A parameter choice strategy for the regularized approximation of Fredholm integral equations of the first kind

Recently, Rajan [A Modified convergence analyis for solving Fredholm integral equations of the first kind, Integral Equ. Oper. Theory, 49 (2004), pp. 511–516.] suggested a modified convergence analysis for solving Fredholm integral equations of the first kind and also considered an a priori parameter choice strategy of choosing the regularization parameter. In this article, as an a posteriori parameter choice strategy for choosing the regularization parameter, a class of discrepancy principle is proposed under the Hilbert space setting and is illustrated numerically.     

Multiphase Soft Segmentation with Total Variation and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Regularization

In this paper, we propose a variational soft segmentation framework inspired by the level set formulation of multiphase Chan-Vese model. We use soft membership functions valued in [0,1] to replace the Heaviside functions of level sets (or characteristic functions) such that we get a representation of regions by soft membership functions which automatically satisfies the sum to one constraint. We give general formulas for arbitrary N-phase segmentation, in contrast to Chan-Vese’s level set method only 2 ^m -phase are studied. To ensure smoothness on membership functions, both total variation (TV) regularization and H ¹ regularization used as two choices for the definition of regularization term. TV regularization has geometric meaning which requires that the segmentation curve length as short as possible, while H ¹ regularization has no explicit geometric meaning but is easier to implement with less parameters and has higher tolerance to noise. Fast numerical schemes are designed for both of the regularization methods. By changing the distance function, the proposed segmentation framework can be easily extended to the segmentation of other types of images. Numerical results on cartoon images, piecewise smooth images and texture images demonstrate that our methods are effective in multiphase image segmentation.     

Numerical solution of linear Fredholm integral equations using sine–cosine wavelets

If we divide the interval [0,1] into N sub-intervals, then sine–cosine wavelets on each sub-interval can approximate any function. This ability helps us to obtain a more accurate approximation of piecewise continuous functions, and, hence, we can obtain more accurate solutions of integral equations. In this article we use a combination of sine–cosine wavelets on the interval [0,1] to solve linear integral equations. We convert the integral equation into a system of linear equations. Numerical examples are given to demonstrate the applicability of the proposed method.     

A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography

In this paper, we consider 3D Bioluminescence tomography (BLT) source reconstruction from Poisson data in three dimensional space. With a priori information of sources sparsity and MAP estimation of Poisson distribution, we study the minimization of Kullback-Leihbler divergence with ℓ ¹ and ℓ ⁰ regularization. We show numerically that although several ℓ ¹ minimization algorithms are efficient for compressive sensing, they fail for BLT reconstruction due to the high coherence of the measurement matrix columns and high nonlinearity of Poisson fitting term. Instead, we propose a novel greedy algorithm for ℓ ⁰ regularization to reconstruct sparse solutions for BLT problem. Numerical experiments on synthetic data obtained by the finite element methods and Monte-Carlo methods show the accuracy and efficiency of the proposed method.     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.     

Control quality enhancement using fractional PIλDμ controller

The proportional–integral–derivative (PID) controllers have remained, by far, the most commonly and practically used in all industrial feedback control applications; therefore, there is a continuous effort to improve the system control quality performances. More recently Podlubny has proposed the fractional PI^λD^μ controller, a generalisation of the classical PID controller, involving an integration action of order λ and differentiation action of order μ. Since then, many researchers have been interested in the use and tuning of this type of controller. In this article, a new conception method of this fractional PI^λD^μ controller is considered. The basic ideas of this new tuning method are based, in the first place, on the classical Ziegler–Nichols tuning method for setting the parameters of the fractional PI^λD^μ controller for λ = μ = 1, which means setting the parameters of the classical PID controller, and on the minimum integral squared error criterion by using the Hall–Sartorius method for setting the fractional integration action order λ and the fractional differentiation action order μ. Illustrative examples and simulation results are presented to show the control quality enhancement of this proposed fractional PI^λD^μ controller conception method compared to the PID controller conception using Ziegler–Nichols tuning method.     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H _c (x) on the collocation points is used for the approximation of singular source terms δ(x−c) or δ ⁽ⁿ⁾(x−c) without any regularization. The direct projection of H _c (x) yields highly oscillatory approximations of δ(x−c) and δ ⁽ⁿ⁾(x−c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided. The current address: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.     

On the method of near equations

Initiated by H. Bateman, F. Tricomi, E. J. Nyström and L. V. Kantorovitch, the method of near equations is extended to linear and continuous operators acting in an arbitrary normed space. This method is applied to linear integral equations of second kind in the spaces C[a,b] and L²[a,b]. The main feature of this method appears in some examples, for which its continuous variant fails but the L²-variant works, and then a subsequent «regularization procedure» leads to a continuous solution satisfying pointwise both the given integral equation and the corresponding error estimation.     

