Heiko Gimperlein
Engineering Mathematics
LeopoldFranzensUniversität Innsbruck
6020 Innsbruck, Austria
phone: (0043) 512 507 61330
email: heiko.gimperlein at uibk.ac.at 
Publications
 Heiko Gimperlein, Bernhard Krötz, Luz Roncal and Sundaram Thangavelu,
Poisson transform and unipotent complex geometry.
preprint.
Our concern is with Riemannian symmetric spaces Z=G/K of the noncompact type and more precisely with the Poisson transform which maps generalized functions on the boundary of Z to eigenfunctions on Z. We give a novel characterization of the image of L^2(N) under the Poisson transform in terms of complex analysis.
 Alessandra Aimi, Giulia Di Credico, Heiko Gimperlein and Ernst P. Stephan,
Higherorder time domain boundary elements for elastodynamics  graded meshes and hpversions.
preprint.
The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasioptimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for polygonal domains in 2d. They exhibit the expected quasioptimal convergence rates and the singular behavior of the solutions.
 Heiko Gimperlein, Matthias Maischak and Ernst P. Stephan,
Coupling of finite and boundary elements for singularly nonlinear transmission and contact problems.
preprint, dedicated to W. Wendland, slides of talk.
This article discusses the wellposedness and error analysis of the coupling of finite and boundary elements for transmission or contact problems in nonlinear elasticity. It concerns W^{1,p}monotone Hencky materials with an unbounded stressstrain relation, as they arise in the modelling of ice sheets, nonNewtonian fluids or porous media. For pseudoplastic materials, where p is smaller than 2, the bilinear form of the boundary element method fails to be continuous in natural function spaces associated to the nonlinear operator. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality. The a posteriori estimate complements recent estimates obtained for mixed finite element formulations of friction problems in linear elasticity.
 Heiko Gimperlein, Ernst P. Stephan and Jakub Stocek,
Corner Singularities for the Fractional Laplacian and Finite Element Approximation.
preprint, slides of talk.
summarized in Oberwolfach Reports, to appear: 1 / 2.
The solutions to pseudodifferential boundary problems, such as for the fractional Laplacian, exhibit singular behavior at the boundary, resulting in slow convergence of Galerkin approximations by an hmethod on quasiuniform meshes. We discuss the asymptotic expansion of the solution at boundaries and corners and, in particular, consider in detail how the corner singularity depends on the order of the operator and the corner geometry. The results imply quasioptimal convergence rates for Galerkin approximations on graded meshes. Numerical experiments underline the theoretical results.
 Heiko Gimperlein, Runan He and Andrew A. Lacey,
Quenching for a semilinear wave equation for MEMS.
preprint.
We consider the formation of finitetime quenching singularities for solutions of semilinear wave equations with negative power nonlinearities, as can model microelectromechanical systems (MEMS). For radial initial data we
obtain, formally, quenching selfsimilar solutions. From formal asymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin approaches
the ODE profile near the quenching time. The analysis and numerical experiments suggest a detailed conjecture for the singular behaviour.
 Heiko Gimperlein, Runan He and Andrew A. Lacey,
Stable quenching for semilinear wave equations.
preprint.
We consider semilinear wave equations with negative power p nonlinearities in dimension n. We prove mode stability and, for n,p>3 and n odd, nonlinear stability of the constant selfsimilar solution in an open set of radial initial data. In particular, near the quenching time the solution approaches the ODE profile within the backward light cone.
 Heiko Gimperlein, Nikoletta Louca and Magnus Goffeng,
The magnitude and spectral geometry.
preprint, python code,
blog post at nCategory Cafe.
We study the geometric significance of Leinster's notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in the plane. For a large class of distance functions, including embedded submanifolds of Euclidean space and Riemannian manifolds satisfying a technical condition, we show that the magnitude function is well defined for sufficiently large R and admits a meromorphic continuation to sectors in the complex plane. In the semiclassical limit of large R, the magnitude function admits an asymptotic expansion, which determines the volume, surface area and integrals of generalized curvatures. Lowerorder terms are computed by black box computer algebra. We initiate the study of magnitude analogues to classical questions in spectral geometry and prove an asymptotic variant of the LeinsterWillerton conjecture.
 Heiko Gimperlein, Nikoletta Louca and Magnus Goffeng,
Semiclassical analysis of a nonlocal boundary value problem related to magnitude.
preprint.
We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in shorttime expansions for heat kernels.
