Integral Equations

Uniform and pointwise numerical approximation of the weighted Hilbert transform on the real line
M. R. Capobianco & G. Criscuolo
(Istituto per Applicazioni della Matematica ­ CNR, Napoli, Italy)

The importance of the Hilbert transform coming from its many applications, justifies some interest in its numerical evaluation. Besides appearing in several physical and engineering problems, the Hilbert transform is the main part of singular integral equations on $\mathbf{R}$.
We propose an algorithm to compute the weighted Hilbert transform $H(wf)$ assuming that the density function $f$ has good integration properties at the limits of the integration interval; these assumptions are the same ones to assure the boundedness of $H(wf)$. The proposed procedure is of interpolatory type and it uses as quadrature nodes the zeros of the orthogonal polynomials $p_m(w), m=1,2,\dots$ with respect to the weight function $w$ on $\mathbf{R}$; namely we approximate $H(wf)$ by $H_m(wf)=H(w\mathcal{L}_ m(w;f))$, where $\mathcal{L}_m(w)$ denotes the Lagrange interpolating operator corresponding to the matrix having as elements the zeros of $p_m(w), m=1,2,\dots$.
The results obtained in [1] are stronger than those ones proved in [4] where, in order to prove the main result for the quadrature rule, the boundedness of the operator H and Lm(w) are examined in unrelated way. Indeed, the authors of [4] do not prove the uniform converge and the stability of the quadrature rule $H(w \mathcal{L}_m(w;f))$ as we show, and they appealed to more complicated interpolatory procedure. Further, in [4] only the case of the Hermite weight is examined. The main observation is that if one wants to use only the zeroes of the orthogonal polynomials for interpolation, one can use instead good bounds on the functions of the second kind. The same formula of [4], based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration, augmented by two carefully chosen extra points, has been proved in [3] for a much class of weights that have varing rates of decay. Furthermore, under the same assumptions on the functions f and w, we propose a Gauss type quadrature to evaluate H(wf). We give results about the convergence and the stability of the proposed procedure with respect to the distance of the singularity from the quadrature nodes. Taking into account these results, we describe the principal computational aspects of a new algorithm to evaluate $H(wf)$ (see also [2]).

References

[1]

M.R. Capobianco, G. Criscuolo, R. Giova, Approximation of the weighted Hilbert transform on the real line by an interpolatory process, to appear on BIT.

[2]

M. R. Capobianco, G. Criscuolo, R. Giova A stable and convrgent algorithm to evaluate Hilbert transform, to appear on Numerical Algorithms.

[3]

S. B. Damelin, K. Diethelm, Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line, to appear on J. Functional Analysis and Optimization.

[4]

C. De Bonis, B. Della Vecchia, G. Mastroianni, Approximation of the weghted Hilbert transform on the real line, Università della Basilicata (1998).

Iterative solution and preconditioning of boundary integral equations in electromagnetism
B. Carpentieri¹ & I.S. Duff, L. Giraud and G. Sylvand
(CERFACS, Toulouse, France)

In recent years, there has been a significant amount of work on the simulation of electromagnetic wave propagation phenomena, addressing various topics ranging from radar cross section to electromagnetic compatibility, to absorbing materials, and antenna design. The Boundary Element Method ($BEM$) has been successfully employed in the numerical solution of this class of problems, proving to be an effective alternative to common discretization schemes like Finite Element Methods ($FEM$'s) and Finite Difference Methods ($FDM$'s). The idea of $BEM$ is to shift the focus from solving a $PDE$ defined on a closed or unbounded domain to solving a boundary integral equation over the finite part of the boundary. The discretization by the boundary element method results in linear systems with dense complex matrices which are very challenging to solve. With the advent of parallel processing, this approach has become viable for large problems and the typical problem size in the electromagnetics industry is continually increasing. Direct dense methods based on Gaussian elimination are often the method of choice because they are reliable and predictable both in terms of accuracy and cost. However, for large-scale problems, they become impractical even on large parallel platforms because they require storage of $n^{2}$ double precision complex entries of the coefficient matrix and ${\cal O}(n^{3})$ floating-point operations to compute the factorization, where $n$ denotes the size of the linear system. Iterative Krylov subspace based solvers are a promising alternative provided we have fast matrix-vector multiplications and robust preconditioners.