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

     

A Multi-Scale Vectorial L<Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript>-TV Framework for Color Image Restoration

A general multi-scale vectorial total variation model with spatially adapted regularization parameter for color image restoration is introduced in this paper. This total variation model contains an L ^τ -data fidelity for any τ∈[1,2]. The use of a spatial dependent regularization parameter improves the reconstruction of features in the image as well as an adequate smoothing for the homogeneous parts. The automated adaptation of this regularization parameter is made according to local statistical characteristics of the noise which contaminates the image. The corresponding multiscale vectorial total variation model is solved by Fenchel-duality and inexact semismooth Newton techniques. Numerical results are presented for the cases τ=1 and τ=2 which reconstruct images contaminated with salt-and-pepper noise and Gaussian noise, respectively.     

Numerical analysis and computations of a high accuracy time relaxation fluid flow model

We present a numerical study based on continuous finite element analysis for a time relaxation regularization of Navier–Stokes equations. This regularization is based on filtering and deconvolution. We study the convergence of the regularized equations using a fully discretized filter and deconvolution algorithm. Velocity and pressure error estimates and the L ₂ Aubin–Nitsche lift technique are proved for the equilibrium problem, and this analysis is accompanied by the velocity error estimate for the time-dependent problem, too. Thus, optimal error estimates in L ₂ and H ¹ norms are derived and followed by their computational verification. Also, computational results of the vortex street are presented for the two-dimensional cylinder benchmark flow problem. Maximum drag and lift coefficients and difference in pressure between the front and back of the cylinder at the final time were investigated as well, showing that the time relaxation regularization can attain the benchmark values.     

An adaptive Huber method for weakly singular second kind Volterra integral equations with non-linear dependencies between unknowns and their integrals

Numerical methods for weakly singular Volterra integral equations with non-linear dependencies between unknowns and their integrals, are almost non-existent in the literature. In the present work an adaptive Huber method for such equations is proposed, by extending the method previously formulated for the first kind Abel equations. The method is tested on example integral equations involving integrals with kernels K(t, τ) = (t ? τ)^?1/2, K(t, τ) = exp[?λ(t ? τ)](t ? τ)^?1/2 (where λ > 0), and K(t, τ) = 1. By controlling estimated local discretisation errors, the integral equation can be solved adaptively on a discrete grid of nodes in the independent variable domain, in a step-by-step fashion. The practical accuracy order is close to 2. The accuracy can be varied by varying the prescribed local error tolerance parameter tol, although the actual errors tend to be larger than tol. Approximations to off-nodal solution values can also be computed, with a comparable accuracy. The method appears numerically stable when partial derivatives, of the non-linear function representing the equation, with respect to the unknown and its integral(s), are of the same sign. The stability of the method in the opposite case may be debated.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

Gene Expression Data to Mouse Atlas Registration Using a Nonlinear Elasticity Smoother and Landmark Points Constraints

This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L ² dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.     

Approximate solution of systems of Volterra integral equations with error analysis

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(?c N ^1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.     

Additive Schwarz preconditioners for the h-p version boundary-element approximation to the hypersingular operator in three dimensions

We study different preconditioners for the h-p version of the Galerkin boundary-element method when used to solve hypersingular integral equations of the first kind on a surface in ?³. These integral equations result from Neumann problems for the Laplace and Lamé equations in the exterior of the surface. The preconditioners are of additive Schwarz type (non-overlapping and overlapping). In all cases, we prove that the condition numbers grow at most logarithmically with the degrees of freedom.     

Compact alternating direction implicit method to solve two-dimensional nonlinear delay hyperbolic differential equations

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously in L²- and H¹-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes.     

The numerical solution of fourth order mildly quasi-linear parabolic initial boundary value problem of second kind

In this article, we report on three-level implicit stable finite difference formulas of O(k ² + h ²) and O(k ² + h ⁴) for the numerical integration of certain mildly quasi-liner fourth order parabolic partial differential equations in one-space dimension. The numerical solution of u _xx is obtained as a by-product of the method. In all cases, we use only (3 + 3 + 3)-grid points and a single computational cell. Difference schemes for the fourth order linear parabolic equation in polar coordinates are also discussed. The stability analysis for the model linear problem is given as a representative example. Numerical results are presented to demonstrate the order and accuracy of the proposed methods.     

Least squares support vector regression for solving Volterra integral equations

In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method.