The defense and reception took place at the Arenberg castle in Heverlee (Leuven).

Jury:	Prof. dr. ir. L. Froyen, chairman,
	Prof. dr. ir. S. Vandewalle, promotor,
	Prof. dr. A. Bultheel,
	Prof. dr. ir. G. Vandenbosch,
	Prof. dr. ir. R. Cools,
	Prof. dr. ir. D. Roose,
	Prof. dr. A. Iserles, (University of Cambridge),
	Prof. dr. R. Stevenson, (Universiteit Utrecht)

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The text is available in postscript or pdf.

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Waves and oscillatory phenomena abound in many disciplines of science and engineering. Prime examples are electromagnetic and acoustic waves that permeate the atmosphere. In this thesis, we analyse and develop algorithms for the efficient numerical simulation of the scattering of such waves.

Time-harmonic scattering problems are modelled by an integral equation formulation. We consider three multiscale methods for the efficient solution of the resulting oscillatory integral equation: methods based on wavelets, methods based on hierarchical matrices and fast multipole methods. Although the discretisation matrix for integral equations is a dense matrix, each of these methods yields a fast matrix-vector product, where the number of operations scales approximately linearly in the number of unknowns. The solution can then be obtained efficiently in combination with an iterative Krylov subspace solver.

We show that wavelet based methods are not suitable for high frequency problems, where the number of oscillations is large with respect to the size of the scattering obstacle. We quantify the behaviour of the method in the oscillatory setting, and propose an improvement based on wavelet packets. Quadrature techniques are constructed for the efficient implementation of wavelet Galerkin discretisations. Methods based on hierarchical matrices and fast multipole methods are discussed for low frequency and high frequency scattering problems, and their applicability is compared.

Due to their ubiquitous nature in wave phenomena, oscillatory integrals are studied. A new method is proposed for the evaluation of univariate and multivariate oscillatory integrals, based on an extension of the method of steepest descent. Contrary to traditional methods, the accuracy of the new method increases rapidly with increasing frequency of the integrand, and it is shown that its computational cost is very low.

Finally, the insights in the behaviour of oscillatory integrals lead to the formulation of a novel method for highly oscillatory integral equations. We propose a hybrid method that combines asymptotic estimates of the solution with a classical boundary element discretisation. The hybrid asymptotic method requires a number of operations that is fixed with respect to the frequency. Results are given for the case of smooth and convex scattering obstacles. We show that the discretisation matrix in this case is small and highly sparse.

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We are surrounded by waves in our everyday lives: think of the microwave oven, cellullar phones and the radio, medical imaging techniques such as sonography and CT-scans, but also of sound and even light. The goal of this PhD is to research efficient methods for the simulation of the propagation and scattering of such waves on obstacles. A computer-based simulation enables many applications: CT (Computed Tomography) scans for example are based completely on numerical software and integral equations. The scattering obstacle in this case is a human being, or, better yet, certain tissues in the human body.

As the complexity of modern applications of waves increase, the complexity of the corresponding mathematical model increases as well. The exact solution of the model using a computer is no longer feasible, even when taking into account the increase of computing power. The multiscale methods that are considered in this PhD yield approximations. The algorithms are constructed to keep relevant information, and to neglect unnecessary details. A representation on different scales is very helpful: de scattering of a radar wave by an airplane is determined mostly by the shape of the hull and the wings of the plane, not by the shape of the gasoline valve. The result of these approximations is that the computations can be faster.

The simulation of waves at higher frequencies is more challenging. At high frequencies, details on small scales do matter for the overall solution. In our work, we propose an originaly hybrid method that can simulate certain scattering problems regardless of the frequency of the waves. The method even becomes better with increasing frequency. This contrasts with the multiscale methods discussed earlier, that require more computing time for higher frequency problems.

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