Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Numerical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.     

Domain decomposition preconditioners for the spectral collocation method

We propose and analyze several block iteration preconditioners for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate that does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism.     

Spectral norm and trace bounds of algebraic matrix Riccatiequations

Upper bounds on spectral norm and trace for the solution of continuous algebraic matrix Riccati equations are proposed. The proposed bounds are based on a form of the general analytic solution. It is shown that, in many cases, the spectral norm and the trace bounds are better than other results     

Numerical Convergence Study of Nearly Incompressible,Inviscid Taylor–Green Vortex Flow

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor–Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor–Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.     

Steganographical Algorithms Based on the Convolution Theorem

A solution to the problem of information safety by the methods of computer steganography is considered. New spectral steganographical algorithms are presented that construct digital containers using the convolution theorem. A transition to spectral representation of the containers is carried out in Fourier basis.     

Discrete Legendre spectral Galerkin method for Urysohn integral equations

In this paper, we consider the discrete Legendre spectral Galerkin method to approximate the solution of Urysohn integral equation with smooth kernel. The convergence of the approximate and iterated approximate solutions to the actual solution is discussed and the rates of convergence are obtained. In particular we have shown that, when the quadrature rule is of certain degree of precision, the superconvergence rates for the iterated Legendre spectral Galerkin method are maintained in the discrete case.     

Shock capturing by the spectral viscosity method

A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods then pollutes the unstable Gibbs oscillations over all the computational domain, and the lack of entropy dissipation prevents convergences in these cases. In this paper, we discuss the spectral viscosity method, which is based on high frequency-dependent vanishing viscosity regularization of the classical spectral methods. We show that this method enforces the convergence of nonlinear spectral approximations without sacrificing their overall spectral accuracy.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

     

Extension of the Class of Queueing Systems with Delay

We consider the problem of determining the characteristics of queuing systems with delay by the classical spectral decomposition method for the solution of the Lindley integral equation. As input distributions for the systems we choose mixtures of exponential distributions shifted to the right of the zero point, for which the spectral decomposition approach allows one to obtain a solution in closed form. We show that in such systems with delay, the average waiting time is shorter than in conventional systems.     

A noniterative solution approach for parallel pseudospectral domain decomposition

A simple superposition technique is proposed for the solution of spectral collocation equations in multi-nonoverlapping subdomains. It is based on a property of linear differential equations that allows the interface conditions to be fully decoupled; thus yielding a strategy with a very high level of concurrency suitable for parallel computations. Numerical experiments indicate, for a fixed total number of collocation points, a significant degradation of spectral accuracy as the number of subdomains increases. While the technique generally yields reasonably good solution, a compromise between accuracy and geometric flexibility must be realized.     

Ray-tracing polymorphic multidomain spectral/hp elements for isosurface rendering

The purpose of this paper is to present a ray-tracing isosurface rendering algorithm for spectral/hp (high-order finite) element methods in which the visualization error is both quantified and minimized. Determination of the ray-isosurface intersection is accomplished by classic polynomial root-finding applied to a polynomial approximation obtained by projecting the finite element solution over element-partitioned segments along the ray. Combining the smoothness properties of spectral/hp elements with classic orthogonal polynomial approximation theory, we devise an adaptive scheme which allows the polynomial approximation along a ray-segment to be arbitrarily close to the true solution. The resulting images converge toward a pixel-exact image at a rate far faster than sampling the spectral/hp element solution and applying classic low-order visualization techniques such as marching cubes.     

Explicit solution to the singular LQ regulation problem

An explicit solution to the multivariable discrete linear quadratic (LQ) regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero. Such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable and that its transfer function matrix is of full rank. The suitable spectral factor is shown to be the product of the system's minimum-phase image and its unitary interactor matrix. The unitary interactor matrix defined is a special case of the nilpotent interactor matrix defined by M.W. Rogozinski et al. (1987)     

Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.Submitted toJ. Scientific Computing, August 25, 1994.     