 Gissell EstradaRodriguez, Heiko Gimperlein and Jakub Stocek,
Nonlocal interface problems: Modelling, regularity, finite element approximation.
preprint.
We study interface problems for nonlocal elliptic operators like the fractional Laplacian. We derive relevant strong formulations from microscopic movement models, as relevant to the movement of immune cells in the white and gray matter in the brain. For the resulting problems we obtain precise asymptotic expansions for the behaviour of solutions near the interface, depending on the fractional orders in the neighboring domains and the diffusion coefficients. The expansions allow to prove optimal convergence rates for finite element approximations on suitably graded meshes. Numerical experiments underline the analysis and show the rich behavior of solutions. This article is the first in a series on the local behavior of solutions near boundaries, interfaces and corner and their numerical approximation.
 Heiko Gimperlein, Ceyhun Ozdemir and Ernst P. Stephan,
Error estimates for FEBE coupling of scattering of waves in the time domain.
Computational Methods in Applied Mathematics (2022), online first, preprint.
This article considers a coupled finite and boundary element method for
an interface problem for the acoustic wave equation. Wellposedness, a priori and
a posteriori error estimates are discussed for a symmetric spacetime Galerkin
discretization related to the energy. Numerical experiments in 3 dimensions
illustrate the performance of the method in model problems.
 Heiko Gimperlein, Bernhard Krötz, Job J. Kuit and Henrik Schlichtkrull,
A PaleyWiener Theorem for HarishChandra modules.
Cambridge Journal of Mathematics 10 (2022), 689742, preprint.
We formulate and prove a PaleyWiener theorem for HarishChandra modules for
a real reductive group. As a corollary we obtain a new and elementary proof of
the Helgason conjecture.
 Heiko Gimperlein and Magnus Goffeng,
The Willmore energy and the magnitude of Euclidean domains.
Proceedings of the American Mathematical Society (2022), preprint.
We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain in an odddimensional Euclidean space, we show that the asymptotic expansion of the magnitude function at infinity determines the Willmore energy of the boundary. This disproves the LeinsterWillerton conjecture for a compact convex body in all odd dimensions.
 Siobhan Duncan*, Gissell EstradaRodriguez*, Jakub Stocek*, Mauro Dragone, Patricia Vargas and Heiko Gimperlein,
Efficient quantitative assessment of robot swarms: Coverage and targeting Levy strategies.
Bioinspiration and Biomimetics 17 (2022), 036006, preprint, slides of talk.
Bioinspired strategies have long been adapted to swarm robotic systems, including biased random walks, reaction to chemotactic cues and longrange coordination. In this paper we show the opportunities provided by analysis tools developed for biological systems: continuum models for the efficient quantitative characterization of robot swarms. They lead to fast methods for the optimization of robot movement laws to achieve a prescribed collective behavior. We show how to compute quantities of robotic interest and illustrate the accuracy and efficiency of our approach for target search and coverage problems. We compute area coverage, respectively hitting times, for Brownian as well as Levy strategies with a characteristic longrange movement. Comparisons between the continuum model and robotic simulations confirm the quantitative agreement and speed up $>100$ of our approach. They show the advantage of Levy strategies over Brownian motion for target search and coverage problems.
 Sara Bernardi, Gissell EstradaRodriguez, Heiko Gimperlein and Kevin Painter,
Macroscopic descriptions of followerleader systems.
Kinetic and Related Models 14 (2021), 9811002, preprint.
The fundamental derivation of macroscopic model equations to describe swarms based on
microscopic movement laws and mathematical analyses into their selforganisation
capabilities remains a challenge from both modelling and analysis perspectives. In this paper we
clarify relevant continuous macroscopic model equations that describe followerleader
interactions for a swarm where these two populations are fixed, using the case example
of honeybee swarming to motivate our model. We study the behaviour
of the swarm over long and short time scales to shed light on the number of leaders needed
to initiate swarm movement, according to the homogeneous or inhomogeneous nature of the
interaction (alignment) kernel. The results indicate the crucial role played by the
interaction kernel to model transient behaviour.
 Heiko Gimperlein, Jakub Stocek and Carolina UrzuaTorres,
Optimal operator preconditioning for pseudodifferential boundary problems.
Numerische Mathematik 148 (2021), 141, preprint, slides of talk.
We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain in either Rn or a Riemannian manifold. For the linear systems of equations arising from loworder GalerkinPetrov discretizations, we prove that the condition number is independent of the mesh size and of the
choice of bases for trial and test spaces. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach recovers and unifies a series of results by Hiptmair, JerezHanckes, Nedelec and UrzuaTorres, with an independent proof which does not require new explicit solution formulas. Numerical examples illustrate the performance of the proposed method.
