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19
Lattice sums for the Helmholtz equation
 SIAM Review
"... Abstract. A survey of different representations for lattice sums for the Helmholtz equation is given. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and ..."
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Abstract. A survey of different representations for lattice sums for the Helmholtz equation is given. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension dΛ. Lattice sums are related to, and can be calculated from, the quasiperiodic Green’s function and this object serves as the starting point of the analysis.
Analysis and synthesis of soundradiation with spherical arrays
, 2009
"... ii This work demonstrates a comprehensive methodology for capture, analysis, manipulation, and reproduction of spatial soundradiation. As the challenge herein, acoustic events need to be captured and reproduced not only in one but in a preferably complete multiplicity of directions, instead. The s ..."
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Cited by 15 (6 self)
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ii This work demonstrates a comprehensive methodology for capture, analysis, manipulation, and reproduction of spatial soundradiation. As the challenge herein, acoustic events need to be captured and reproduced not only in one but in a preferably complete multiplicity of directions, instead. The solutions presented in this work are using the soapbubble model, a working hypothesis about soundradiation, and are based on fundamental mathematical descriptions of spherical acoustic holography and holophony. These descriptions enable a clear methodic approach of soundradiation capture and reproduction. In particular, this work illustrates the implementation of surrounding spherical microphone arrays for the capture of soundradiation, as well as the analysis of soundradiation with a functional model. Most essential, the thesis shows how to obtain holophonic reproduction of soundradiation. For this purpose, a physical model of compact spherical loudspeaker arrays is established alongside with its electronic control. iii iv
Efficient FMM accelerated vortex methods in three dimensions via the LambHelmholtz decomposition
 J. Comput. Phys
"... Vortex methods are used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (di ..."
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Cited by 4 (1 self)
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Vortex methods are used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions. We present a formulation of vortex methods based on the LambHelmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing “conversion ” operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented. 1
A fast and stable method for rotating spherical harmonic expansions
 J. Comput. Phys
, 2009
"... a b s t r a c t In this paper, we present a simple and efficient method for rotating a spherical harmonic expansion. This is a wellstudied problem, arising in classical scattering theory, quantum mechanics and numerical analysis, usually addressed through the explicit construction of the Wigner ro ..."
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Cited by 4 (0 self)
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a b s t r a c t In this paper, we present a simple and efficient method for rotating a spherical harmonic expansion. This is a wellstudied problem, arising in classical scattering theory, quantum mechanics and numerical analysis, usually addressed through the explicit construction of the Wigner rotation matrices. We show that rotation can be carried out easily and stably through ''pseudospectral" projection, without ever constructing the matrix entries themselves. Existing fast algorithms, based on recurrence relations, are subject to a variety of instabilities, limiting the effectiveness of the approach for expansions of high degree.
Multiple scattering by multiple spheres: a new proof of the lloydberry formula for the effective wavenumber
 SIAM J. Appl. Math
"... Abstract. We provide the first classical derivation of the Lloyd–Berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers. Our approach clarifies the assumptions under which the Lloyd–Berry formula is valid. More precisely, we d ..."
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Abstract. We provide the first classical derivation of the Lloyd–Berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers. Our approach clarifies the assumptions under which the Lloyd–Berry formula is valid. More precisely, we derive an expression for the effective wavenumber which assumes the validity of Lax’s quasicrystalline approximation but makes no further assumptions about scatterer size, and then we show that the Lloyd–Berry formula is obtained in the limit as the scatterer size tends to zero.
A scalar potential formulation and translation theory for the timeharmonic Maxwell equations
, 2006
"... We develop a computational method based on a scalar potential representation, which efficiently reduces the solution of Maxwell’s equations to the solution of two scalar Helmholtz equations. One of the key contributions of this paper is a theory for the translation of Maxwell solutions using such a ..."
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Cited by 3 (1 self)
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We develop a computational method based on a scalar potential representation, which efficiently reduces the solution of Maxwell’s equations to the solution of two scalar Helmholtz equations. One of the key contributions of this paper is a theory for the translation of Maxwell solutions using such a representation, since the scalar potential form is not invariant with respect to translations. The translation theory is developed by introducing “conversion ” operators, which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame. Advantages of this representation include the fact that only two Helmholtz equations need be solved, and moreover, the divergence free constraints are satisfied automatically by construction. The availability of a translation theory for this representation can find application in methods such as the Fast Multipole Method. For illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of Mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the Tmatrix method. The resulting system was solved using an iterative method based on GMRES. The results of the computations agree well with previous computational and experimental results. 1
Fast, Parallel Techniques for TimeDomain Boundary Integral Equations
, 2014
"... First of all, I would like to express my deepest gratitude to my PhD advisor, Dr. Lehel Banjai, for his guidance, support and patience. I am indebted to him for excellent scientific advice, constructive criticism, as well as a great deal of time he dedicated for correcting my scientific writing and ..."
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First of all, I would like to express my deepest gratitude to my PhD advisor, Dr. Lehel Banjai, for his guidance, support and patience. I am indebted to him for excellent scientific advice, constructive criticism, as well as a great deal of time he dedicated for correcting my scientific writing and presentation. It is my pleasure to thank Prof. Dr. Dr. h.c. Wolfgang Hackbusch for providing me with an opportunity to work at the Max Planck Institute, as well as the International Max Planck Research School for the fi
Hierarchical matrices and the . . .
, 2014
"... The solution of boundaryvalue problems for the Helmholtz equation with decay is required by many physical applications, in particular viscoelastodynamics and electromagnetics. The boundary integral equation method allows to reduce the dimensionality of the problem by expressing the unknown quantity ..."
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The solution of boundaryvalue problems for the Helmholtz equation with decay is required by many physical applications, in particular viscoelastodynamics and electromagnetics. The boundary integral equation method allows to reduce the dimensionality of the problem by expressing the unknown quantity with the help of a boundary integral operator of a density given on the boundary of the domain. However, the BEM discretization of boundary integral formulations typically leads to densely populated matrices. In the last three decades a new generation of datasparse methods for the approximation of BEM matrices was designed. Among those are panelclustering, hierarchical matrices (Hmatrices), H2matrices and fast multipole methods (FMM). In this work we review main concepts of datasparse techniques. We present a description of the highfrequency fast multipole method (HF FMM) with some technical details, both for a real and complex wavenumber. A significant part of the report is dedicated to the error analysis of the HF FMM applied to the Helmholtz equation with a complex wavenumber. We compare the performance of the multilevel highfrequency fast multipole method and Hmatrices for the approximation of the
Recent advances and emerging applications . . .
, 2011
"... was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts ..."
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was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green’s functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the