Timo Betcke

NLEVP: A Collection of Nonlinear Eigenvalue Problems

We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.

Solving Boundary Integral Problems with BEM

Many important partial differential equation problems in homogeneous media, such as those of acoustic or electromagnetic wave propagation, can be represented in the form of integral equations on the boundary of the domain of interest. In order to solve such problems, the boundary element method (BEM) can be applied. The advantage compared to domain-discretisation-based methods such as finite element methods is that only a discretisation of the boundary is necessary, which significantly reduces the number of unknowns. Yet, BEM formulations are much more difficult to implement than finite element methods. In this article, we present BEM++, a novel open-source library for the solution of boundary integral equations for Laplace, Helmholtz and Maxwell problems in three space dimensions. BEM++ is a C++ library with Python bindings for all important features, making it possible to integrate the library into other C++ projects or to use it directly via Python scripts. The internal structure and design decisions for BEM++ are discussed. Several examples are presented to demonstrate the performance of the library for larger problems.

A Fast Parallel Solver for the Forward Problem in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a noninvasive imaging modality, where imperceptible currents are applied to the skin and the resulting surface voltages are measured. It has the potential to distinguish between ischaemic and haemorrhagic stroke with a portable and inexpensive device. The image reconstruction relies on an accurate forward model of the experimental setup. Because of the relatively small signal in stroke EIT, the finite-element modeling requires meshes of more than 10 million elements. To study the requirements in the forward modeling in EIT and also to reduce the time for experimental image acquisition, it is necessary to reduce the run time of the forward computation. We show the implementation of a parallel forward solver for EIT using the Dune-Fem C++ library and demonstrate its performance on many CPUs of a computer cluster. For a typical EIT application a direct solver was significantly slower and not an alternative to iterative solvers with multigrid preconditioning. With this new solver, we can compute the forward solutions and the Jacobian matrix of a typical EIT application with 30 electrodes on a 15-million element mesh in less than 15 min. This makes it a valuable tool for simulation studies and EIT applications with high precision requirements. It is freely available for download.

Stroke type differentiation using spectrally constrained multifrequency EIT: evaluation of feasibility in a realistic head model

We investigate the application of multifrequency electrical impedance tomography (MFEIT) to imaging the brain in stroke patients. The use of MFEIT could enable early diagnosis and thrombolysis of ischaemic stroke, and therefore improve the outcome of treatment. Recent advances in the imaging methodology suggest that the use of spectral constraints could allow for the reconstruction of a one-shot image. We performed a simulation study to investigate the feasibility of imaging stroke in a head model with realistic conductivities. We introduced increasing levels of modelling errors to test the robustness of the method to the most common sources of artefact. We considered the case of errors in the electrode placement, spectral constraints, and contact impedance. The results indicate that errors in the position and shape of the electrodes can affect image quality, although our imaging method was successful in identifying tissues with sufficiently distinct spectra.

Correcting electrode modelling errors in EIT on realistic 3D head models.

Electrical impedance tomography (EIT) is a promising medical imaging technique which could aid differentiation of haemorrhagic from ischaemic stroke in an ambulance. One challenge in EIT is the ill-posed nature of the image reconstruction, i.e., that small measurement or modelling errors can result in large image artefacts. It is therefore important that reconstruction algorithms are improved with regard to stability to modelling errors. We identify that wrongly modelled electrode positions constitute one of the biggest sources of image artefacts in head EIT. Therefore, the use of the Fréchet derivative on the electrode boundaries in a realistic three-dimensional head model is investigated, in order to reconstruct electrode movements simultaneously to conductivity changes. We show a fast implementation and analyse the performance of electrode position reconstructions in time-difference and absolute imaging for simulated and experimental voltages. Reconstructing the electrode positions and conductivities simultaneously increased the image quality significantly in the presence of electrode movement.

Computationally Efficient Boundary Element Methods for High-Frequency Helmholtz Problems in Unbounded Domains

This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation.