 Heiko Gimperlein and Bernhard Krötz,
On Sobolev norms for Lie group representations.
Journal of Functional Analysis 280 (2021), 108882, preprint.
We define Sobolev norms of arbitrary real order for a Banach representation
of a Lie group, with regard to a single differential operator D. Here, D is a Laplace element in the universal
enveloping algebra plus a multiple R of the identity which depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for D on the
space of smooth vectors. The main tool is a novel factorization for the delta distribution on a Lie group.
 Heiko Gimperlein and Magnus Goffeng,
On the magnitude function of domains in Euclidean space.
American Journal of Mathematics 143 (2021),
939  967,
preprint, blog posts 1, 2, 3, 4, slides of talk.
We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X in an odddimensional Euclidean space, we find geometric significance in the function M(R) = mag(R X). M(R) extends from the positive halfline to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit M(R) admits an asymptotic expansion. The three leading terms are given by the volume, surface area and integral of the mean curvature. In particular, for convex X the three leading terms are given by the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients. The fourth term is the Willmore energy, in contrast to the conjectured Euler characteristic in 3d.
 Heiko Gimperlein, Matthias Maischak and Ernst P. Stephan,
FEBE coupling for a transmission problem involving microstructure,
Journal of Applied and Numerical Optimization 3 (2021), 315  331, preprint.
special issue in honor of J. Gwinner.
In this second article on the theoretical numerical analysis of nonlinear transmission
problems, we extend our previously introduced methods to a nonconvex doublewell potential coupled to the linear Laplace equation via contact conditions.
This problem arises in phenomenological models for phase transitions and does not even admit a weak solution.
However, certain macroscopic quantities like the stress, the region of fine oscillations,
socalled microstructure, or the displacement outside this region are captured in a Young measure,
which may be computed efficiently.
 Heiko Gimperlein, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
A residual a posteriori estimate for the timedomain boundary element method.
Numerische Mathematik 146 (2020), 239280, preprint, slides of talk.
This article investigates residual a posteriori error estimates and adaptive mesh refinements for timedependent boundary element methods for the wave equation. We
obtain reliable estimates for Dirichlet and acoustic boundary conditions which hold for
a large class of discretizations. Efficiency of the error estimate is shown for a natural
discretization of low order. Numerical examples confirm the theoretical results. The
resulting adaptive mesh refinement procedures in 3d recover the adaptive convergence
rates known for elliptic problems.
 Ernesto Estrada, Gissell EstradaRodriguez and Heiko Gimperlein,
Metaplex networks: Influence of the exoendo structure of complex systems on diffusion.
SIAM Review 62 (2020), 617645, Research Spotlights, preprint, slides of talk.
In a complex system the interplay between the internal structure of its entities and their interconnection may play a fundamental role in the global functioning of the system. Here, we define the concept of metaplex, which describes such tradeoff between internal structure of entities and their interconnections. We then define a dynamical system on a metaplex and study diffusive processes on them. We provide analytical and computational evidences about the role played by the size of the nodes, the location of the internal coupling areas, and the strength and range of the coupling between the nodes on the global dynamics of metaplexes. Finally, we extend our analysis to two realworld metaplexes: a landscape and a brain metaplex. We corroborate that the internal structure of the nodes in a metaplex may dominate the global dynamics (brain metaplex) or play a regulatory role (landscape metaplex) to the influence of the interconnection between nodes.
 Gissell EstradaRodriguez and Heiko Gimperlein,
Interacting particles with Levy strategies: limits of transport equations for swarm robotic systems.
SIAM Journal on Applied Mathematics 80 (2020), 476498, preprint, slides of talk movies: Levy robots with alignment interactions.
Levy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE descriptions from individual robot strategies. Starting from a kinetic equation which describes the movement of robots based on alignment, collisions and occasional long distance runs according to a Levy distribution, we obtain a system of evolution equations for the fractional diffusion for long times. We show that the system allows efficient parameter studies for a target search problem, addressing basic questions like the optimal number of robots needed to cover an area in a certain time. For shorter times, in the hyperbolic limit of the kinetic equation the PDE model is dominated by alignment, irrespective of the longrange movement. This is in agreement with previous results in swarming of selfpropelled particles. The article indicates the novel and quantitative modeling opportunities which swarm robotic systems provide for the study of both emergent collective behaviour and anomalous diffusion, on the respective time scales.
