63 research outputs found
On polynomial collocation for second kind integral equations with fixed singularities of Mellin type
We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gauss quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with O(n^2[log n]^2) resp. O(n^2) operations, where n is the dimension of the polynomial trial space
On polynomial collocation for Cauchy singular integral equations with fixed singularities
In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weighted L2 spaces, we derive necessary and sufficient conditions
On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces
This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown
The numerical solution of Cauchy singular integral equations with additional fixed singularities
In this paper we propose a quadrature method for the numerical solution of Cauchy singular integral
equations with additional fixed singularities. The unknown function is approximated by a weighted
polynomial which is the solution of a finite dimensional equation obtained discretizing the involved
integral operators by means of a Gauss-Jacobi quadrature rule. Stability and convergence results for the
proposed procedure are proved. Moreover, we prove that the linear systems one has to solve, in order to
determine the unknown coefficients of the approximate solutions, are well conditioned. The efficiency of
the proposed method is shown through some numerical examples
ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM
This thesis is concerned with the numerical solution of boundary integral equations
and the numerical analysis of iterative methods. In the first part, we assume
the boundary to be smooth in order to work with compact operators; while in the
second part we investigate the problem arising from allowing piecewise smooth
boundaries. Although in principle most results of the thesis apply to general problems
of reformulating boundary value problems as boundary integral equations
and their subsequent numerical solutions, we consider the Helmholtz equation
arising from acoustic problems as the main model problem.
In Chapter 1, we present the background material of reformulation of Helmhoitz
boundary value problems into boundary integral equations by either the indirect
potential method or the direct method using integral formulae. The problem of
ensuring unique solutions of integral equations for exterior problems is specifically
discussed. In Chapter 2, we discuss the useful numerical techniques for
solving second kind integral equations. In particular, we highlight the superconvergence
properties of iterated projection methods and the important procedure
of Nystrom interpolation.
In Chapter 3, the multigrid type methods as applied to smooth boundary
integral equations are studied. Using the residual correction principle, we are
able to propose some robust iterative variants modifying the existing methods to
seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient
method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease,
faster convergence of multigrid type methods and fixed step convergence of the
conjugate gradient method.
In the case of non-smooth integral boundaries, we first derive the singular
forms of the solution of boundary integral solutions for Dirichlet problems and
then discuss the numerical solution in Chapter 5. Iterative methods such as two
grid methods and the conjugate gradient method are successfully implemented
in Chapter 6 to solve the non-smooth integral equations. The study of two
grid methods in a general setting and also much of the results on the conjugate
gradient method are new. Chapters 3, 4 and 5 are partially based on publications
[4], [5] and [35] respectively.Department of Mathematics and Statistics,
Polytechnic South Wes
An optimal order collocation method for first kind boundary integral equations on polygons.
This paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in ℝ2. Before discretization the equation is transformed to an equivalent equation over [-π,π] using a nonlinear parametrization of the polygon which varies more slowly than arc-length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on [-π,π]. This latter integral equation is shown to be well-posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non-standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order k (degree k-1) converges with optimal order O(hk). (The collocation points are the midpoints of subintervals when k is odd and the break-points when k is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples
A range description for the planar circular Radon transform
The transform considered in the paper integrates a function supported in the
unit disk on the plane over all circles centered at the boundary of this disk.
Such circular Radon transform arises in several contemporary imaging
techniques, as well as in other applications. As it is common for transforms of
Radon type, its range has infinite co-dimension in standard function spaces.
Range descriptions for such transforms are known to be very important for
computed tomography, for instance when dealing with incomplete data, error
correction, and other issues. A complete range description for the circular
Radon transform is obtained. Range conditions include the recently found set of
moment type conditions, which happens to be incomplete, as well as the rest of
conditions that have less standard form. In order to explain the procedure
better, a similar (non-standard) treatment of the range conditions is described
first for the usual Radon transform on the plane.Comment: submitted for publicatio
The h-p-version of spline approximation methods for Mellin convolution equations.
We consider the numerical solution of Mellin convolution equations on an interval by the h-p-version of spline approximation methods. Using a geometric mesh refinement towards the singularity of the integral equation, we prove stability and exponential convergence in the Lq norm, 1 ≤ q ≤ ∞, for Galerkin, collocation and Nyström methods based on piecewise polynomials
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