Mini-Workshop: Fast Solvers for Highly Oscillatory Problems

The efficient numerical solution of highly oscillatory problems is one of the grand challenges of Applied Mathematics with diverse applications across the natural sciences and engineering. This workshop brings together experts in domain based methods and integral equation methods to share novel ideas and to discuss challenges on the way to developing efficient solvers at high frequencies. Mathematics Subject Classification (2010): 65xx. Introduction by the Organisers The fast solution of highly oscillatory problems remains one of the great challenges of applied and computational mathematics. This workshop brought together experts working on fast direct solvers for integral equations, preconditioning and domain decomposition methods to share novel ideas for the development of scalable frequency domain solvers for acoustic and electromagnetic problems. The workshop was roughly divided into three broad subject areas, namely 1.) fast direct solvers for Helmholtz problems, 2.) fast iterative methods and preconditioning for oscillatory integral equations, and 3.) domain decomposition methods for volume problems. The first day started off with an overview talk by Per-Gunnar Martinsson, outlining the challenges of developing fast direct solvers for high-frequency problems. We then had talks by Steffen Börm and Markus Melenk on novel directional H2 matrix techniques for highly oscillatory problems. 2870 Oberwolfach Report 50/2016 In the evening Timo Betcke led a discussion on large-scale industrial challenges for high-frequency solvers and the need to develop large-scale coupled FEM/BEM domain decomposition frameworks to address them. The second day saw talks by Adrianna Gillman and Alex Barnett on fast direct solvers for oscillatory problems, and on the fast solution of periodic problems, respectively. The algorithms presented in these talks produced stunning results and were backed up by beautiful graphical visualizations of solutions of oscillatory problems in two and three space dimensions (see also the respective extended abstracts). Significant discussions were created by Mike O’Neil’s talk. He presented novel results on butterfly algorithms, and it was decided to devote the whole Wednesday afternoon to a more detailed understanding of butterfly algorithms. Butterfly compression has the potential to significantly improve the efficiency of fast direct solvers for oscillatory problems and a lot of work is currently going into the development of novel algorithms based on butterfly ideas. A remarkable result of the butterfly discussions during the week was that directional H2 structures applied to individual admissible blocks lead to a butterfly representation. This opens up the potential to apply algorithmic developments for H2 matrices to butterfly decompositions. Due to weather changes the traditional tour was done together with the other workshop groups already on Tuesday afternoon. On Wednesday the focus shifted to fast multipole methods and preconditioning. Stéphanie Chaillat started with an overview talk on Fast Multipole Methods and various applications in elastodynamics, followed by a talk by Marion Darbas on novel analytic preconditioners for high-frequency elastic problems. The talks were concluded on Wednesday by an overview presentation by Timo Betcke on the BEM++ software framework which provides solvers for a wide range of electrostatic, acoustic and electromagnetic problems. In the afternoon the aforementioned discussions on butterfly algorithms, led by Mike O’Neil took place. Thursday started with the second part of Stéphanie Chaillat’s overview talk on fast solvers for elastodynamics. This was followed by an overview by Sabine Le Borne on the use of fast hierarchical matrix solver techniques for integral equations in scattered data approximation problems. The final talk of the day was an overview talk by Martin Gander on variants of optimized Schwarz domain decomposition solvers for high-frequency problems. His framework generalizes a range of methods, including sweeping preconditioners, polarized traces, and multitrace formulations. This sparked many discussions in the afternoon, leading over to Friday which concluded with talks by Ivan Graham on shifted Laplacian preconditioners and an introduction to the ideas behind polarized traces by Laurent Demanet, which provided a fitting conclusion to the workshop. The workshop created a unique atmosphere to bring together fast solver experts from the domain decomposition and the boundary integral equation community. Mini-Workshop: Fast Solvers for Highly Oscillatory Problems 2871 The main threads that developed throughout the workshop were the efficient use of directional approximations and butterfly ideas, numerical and analytic DtN approximations for preconditioning and as transmission conditions in domain decomposition methods, and unifying domain decomposition frameworks that can incorporate a range of currently investigated methods. The format of the workshop allowed to exchange these ideas and give strong impulses for future research into fast high-frequency solvers. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mini-Workshop: Fast Solvers for Highly Oscillatory Problems 2873 Mini-Workshop: Fast Solvers for Highly Oscillatory Problems

An efficient boundary element solver for trans-abdominal high-intensity focused ultrasound treatment planning

High-intensity focused ultrasound (HIFU) is a promising treatment modality for the non-invasive ablation of pathological tissue in many organs, including the liver. Since many patients are not suitable candidates for liver surgery, the possibility to locally deposit thermal energy in a non-invasive way would bear significant clinical impact. Optimal treatment planning strategies based on high-performance computing numerical methods are expected to form a vital component of a successful clinical outcome in which healthy tissue is preserved and optimal focusing achieved, thus compensating for soft tissue heterogeneity and the presence of ribs. The boundary element method (BEM) is an effective approach for this purpose because only the boundaries of the ribs and soft tissue regions require discretization, as opposed to standard approaches which require the entire volume around the ribcage to be meshed. A Galerkin discretized Burton-Miller formulation used in combination with preconditioning and matrix compre...

Software frameworks for integral equations in electromagnetic scattering based on Calderón identities

Abstract In recent years there have been tremendous advances in the theoretical understanding of boundary integral equations for Maxwell problems. In particular, stable dual pairings of discretisation spaces have been developed that allow robust formulations of the preconditioned electric field, magnetic field and combined field integral equations. Within the BEM++ boundary element library we have developed implementations of these frameworks that allow an intuitive formulation of the typical Maxwell boundary integral formulations within a few lines of code. The basis for these developments is an efficient and robust implementation of Calderon identities together with a product algebra that hides and automates most technicalities involved in assembling Galerkin boundary integral equations. In this paper we demonstrate this framework and use it to derive very simple and robust software formulations of the standard preconditioned electric field, magnetic field and regularised combined field integral equations for Maxwell.

Building integral equation methods with the open-source library BEM++

Surface Integral Equations are often used to model electromagnetic scattering phenomena. Large-scale problems can efficiently be solved with Boundary Element Methods (BEM) of which the Method of Moments (MoM) has found widespread use in the computational electromagnetics community. The framework of integral equations allows for the design of many different formulations for a wide range of scattering problems. Most of them are a clever combination of the basic electric and magnetic field integral operators. In this paper, the open-source library BEM++ will be used as a powerful tool to build different integral equation formulations and preconditioners.