 Muhammad Iqbal, David Stark, Heiko Gimperlein, M. Shadi Mohamed and Omar Laghrouche,
Local adaptive qenrichments and generalized finite elements for transient heat diffusion problems.
Computer Methods in Applied Mechanics and Engineering 372 (2020), 113359, preprint.
 Muhammad Iqbal, Heiko Gimperlein, Omar Laghrouche, Khurshid Alam and M. Shadi Mohamed,
A residual a posteriori error estimate for partition of unity finite elements for three dimensional transientheat diffusion problems using multiple global enrichment functions.
International Journal for Numerical Methods in Engineering 121 (2020), 27272746, preprint.
 Muhammad Iqbal, Khurshid Alam, Heiko Gimperlein, Omar Laghrouche and M. Shadi Mohamed,
Effect of enrichment functions on GFEM solutions of timedependent conduction heat transfer problems.
Applied Mathematical Modelling 85 (2020), 89106, preprint.
 Heiko Gimperlein, Ceyhun Ozdemir and Ernst P. Stephan,
A timedependent FEMBEM coupling method for fluidstructure interaction in 3d.
Applied Numerical Mathematics 152 (2020), 4965, preprint.
We consider the wellposedness and a priori error estimates of a 3d FEMBEM coupling method for fluidstructure interaction in the time domain. For an elastic body immersed in a fluid, the exterior linear wave equation for the fluid is reduced to an integral equation on the boundary involving the PoincareSteklov operator. The resulting problem is solved using a Galerkin boundary element method in the time domain, coupled to a finite element method for the Lame equation inside the elastic body. Based on ideas from the timeindependent coupling formulation, we discuss the a priori error analysis and implementation of the proposed method. Numerical experiments underline the theoretical results for model problems.
 Heiko Gimperlein, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
hpversion time domain boundary elements for the wave equation on quasiuniform meshes.
Computer Methods in Applied Mechanics and Engineering 356 (2019), 145174, preprint.
slides of talk.
Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R3 exhibit singular behavior from the edges and corners.
We present quasioptimal hpexplicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasiuniform meshes on the boundary. The results are applied to an hpversion of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.
 Heiko Gimperlein and Jakub Stocek,
Spacetime adaptive finite elements for nonlocal parabolic variational inequalities.
Computer Methods in Applied Mechanics and Engineering 352 (2019), 137171, preprint. slides of talk.
This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting spacetime adaptive meshrefinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for 2dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of spacetime adaptive, as well as uniform and graded discretizations.
 Heiko Gimperlein and Magnus Goffeng,
Commutator estimates on contact manifolds and applications.
Journal of Noncommutative Geometry 13 (2019), 363406, preprint, slides of talk.
This article studies sharp norm estimates for the commutator of pseudodifferential operators with nonsmooth functions on closed subRiemannian manifolds. In particular, we obtain a Calderon commutator estimate: If D is a firstorder operator in the Heisenberg calculus and f is Lipschitz in the CarnotCaratheodory metric, then [D,f] extends to an L^2bounded operator. Using interpolation, it implies sharp weakSchatten class properties for the commutator between zerothorder operators and Holder continuous functions. We present applications to subRiemannian spectral triples on Heisenberg manifolds as well as to the regularization of a functional studied by EnglisGuoZhang.
 Gissell EstradaRodriguez, Heiko Gimperlein, Kevin Painter and Jakub Stocek,
Spacetime fractional diffusion in cell movement models with delay.
Mathematical Models and Methods in Applied Sciences 29 (2019), 6588, preprint, slides of talk.
The movement of organisms and cells can be governed by long distance runs according to an approximate Levy walk. In the case of T cells in infected brain tissue the runs are further interrupted by long delays. This article clarifies the form of continuous model equations for such movement:
In the biologically relevant regime we derive nonlocal diffusion equations, involving fractional Laplacians in space and fractional time derivatives,
from the microscopic velocity jump models used to interpret the experiments. Starting from powerlaw distributions of run and waiting times,
we investigate the relevant parabolic limit of a kinetic equation for the resting and moving individuals. The analysis and numerical experiments
shed light on how chemokines shorten the average time taken by T cells to find rare targets, and on the absence of chemotaxis. In addition to immunology, our work has implications for
similar Levy diffusion observed in other organisms, such as mussels, marine predators and monkeys.
 Heiko Gimperlein and David Stark,
Algorithmic aspects of enriched time domain boundary element methods.
Engineering Analysis with Boundary Elements 100 (2019), 118124, preprint.