In this talk we present results concerning the use of iterative Krylov solvers for the numerical solution of boundary integral equations in electromagnetism with special emphasis on the design of robust preconditioners, a crucial component of Krylov methods in this context. We consider in particular a sparse approximate inverse preconditioner based on Frobenius-norm minimization that use a static nonzero pattern selection. We report on results on the numerical scalability of the preconditioner on large problems in collaboration with AEDS, implemented in its FMM code. Finally we consider two possible multilevel extensions for the sparse approximate inverse. The first is based on the introduction of low-rank corrections which intend to improve the quality of the sparse approximate inverse computed on difficult problems. The second is based on inner-outer solution schemes in the FMM context which use different level of accuracy for the M-V products.

References

[1]

B. Carpentieri, I. S. Duff and L. Giraud, Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism, Numerical Linear Algebra with Applications, 2000, vol. 7, n. 7-8, pages 667-685

[2]

B. Carpentieri, I. S. Duff, L. Giraud and G. Sylvand, Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations, 2001, to appear.

Numerical approximation of some BEM singular integrals
G. Criscuolo
(Università degli Studi di Napoli "Federico II", Italy)

Very recently, Mastroianni and Monegato have derived error estimates for the numerical approach to evaluate the integral

$$(1)\qquad\qquad\qquad\qquad\int_a^b\int_{-1}^1\frac{f(x,y)}{x-y}w(x)dxdy,\qquad\qquad\qquad \qquad\qquad\qquad$$

where $(a,b)\equiv (-1,1)$, or $(a,b)\equiv (a,-1)$, or $(a,b)\equiv (1,b)$, $f(x,y)$ is a smooth function on both variables, and $w(x)=w^\alpha(x)=(1-x^2)^\alpha, \alpha>-1$. When $y\in (-1,1)$, the inner integral is defined in the Cauchy principal value sense. The results shown in [2] have wide interest in the applications. Indeed, in the applications of Galerkin boundary element methods, for the solution of one­dimensional singular and hypersingular equations, one has to deal with integrals which after proper normalization are of the type (1). Further, two­dimensional singular integrals of form (1) arise in some aeroelasticity problems (see the references in [2]). Using the same numerical approach to evaluate (1) proposed in [2], we improve the corresponding error estimates. Furthermore, we generalize the results in [2] for a wider class of weights and for different choice of the quadrature nodes. The more general results about the convergence of the numerical approach to evaluate (1) are of interest in the applications (see also [1]).

References

[1]

G. Criscuolo, Numerical integration of BEM singular integrals, submitted.

[2]

G. Mastroianni, G. Monegato, Error esimates in the numerical] evaluation of some BEM singular integrals, Math. Comp. 70 (2000), pp. 251­267.

High Performance Methods for Volterra Equations with Weakly Singular Kernels
E. Russo & G. Capobianco & M.R. Crisci
(Dipartimento di Matematica e Informatica, Università di Salerno, Italy)

Large systems of Volterra Integral Equations with weakly singular kernels arise in many branches of applications such as, for example, reaction-diffusion problems in small cells.

In order to get accurate solutions in a reasonable time frame, high performances numerical methods are required. Methods of this kind are the iterative Waveform Relaxation that split the system into uncoupled subsystem and so realize a massive parallelism across the system. However, fully parallel WR methods are usually slowly convergent.

In order to construct fast convergent fully parallel WR methods, the authors introduce non-stationary continuous and discrete time WR methods. The convergence analysis is performed.

Non-stationary Richardson WR method is constructed in such a way to optimize the speed of convergence. A significant error bound is obtained, which allow to predict the number of iterations required to get a given tolerance and to determine the class of problems for which the method is more suitable.

A parallel code based on this method is under construction.