Non-separable Kernel Research Articles

An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind

In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels.

Solving nonlinear integral equations with non-separable kernel via a high-order iterative process

In this work we focus on location and approximation of a solution of nonlinear integral equations of Hammerstein-type when the kernel is non-separable through a high order iterative process. For this purpose, we approximate the non-separable kernel by means of a separable kernel and then, we perform a complete study about the convergence criteria for the approximated solution obtained to the solution of our first problem. Different examples have been tested in order to apply our theoretical results.

Iterative schemes for solving the Chandrasekhar [formula omitted]-equation using the Bernstein polynomials

In this work, we use Newton-type iterative schemes to obtain a domain of existence of solution, approximate the solution of Chandrasekhar H-equations and deal with the case of nonlinear integral equations with non-separable kernels. A change of variable in the Chandrasekhar H-equation allows us to apply a previous study by describing nonlinear integral equations of Hammerstein-type with non-separable kernel. We use the Bernstein polynomials for approximating the non-separable kernel and then we apply a semilocal converge study done previously to the Chandrasekhar H-equation. Moreover, we apply Newton-type iterative schemes for some specific Chandrasekhar H-equations to approximate the H-function solution and compare our results with others obtained previously.

Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.

A reliable treatment to solve nonlinear Fredholm integral equations with non-separable kernel

This work is devoted to solve integral equations formulated in terms of the kernel functions and Nemytskii operators. This type of equations appear in different applied problems such as electrostatics and radiative heat transfer problems. We deal with both cases separable and non-separable kernels by setting the theoretical semilocal convergence results for an adequate iterative scheme that can be useful for approximating the solution of the infinite dimensional problem. We pay special attention to non-separable kernels avoiding the solution given in previous works where the original nonlinear integral equation has been approximated by means of an equation with separable kernel. However, in this case, we introduce an approximation of the derivative operator that it is needed for applying the iterative scheme considered. Moreover, we study the localization and separation of possible solutions of nonlinear integral equation by means of a result of semilocal convergence for the iterative scheme considered. The theoretical results obtained have been tested with some applied problems showing competitive results.

Nonlinear Fredholm integral equations and majorant functions

From the majorant principle of Kantorovich and the theoretical significance of Newton’s method, we obtain domains of existence and uniqueness of solution for nonlinear Fredholm integral equations, so that these domains are defined from the positive real zeros of a scalar function that we call majorant function. We illustrate this study with three Fredholm integral equations, where separable and nonseparable kernels are involved, by obtaining domains of existence and uniqueness of solution, approximating solutions numerically and giving a priori and a posteriori error estimates of the approximations.

The majorant principle applied to Hammerstein integral equations

We use the majorant principle of Kantorovich to analyze the semilocal convergence of Newton’s method when it is applied to some Hammerstein integral equations. Moreover, from using the theoretical significance of the result obtained, we draw conclusions about the existence and uniqueness of solution of the equations. Furthermore, a solution of a Hammerstein integral equation with non-separable kernel is approximated.

Multidimensional NMR inversion without Kronecker products: Multilinear inversion

Multidimensional NMR inversion using Kronecker products poses several challenges. First, kernel compression is only possible when the kernel matrices are separable, and in recent years, there has been an increasing interest in NMR sequences with non-separable kernels. Second, in three or more dimensions, the singular value decomposition is not unique; therefore kernel compression is not well-defined for higher dimensions. Without kernel compression, the Kronecker product yields matrices that require large amounts of memory, making the inversion intractable for personal computers. Finally, incorporating arbitrary regularization terms is not possible using the Lawson–Hanson (LH) or the Butler–Reeds–Dawson (BRD) algorithms. We develop a minimization-based inversion method that circumvents the above problems by using multilinear forms to perform multidimensional NMR inversion without using kernel compression or Kronecker products. The new method is memory efficient, requiring less than 0.1% of the memory required by the LH or BRD methods. It can also be extended to arbitrary dimensions and adapted to include non-separable kernels, linear constraints, and arbitrary regularization terms. Additionally, it is easy to implement because only a cost function and its first derivative are required to perform the inversion.