 Heiko Gimperlein, Fabian Meyer, Ceyhun Ozdemir, David Stark and Ernst P. Stephan,
Boundary elements with mesh refinements for the wave equation.
Numerische Mathematik 139 (2018), 867912, preprint, slides of talk.
The solution of the wave equation in a polyhedral domain admits an asymptotic expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as an equivalent boundary integral equations in time domain and study the regularity properties and numerical approximation of the solution. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates expected for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet problem, the DirichlettoNeumann operator and applications to the sound emitted by a tire.
 Gissell EstradaRodriguez, Heiko Gimperlein and Kevin Painter,
Fractional PatlakKellerSegel equations for
chemotactic superdiffusion.
SIAM Journal on Applied Mathematics 78 (2018), 11551173, preprint, slides of talk..
The long range movement of certain organisms in the presence of a chemoattractant
can be governed by long distance runs, according to an approximate Levy distribution. This article
clarifies the form of biologically relevant model equations: We derive PatlakKellerSegellike equations
involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting
from a powerlaw distribution of run times, we derive a kinetic equation in which the collision term
takes into account the long range behaviour of the individuals. A fractional chemotactic equation
is obtained in a biologically relevant regime. Apart from chemotaxis, our work has implications for
biological diffusion in numerous processes.
 Heiko Gimperlein and David Stark,
On a preconditioner for time domain boundary element methods.
Engineering Analysis with Boundary Elements 96 (2018), 109114, preprint.
We propose a time stepping scheme for the spacetime systems obtained from Galerkin timedomain boundary element methods. Based on extrapolation, the method proves stable, becomes exact for increasing degrees of freedom and can be used either as a preconditioner, or as an efficient standalone solver for scattering problems with smooth solutions. It also significantly reduces the number of GMRES iterations for screen problems, with less regularity, and we explore its limitations for enriched methods based on nonpolynomial approximation spaces.
 Heiko Gimperlein, Fabian Meyer, Ceyhun Ozdemir and Ernst P. Stephan,
Time domain boundary elements for dynamic contact problems.
Computer Methods in Applied Mechanics and Engineering 333 (2018), 147175. preprint, slides of talk.
This article considers a unilateral contact problem for the scalar wave equation. The problem is reduced to a variational inequality for the DirichlettoNeumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed method. They indicate the stability and convergence beyond flat geometries.
 Heiko Gimperlein, Ceyhun Ozdemir and Ernst P. Stephan,
Time domain boundary element methods for the Neumann problem: Error estimates and acoustic problems.
Journal of Computational Mathematics 36 (2018), 7089, preprint, slides of talk.
special issue in honor of Hsiao, Nedelec and Wendland
We investigate time domain boundary element methods for the wave equation, with a view towards sound emission problems in computational acoustics. The Neumann problem is reduced to a time dependent integral equation for the hypersingular operator, and
we present a priori and a posteriori error estimates for conforming Galerkin approximations. Numerical experiments validate the convergence of our boundary element scheme and compare it with the numerical approximations obtained from an integral equation of the second kind. Computations in a halfspace, for varying acoustic properties of the street, illustrate the role of our approach in the study of traffic noise.
 Heiko Gimperlein, Quoc Thong Le Gia, Matthias Maischak and Ernst P. Stephan,
Solving approximate cloaking problems using finite element methods.
ANZIAM Journal 58 (2017), C162C174, Proceedings of CTAC 2016, preprint.
Motivated by approximate cloaking problems, we consider
a variable coefficient Helmholtz equation with a fixed wave number.
We use finite element methods to discretize the equation. Numerical
results investigate the cloaking behaviour in the numerical solutions.
 Claudio Dappiaggi, Heiko Gimperlein, Simone Murro and Alexander Schenkel,
Wavefront sets and polarizations on supermanifolds.
Journal of Mathematical Physics 58 (2017), 023504, 16 pages, preprint.
We develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable concept of superwavefront set for superdistributions which generalizes Dencker's polarization sets for vectorvalued distributions to supergeometry. In particular, our superwavefront sets are able to detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application, we study the singularities of distributional solutions of a supersymmetric field theory.
 Heiko Gimperlein and Magnus Goffeng,
Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds.
Forum of Mathematics, Sigma 5 (2017), e3, 57 pages, preprint, slides of talk.
We study the spectral behavior and noncommutative geometry of commutators [P,f], where P is an operator of order 0 with geometric origin and f a multiplication operator by a function. When f is Holder continuous, the spectral asymptotics is governed by singularities. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Holder continuous functions f, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
 Heiko Gimperlein and Alden Waters,
A deterministic optimal design problem for the heat equation.