Spectral projective newton-methods for quasilinear elliptic boundary value problems

We consider Newton-like methods for the solution of quasilinear elliptic boundary value problems. The quasilinear problems are linearized by a Newton-method and the linear problems are approximately solved by a spectral projection method (e.g., the Ritz-Galerkin or the collocation method). convergence results are derived that show the spectral accuracy of this method. The results are of a local type which means that we assume the starting approximation to be sufficiently near to the exact solution.     

Rational approximations of spectral densities based on the Alpha divergence

We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by selecting the spectral density having minimum “distance” from under the constraint corresponding to imposing the given second-order statistics. We analyze the structure of the optimal solutions as the minimized “distance” varies in the Alpha divergence family. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback–Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information.     

Accelerating spectral clustering with partial supervision

Spectral Clustering is a popular learning paradigm that employs the eigenvectors and eigenvalues of an appropriate input matrix for approximating the clustering objective. Albeit its empirical success in diverse application areas, spectral clustering has been criticized for its inefficiency when dealing with large-size datasets. This is mainly due to the fact that the complexity of most eigenvector algorithms is cubic with respect to the number of instances and even memory efficient iterative eigensolvers (such as the Power Method) may converge very slowly to the desired eigenvector solutions. In this paper, inspired from the relevant work on Pagerank we propose a semi-supervised framework for spectral clustering that provably improves the efficiency of the Power Method for computing the Spectral Clustering solution. The proposed method is extremely suitable for large and sparse matrices, where it is demonstrated to converge to the eigenvector solution with just a few Power Method iterations. The proposed framework reveals a novel perspective of semi-supervised spectral methods and demonstrates that the efficiency of spectral clustering can be enhanced not only by data compression but also by introducing the appropriate supervised bias to the input Laplacian matrix. Apart from the efficiency gains, the proposed framework is also demonstrated to improve the quality of the derived cluster models.     

Fields of non-linear regression models for atmospheric correction of satellite ocean-color imagery

Remote sensing of ocean color from space, a problem that consists of retrieving spectral marine reflectance from spectral top-of-atmosphere reflectance, is considered as a collection of similar inverse problems continuously indexed by the angular variables influencing the observation process. A general solution is proposed in the form of a field of non-linear regression models over the set T of permitted values for the angular variables, i.e., as a map from T to some function space. Each value of the field is a regression model that performs a direct mapping from the top-of-atmosphere reflectance to the marine reflectance. Since the spectral components of the field take values in the same variable vector space, the retrievals in individual spectral bands are not independent, i.e., the solution is not just a juxtaposition of independent models for each spectral band. A scheme based on ridge functions is developed to approximate this solution to an arbitrary accuracy, and is applied to the retrieval of marine reflectance in Case 1 waters, for which optical properties are only governed by biogenic content. The statistical models are evaluated on synthetic data as well as actual data originating from the SeaWiFS instrument, taking into account noise in the data. Theoretical performance is good in terms of accuracy, robustness, and generalization capabilities, suggesting that the function field methodology might improve atmospheric correction in the presence of absorbing aerosols and provide more accurate estimates of marine reflectance in productive waters. When applied to SeaWiFS imagery acquired off California, the function field methodology gives generally higher estimates of marine reflectance than the standard SeaDAS algorithm, but the values are more realistic.     

Minimax linear smoothing for capacities: The case of correlated signals and noise

Results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties, are extended to the case in which the signal and noise fields are arbitrarily correlated. As before, spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two-alternating Choquet capacities. It is demonstrated that this problem admits a general solution in terms of the Huber-Strassen derivative between the capacities that generate the uncertainty sets, and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise. The resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in (nonparametric) minimax hypothesis testing. These new results extend the applicability of minimax robust smoothing techniques to application involving signal-dependent noise phenomena, such as multipath and clutter, which are usually difficult to model precisely.     

Modeling of first-order photobleaching kinetics using Krylov subspace spectral methods

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate.     

An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations

We present a double Legendre spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomo-geneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. One numerical application of how to use these methods is described. Numerical results obtained compare favorably with those of the analytical solution. Accurate double Legendre spectral approximations for Poisson' and Helmholtz' equations are also noted.