical results demonstrate the spectral accuracy in space, and which confirm theoretical analysis well.

Adaptive Trajectory Tracking Control of Dynamics of Nonholonomic Mobile Robot based on Orthogonal Function Approximation Technique

Adaptive Trajectory Tracking Control of Dynamics of Nonholonomic Mobile Robot based on Orthogonal Function Approximation Technique

Symmetric orthogonal approximation to symmetric tensors with applications to image reconstruction

SummaryThe main objective of this paper is to study an approximation of symmetric tensors by symmetric orthogonal decomposition. We propose and study an iterative algorithm to determine a symmetric orthogonal approximation and analyze the convergence of the proposed algorithm. Numerical examples are reported to demonstrate the effectiveness of the proposed algorithm. We also apply the proposed algorithm to represent correlated face images. We demonstrate better face image reconstruction results by combining principal components and symmetric orthogonal approximation instead of combining principal components and higher‐order SVD results.

Two-step variable selection in partially linear additive models with time series data

ABSTRACTLots of semi-parametric and nonparametric models are used to fit nonlinear time series data. They include partially linear time series models, nonparametric additive models, and semi-parametric single index models. In this article, we focus on fitting time series data by partially linear additive model. Combining the orthogonal series approximation and the adaptive sparse group LASSO regularization, we select the important variables between and within the groups simultaneously. Specially, we propose a two-step algorithm to obtain the grouped sparse estimators. Numerical studies show that the proposed method outperforms LASSO method in both fitting and forecasting. An empirical analysis is used to illustrate the methodology.

A Geometric Approach to Dynamical Model Order Reduction

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum.

Mixed generalized Hermite–Fourier spectral method for Fokker–Planck equation of periodic field

In this paper, we develop a mixed Hermite–Fourier spectral method for the Fokker–Planck equation of periodic field. Radical to the numerical solutions of partial differential equations of the sort, results on mixed generalized Hermite–Fourier orthogonal approximation are established. The convergence of the constructed spectral scheme is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.

Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions

In this paper we propose a method for the strong imposition of random Dirichlet boundary conditions in the Dynamical Low Rank (DLR) approximation of parabolic PDEs and, in particular, incompressible Navier Stokes equations. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low rank format, with rank smaller then S. We characterize the tangent space to the constrained manifold by means of a Dual Dynamically Orthogonal (Dual DO) formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The Dual DO formulation is also convenient to include the incompressibility constraint, when dealing with incompressible Navier Stokes equations. We show the performance of the proposed Dual DO approximation on two numerical test cases: the classical benchmark of a laminar flow around a cylinder with random inflow velocity, and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from MRI data with random inflow conditions coming from Doppler measurements.

Spatiotemporal distribution model for zinc electrowinning process and its parameter estimation

This paper focuses on the distributed parameter modeling of the zinc electrowinning process (ZEWP) to reveal the spatiotemporal distribution of concentration of zinc ions (CZI) and sulfuric acid (CSA) in the electrolyte. Considering the inverse diffusion of such ions in the electrolyte, the dynamic distribution of ions is described by the axial dispersion model. A parameter estimation strategy based on orthogonal approximation has been proposed to estimate the unknown parameters in the process model. Different industrial data sets are used to test the effectiveness of the spatiotemporal distribution model and the proposed parameter estimation approach. The results demonstrate that the analytical model can effectively capture the trends of the electrolysis reaction in time and thus has the potential to implement further optimization and control in the ZEWP.

Determination of continuous relaxation spectrum based on the Fuoss-Kirkwood relation and logarithmic orthogonal power-series approximation

In this study, we suggest a new algorithm for inferring continuous spectrum from dynamic moduli data. The algorithm is based on the Fuoss-Kirkwood relation (Fuoss and Kirkwood, 1941) and logarithmic powerseries approximation. The Fuoss-Kirkwood relation denotes the existence of the uniqueness of continuous spectrum. If we know the exact equation of dynamic moduli, then continuous spectrum can be inferred uniquely. We used the Chebyshev polynomials of the first kind to approximate dynamic moduli data in double-logarithmic scale. After the approximation, a spectrum equation can be derived by use of the complex decomposition method and the Fuoss-Kirkwood relation. We tested our algorithm to both simulated and experimental data of dynamic moduli and compared our result with those obtained from other methods such as the fixed-point iteration (Cho and Park, 2013) and cubic Hermite spline (Stadler and Bailly, 2009).

The best M-term trigonometric approximations of the classes of periodic multivariate functions with bounded generalized derivative in the space L q

The order estimates of the best M-term trigonometric approximations of functions \( {D}_{\beta}^{\psi } \) and the classes of (ψ , β)-differentiable periodic multivariate functions in the space L q , 2 ≤ q < ∞ are obtained. It is shown that, under certain conditions imposed on the parameter q; the best M-term trigonometric approximations \( {e}_M{\left({L}_{\beta, 1}^{\psi}\right)}_q \) have better order than the best orthogonal trigonometric approximations \( e\frac{1}{M}{\left({L}_{\beta, 1}^{\psi}\right)}_q \).