SIAM Journal on Control and Optimization 55 (2017), 5169, preprint.
In everyday language, this article studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using wave packet decompositions from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semianalytic and can be approximated by solving a sequence of finitedimensional optimization problems.
 Lothar Banz, Heiko Gimperlein, Abderrahman Issaoui and Ernst P. Stephan,
Stabilized mixed hpBEM for frictional contact problems in linear elasticity.
Numerische Mathematik 135 (2017), 217263, preprint, slides of talk.
We analyze a highorder, adaptive boundary element method for the approximation of a frictional contact problem. As a basic problem of highorder polynomial ansatz functions, the resulting discretized system degenerates as the polynomial degree tends to infinity. In the 90s, Barbosa and Hughes showed that for finite elements the stability (LBB) condition could be restored by adding certain terms, which vanish for the exact solution. We show that their basic approach can be extended to the boundary element method, prove convergence and obtain an a posteriori error estimate. Numerical experiments confirm the theoretical results and explore the dependence on stabilization parameters.
 Heiko Gimperlein, Matthias Maischak and Ernst P. Stephan,
Adaptive timedomain boundary element methods and engineering applications.
invited survey, Journal of Integral Equations and Applications 29 (2017), 75105, preprint.
A review article about the recent progress in timedomain boundary element methods for the wave equation.
 Muhammad Iqbal, Heiko Gimperlein, M. Shadi Mohamed and Omar Laghrouche,
An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems.
International Journal for Numerical Methods in Engineering 110 (2017), 11031118, preprint.
conference: Proceedings of the 23rd UK Conference of the Association for Computational Mechanics in Engineering (2015), 445448, preprint.
conference: Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2015, 6 pages.
conference: Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2014, 6 pages.
We investigate a posteriori error estimates and their relevance to engineering applications for timedependent partition of unity finite element methods. We prove a simple, reliable a posteriori estimate of residual type for the heat equation. Numerical experiments illustrate its use for heat diffusion problems and engineering applications.
 David Stark and Heiko Gimperlein,
Boundary elements and mesh refinements for the wave equation.
conference: Proceedings of the 25th UK Conference of the Association for Computational Mechanics in Engineering (2017), 309312, preprint.
 Gissell EstradaRodriguez and Heiko Gimperlein,
Generalized finite elements for blowup solutions to reactiondiffusion equations.
conference: Proceedings of the 25th UK Conference of the Association for Computational Mechanics in Engineering (2017), 8184, preprint.
 Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
Corrigendum  Analytic representation theory of Lie groups: General theory and analytic
globalizations of HarishChandra modules,
Compositio Mathematica 153 (2017), 214217, preprint.
 Lothar Banz, Heiko Gimperlein, Zouhair Nezhi and Ernst P. Stephan,
Time domain BEM for sound radiation of tires.
Computational Mechanics 58 (2016), 4557, preprint, slides of talk.
This article presents computational results for certain benchmark problems related to the noise emission of car tires.
We use a time domain single layer potential ansatz to solve the acoustic wave equation outside a tire in the halfspace
above a street. The resulting integral equation on the boundary is solved approximately by a Galerkin boundary element method using piecewise
constant test and trial functions in space and time. After a validation experiment, we present numerical results for
the Aweighted sound pressure of a vibrating tyre, for the noise amplification due to the
hornlike geometry of the tire in conjunction with the road as well as for the Doppler effect.
 Heiko Gimperlein and Alden Waters,
Stability analysis in magnetic resonance elastography II.
Journal of Mathematical Analysis and Applications 434 (2016), 18011812, preprint.
In the sequel to Ammari, Waters and Zhang (2015), we show how the general Fredholm theory of elliptic boundary problems improves the available stability estimates for an inverse problem for the Stokes equation, with very short proofs. The new estimates prove the convergence of practically relevant numerical reconstruction schemes.
 David Stark and Heiko Gimperlein,
A partition of unity boundary element method for transient wave propagation.
conference: Proceedings of the 24th UK Conference of the Association for Computational Mechanics in Engineering (2016), 347351, preprint.
conference (best paper prize): Proceedings of the Infrastructure and Environment Scotland Postgraduate Conference 2015, 6 pages.
We introduce and study a timedomain boundary element method based on planewave basis functions.