IMPULSIVE INTEGRAL INEQUALITIES WITH A NON-SEPARABLE KERNEL

In this paper we present some Gronwall-tpye inequalities with a non-separable kernel and obtain the explicit estimate for solutions of impulsive differential equations. Furthermore, we give an example to illustrate our results.

Variational Image Registration Using Inhomogeneous Regularization

We present a generalization of the convolution-based variational image registration approach, in which different regularizers can be implemented by conveniently exchanging the convolution kernel, even if it is nonseparable or nonstationary. Nonseparable kernels pose a challenge because they cannot be efficiently implemented by separate 1D convolutions. We propose to use a low-rank tensor decomposition to efficiently approximate nonseparable convolution. Nonstationary kernels pose an even greater challenge because the convolution kernel depends on, and needs to be evaluated for, every point in the image. We propose to pre-compute the local kernels and efficiently store them in memory using the Tucker tensor decomposition model. In our experiments we use the nonseparable exponential kernel and a nonstationary landmark kernel. The exponential kernel replicates desirable properties of elastic image registration, while the landmark kernel incorporates local prior knowledge about corresponding points in the images. We examine the trade-off between the computational resources needed and the approximation accuracy of the tensor decomposition methods. Furthermore, we obtain very smooth displacement fields even in the presence of large landmark displacements.

A new multidimensional fast convolution algorithm

The article introduces a new fast convolution-type technique for computing the multidimensional periodic convolutions of signals defined on generic lattices. The method presented does not evaluate multidimensional periodic convolutions via multidimensional FFTs but with an original algorithm. The major advantages of this technique are that it is more efficient than the well-known FFT-based fast convolution method and that its efficiency increases with the signal's dimensionality. One striking indication of its computational efficiency is that the number of operations per output point required by this method with nonseparable kernels of assigned dimensions is always smaller than the number of operations required by classical fast convolution algorithms with separable kernels of identical dimensions.

Linear response function of a bounded jellium metal in the RPA

The integral equation describing the dielectric response of a surface system cannot in general by solved analytically because quantum effects introduce a nonseparable kernel. An approximate separable kernel is proposed with which the static response problem (for the infinite surface barrier model) is readily solved in exact analytic form. The method, when tested for several physical problems, proves rather accurate and is then used to calculate the real self-energy of a test charge at arbitrary positions relative to the surface. Dynamical response are briefly discussed in the context of the dispersion of long-wave surface plasmons, for which an alternative iterative method is also proposed.

Fourier Transform Methods in Linear Transport Theory

We generalize a three-dimensional Fourier transform method presented previously, to solve various forms of the linearized transport equation in planar geometry. Infinite-space problems are in general easily treated by our procedures. Half-space problems may also be solved analytically in, at least, the following cases: (i) separable scattering kernel (arbitrary particle speed); (ii) one-speed, anistropic scattering with rotation-invariant kernel; (iii) one-dimensional energy-dependent kernel of Kac; (iv) multigroup transport with down scattering only. The general importance of recursion relations to problems with nonseparable kernels is emphasized. A comparison is made between our methods and the two-dimensional singular eigenfunction approach, and a criterion is presented for the analytic solubility of any problem of the general form considered.

Scattering by non-local potentials

The kernels of a class of non-local potentials are related to the Green functions of the second order differential equations. For these potentials, the integro-differential wave equation may be transformed into a set of coupled ordinary differential equations, which can be solved by a factorization method. A solvable example of the non-separable kernel is given and the variation of the exact wave function with the non-locality of the interaction is studied. The effective mass approximation of Frahn and Lemmer for the optical potential is derived and the connection between the wave function in this approximation and the wave function for the equivalent local potential is discussed.