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis.

Efficient and Adaptive Orthogonal Finite Element Representation of the Geopotential

We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10−9 m s−2, globally) are required near the Earth’s surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with efficiency optimized using radial adaptation.

Orthogonal Binary Modulation Division for Two-User Uplink Massive MIMO Systems With Noncoherent ML Detection

This letter considers a noncoherent two-user uplink system, where each user has a single antenna and a base station is equipped with a large number of antennas. It is assumed that small scale channel fading is Rayleigh fading and quasi-static in two consecutive time slots and changes to other values independently in the next block. For such massive MIMO uplink system, we propose an orthogonal binary modulation scheme and utilize a noncoherent maximum likelihood receiver rather than a commonly adopted channel orthogonality approximation to analyze its error performance. A fast decoding algorithm is obtained and an exact bit error probability (BEP) formula is derived. When the large scale fading coefficients are available at the transmitters, a closed-form optimal power loading is attained that minimizes a tight upper bound of BEP. Computer simulations demonstrate that in a mild signal-to-noise ratio region, the optimized upper bound is almost equal to the truly optimized BEP. Hence, our derived BEP formula can be used to quantify how many antennas we need for the given BEP. In addition, the extension of all these results to the system with coherence time longer than two is also briefly discussed.

Mixed Spectral Method for Heat Transfer Using Generalised Hermite Functions and Legendre Polynomials

AbstractWe propose amixed spectral method for heat transfer in unbounded domains, using generalised Hermite functions and Legendre polynomials. Some basic results on the mixed generalised Hermite-Legendre orthogonal approximation are established, which plays important roles in spectral methods for various problems defined on unbounded domains. As an example, the mixed generalised Hermite-Legendre spectral scheme is constructed for anisotropic heat transfer. Its convergence is proven, and some numerical results demonstrate the spectral accuracy of this approach.

Identification of elastic-plastic properties of metal materials by using the residual imprint of spherical indentation

In this paper, a novel inverse computation method is proposed to estimate the elastic-plastic properties of metal materials by using only the residual imprint of spherical indentation. The advantage of this method is that it does not need to know the entire loading history. The indentation experiment can be easily implemented on a hardness tester while the residual imprint can be measured using a 3D measuring laser microscope. We correlate the imprint snapshot with material constitutive parameters using proper orthogonal decomposition and parametric approximation, and then we solve the inverse problem using the “Interior-point” optimization algorithm. The effectiveness of this method is verified by application on 2099-T83 Al-Li alloys. Results show the inverse analysis gives well-posed solution of parameters of materials only when the penetration depth or the prescribed indentation load is sufficient, and the elastic-plastic properties obtained from indentation and uniaxial experimental data show good agreement. Besides, the sensitivity investigation indicates the use of weighting imprint obtained from different experiment loads is able to give a more stable and reliable result.

A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation Models with Mixed Effects

ABSTRACTSeveral approaches exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA; Ramsay & Silverman, 2005), generalized local linear approximation (GLLA; Boker, Deboeck, Edler, & Peel, 2010), and generalized orthogonal local derivative approximation (GOLD; Deboeck, 2010). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo (MC) study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children’s self-regulation.

A high accuracy spectral method based on min/max principle for biharmonic eigenvalue problems on a spherical domain

In this study, we develop a high accuracy spectral method based on the min/max principle for biharmonic eigenvalue problems in the spherical domain. By analyzing the orthogonal spherical harmonic and approximation and using the min/max principle, we first deduce the error estimates of approximate eigenvalues. Then we construct an appropriate set of orthogonal spherical functions contained in H02(Ω) and establish the matrix formulations for the discrete variational form, whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently. Finally, we provide some numerical experiments to validate that the theoretical results are correct.

The use of generalized Laguerre functions for solving the equation of magnetohydydinamic flow due to a stretching cylinder

In this paper, we develop dual-Petrov-Galerkin approximations to linear third-order equation defined on unbounded domain. The convergence of proposed scheme is proved strictly. We also establish some basic results on generalized Laguerre orthogonal approximations. The nonlinear ordinary differential equation is solved using generalized Laguerre functions. Using the operational matrix of derivative, the problem is reduced to set of algebraic equations. We also present the comparison of this work with some other numerical results. It shows that the present solution is highly accurate.

Approximating Characteristics of the Classes L β,p ψ of Periodic Functions in the Space L q

We obtain the exact-order estimates of the best m-term and orthogonal trigonometric approximations and establish the order of trigonometric widths for the classes L , in the space L q for some relations between the parameters p and q.