 Muhammad Iqbal, M. Shadi Mohamed, Omar Laghrouche, Heiko Gimperlein, Mohammed Seaid and Jon Trevelyan
Solution of three dimensional transient heat diffusion problems using an enriched finite element method.
conference: Proceedings of the 24th UK Conference of the Association for Computational Mechanics in Engineering (2016), 197200 preprint.
We present a highly efficient generalized finite element method for the heat equation in three dimensions.
 Heiko Gimperlein, Zouhair Nezhi and Ernst P. Stephan,
A priori error estimates for a timedependent boundary element method for the acoustic wave equation in a halfspace.
Mathematical Methods in the Applied Sciences 40 (2017), 448462, special issue in honor of M. Costabel, preprint.
We investigate a timedomain Galerkin boundary element method for the wave equation outside a Lipschitz obstacle in an absorbing halfspace, with application to the sound radiation of tyres. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them.
 Heiko Gimperlein and Gerd Grubb,
Heat kernel estimates for pseudodifferential operators, fractional Laplacians and DirichlettoNeumann operators,
Journal of Evolution Equations 14 (2014), 4983, preprint.
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. Operators satisfying such estimates are known to have a rich spectral theory, and applications include maximal regularity for the associated evolution equation and a functional calculus in L^p. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the DirichlettoNeumann semigroup.
 Adrian Costea, Heiko Gimperlein and Ernst P. Stephan,
A NashHörmander iteration and boundary elements for the Molodensky problem, Numerische Mathematik 127 (2014), 134, preprint.
We investigate the mathematically justified numerical approximation of the nonlinear Molodensky problem, the free boundary problem to determine the unknown surface of the
earth from the gravitational potential and the gravity vector on this surface. While the wellposedness of this problem has been shown by Hörmander, from a numerical perspective only the linearized problem has been understood.
The solution, based on a smoothed NashHörmander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higherorder heat equation to overcome the loss of regularity.
We obtain a quantitative a priori estimate for the error after k steps, legitimize the choice of heatequation smoothing in this context and study the accurate evaluation of second derivatives of the gravitational potential on the boundary, using a representation in terms of a hypersingular integral. Numerical results, using a bondary element method for the exterior problem, compare the error between the approximation and the exact solution in a model problem.
Costea's thesis contains a preliminary, extended discussion.
 Lothar Banz, Adrian Costea, Heiko Gimperlein and Ernst P. Stephan,
Numerical simulations of the nonlinear Molodensky problem, Proceedings of the European Geosciences Union General Assembly 2013,
Studia geophysica et geodaetica 58 (2014), 489504, preprint, conference proceedings.
A complement to the article "A NashHörmander iteration and boundary elements for the Molodensky problem" concerning numerical and geodetic aspects.
 Heiko Gimperlein, Bernhard Krötz and Christoph Lienau,
Analytic factorization of Lie group representations,
Journal of Functional Analysis 262 (2012), 667681, preprint.
For every moderate growth representation of a real Lie group on a Frechet space E,
the article proves a DixmierMalliavin factorization theorem for the space of analytic
vectors E^w: There exists a natural algebra A of superexponentially decreasing analytic
functions on the group such that E^w = A * E^w. This theorem sharpens a classical result by
Nelson about the density of analytic vectors. While the rigid structure of analytic functions
makes a direct approach difficult, the crucial idea is to refine the functional calculus of Cheeger,
Gromov and Taylor to carefully transport selected functions from the real line to the group. The
geometric methods also strengthen certain conclusions from DixmierMalliavin's result, extending
Goodman's characterization that a vector is smooth whenever it is so for the Laplacian to nonunitary
representations and to analytic vectors.
 Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
Analytic representation theory of Lie groups: General theory and analytic
globalizations of HarishChandra modules,
Compositio Mathematica 147 (2011), 15811607, preprint,
Corrigendum.
A representation of a real Lie group on a vector space E is called analytic, if every
vector in E is analytic (i.e. its orbit map is a realanalytic function on the group) and if the natural inductive limit topology on the space of analytic vectors
coincides with the topology of E. The article introduces a coherent general framework to analyze such representations
and systematically studies the resulting categories of representations with moderate or arbitrary growth. For reductive groups, the minimal globalization of Kashiwara and Schmid is analyzed and an elementary proof
of Schmid's globalization theorem is obtained: Every HarishChandra module over the Lie algebra and a maximal compact subgroup has a unique globalization to a moderate growth analytic representation of the group. It is given by a quotient of A^N, where A denotes
the convolution algebra of analytic superexponentially decaying functions on the group from our
previous work with Krötz and Lienau, and the theorem effectively provides an equivalence of categories.
 Heiko Gimperlein, Matthias Maischak, Elmar Schrohe and Ernst P. Stephan,
Adaptive FEBE coupling for strongly nonlinear transmission problems with Coulomb friction,
Numerische Mathematik 117 (2011), 307332, preprint.
This is the first of two articles analyzing adaptive finite element/boundary element procedures
for nonlinear transmission problems with contact. Here we study a nonlinear uniformly monotone
operator such as the pLaplacian in a bounded Lipschitz domain coupled to the linear Laplace equation in the exterior complement.
While elliptic problems have been studied for many years, the proper framework for these degeneratelynonlinear problems
remained unclear. Our analysis shows how mixed L^pL^2Sobolev spaces provide a suitable setting to analyze the convergence
of the Galerkin approximations.
A posteriori error estimates similar to the ones for nonlinear Dirichlet problems allow to localize
the error of the approximation and refine the grid accordingly.
 Heiko Gimperlein, Bernhard Krötz and Henrik Schlichtkrull,
Analytic globalizations of HarishChandra modules, Oberwolfach Reports
7
(2010), 30503052,
preprint, conference proceedings.
This extended abstract summarizes the topics covered in my
talk in Overwolfach, November 2010. It outlines an elementary approach to
aspects of Schmid's and Kashiwara's work on analytic globalizations
of HarishChandra modules.
 Heiko Gimperlein,
Introduction to hyperbolic billiards, Oberwolfach Reports 7 (2010),
961964, preprint, conference proceedings.
This extended abstract summarizes the topics covered in my
survey talk on dispersive billiards and Sinai's local ergodicity theorem in Oberwolfach, April 2010.
 Heiko Gimperlein and Elmar Schrohe,
Spectral projections of boundary problems and the regularity of the eta function,
research announcement: preprint (2009).
In the article (and the announcement), we show that the eta function associated to any
selfadjoint elliptic boundary problem is regular in 0. Also, the value in 0 of the zeta function of
a possibly nonselfadjoint boundary problem is
independent of the spectral cut. Our result extends a subtle topological theorem of AtiyahPatodiSinger, Gilkey and
Wodzicki for closed manifolds. Technically, we prove a vanishing theorem for the noncommutative residue,
an essentially unique trace, on the Ktheory of a new "minimal" algebra of general boundary problems.
The theorem then follows by showing that the spectral projections of a boundary problem belong to
the algebra. This is the crucial result of the paper and, in the context of spectral asymptotics, resolves
a conjecture of Ivrii for secondorder operators.
 Heiko Gimperlein, Matthias Maischak, Elmar Schrohe and Ernst P. Stephan,
A Finite element / boundary element coupling method for strongly nonlinear
transmission problems with contact,
Oberwolfach Reports 5 (2008), 20772080, preprint, conference proceedings.
The announcement gives a numerically efficient variational formulation for the approximation of nonlinear transmission problems with contact. Certain mixed L^pL^2Sobolev spaces
are shown to provide a convenient framework to analyze Galerkin approximations of the solutions to e.g. the pLaplacian or a nonconvex doublewell potential in a bounded Lipschitz domain coupled to a linear equation in the exterior complement.
The analysis applies to systems encountered in nonlinear elasticity as well as to phenomenological models for phase transitions.
 Heiko Gimperlein, Stefan Wessel, Jörg Schmiedmayer and Luis Santos,
Random onsite interactions versus random potential in ultracold atoms in optical lattices,
Applied Physics B 82
(2006), 217224,
preprint.
The article continues our study of bosons with random interactions and provides details which did not fit into the Letter below.
 Heiko Gimperlein, Stefan Wessel, Jörg Schmiedmayer and Luis Santos,
Ultracold atoms in optical lattices with random onsite interactions,
Physical Review Letters 95
(2005), 170401/14,
preprint.
The Letter presents the zerotemperature phase diagram for bosons on a lattice with a random repulsive interaction between particles occupying the same site.
Characteristic qualitative changes are observed when compared to
the nonrandom case or a random chemical potential: The Mottinsulating regions shrink and eventually
vanish for any finite disorder strength beyond a sufficiently large filling factor. Furthermore, at
low values of the chemical potential both the superfluid and Mott insulator are stable towards
formation of a Bose glass leading to a possibly nontrivial tricritical point.
We propose a feasible experimental realization of our scenario in the context of ultracold atoms
on optical lattices, which circumvents certain difficulties encountered in current experiments with disordered cold gases.
