Ralf Hiptmair


Last Name

Hiptmair

First Name

Ralf

ORCID

0000-0002-1378-2668

Organisational unit

03632 - Hiptmair, Ralf / Hiptmair, Ralf

Search Results

Publications1 - 10 of 153
  • Auxiliary space preconditioners for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficients
    Item type: Journal Article
    Ayuso de Dios, Blanca; Hiptmair, Ralf; Pagliantini, Cecilia (2016)
    IMA Journal of Numerical Analysis
    We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric interior penalty discontinuous Galerkin discretization of H(curl,Ω)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners rely on the auxiliary space method (ASM) employing an auxiliary space of H(curl,Ω)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ν and β in the second- and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,Ω)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners.
  • First-Kind Boundary Integral Equations for the Dirac Operator in 3d Lipschitz Domains
    Item type: Report
    Schulz, Erick; Hiptmair, Ralf (2020)
    SAM Research Report
    We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain’s boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H−1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.
  • Natural BEM for the Electric Field Integral Equation on polyhedra
    Item type: Report
    Hiptmair, Ralf; Schwab, Christoph (2001)
    SAM Research Report
    We consider the electric field integral equation on the surface of polyhedral domains and its Galerkin-discretization by means of divergence-conforming boundary elements. With respect to a Hodge decomposition the continuous variational problem is shown to be coercive. However, this does not immediately carry over to the discrete setting, as discrete Hodge decompositions fail to possess essential regularity properties. Introducing an intermediate semidiscrete Hodge decomposition we can bridge the gap and come up with asymptotically optimal a-priori error estimates. Hitherto, those had been elusive, in particular for non-smooth boundaries.
  • Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM
    Item type: Journal Article
    Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria; et al. (2014)
    ESAIM: Mathematical Modelling and Numerical Analysis
    We study the approximation of harmonic functions by means of harmonic polynomials in twodimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a $\delta$-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on $\delta$. We apply the obtained estimates to show exponential convergence with rate $O(exp(-b\sqrt{N}))$, $N$ being the number of degrees of freedom and $b > 0$, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate $O(exp(-b \sqrt[3]{N}))$, and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.
  • Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations
    Item type: Journal Article
    Hiptmair, Ralf; Wu, Haijun; Zheng, Weiying (2012)
    Numerical Mathematics
    We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.
  • Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation
    Item type: Journal Article
    Carrera Vazquez, Almudena; Hiptmair, Ralf; Woerner, Stefan (2022)
    ACM Transactions on Quantum Computing
    We present a quantum algorithm to solve systems of linear equations of the form Ax=b, where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O(1/√ε, poly (log κ, log N)), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O(κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O(1/ε2) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.
  • Convergence of lowest order semi-Lagrangian schemes
    Item type: Report
    Heumann, Holger; Hiptmair, Ralf (2011)
    SAM Research Report
    We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in $R^n$. We provide comprehensive a priori convergence estimates for their spatio-temporal discretization by means of a semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic $L^2$-estimate $O(\tau + h^r + h^{r+1} \tau^{-1/2} + \tau^{1/2})$, where $h$ is the spatial meshwidth, $\tau$ denotes the timestep, and $r$ the polynomial degree of the forms used as trial functions. This estimate can even be improved considerably in a variety of special settings.
  • Multi-trace boundary integral equations
    Item type: Report
    Claeys, Xavier; Hiptmair, Ralf; Jerez-Hanckes, Carlos (2012)
    SAM Research Report
    We consider the scattering of acoustic or electromagnetic waves at a penetrable object composed of different homogeneous materials. This problem can be recast as a firstkind boundary integral equation posed on the interface trace spaces through what we call a single trace boundary integral equation formulation (STF). Its Ritz-Galerkin discretization by means of low-order piecewise polynomial boundary elements on fine interface triangulations leads to ill-conditioned linear systems of equations, which defy efficient iterative solution. Powerful preconditioners for discrete boundary integral equations are provided by the policy of operator preconditioning provided that the underlying trace spaces support a duality pairing with L2 pivot space. This condition is not met by the STF. As a remedy we have proposed two variants of new multi-trace boundary integral equations (MTF); whereas the STF features unique Cauchy traces on material domain interfaces as unknowns, the multi-trace approach tears apart the traces so that local traces are recovered. Local trace spaces are in duality with respect to the L2-pairing, and, thus, operator preconditioning becomes available for MTF.
  • Stable Skeleton Integral Equations for General-Coefficient Helmholtz Transmission Problems
    Item type: Report
    Gräßle, Benedikt; Hiptmair, Ralf; Sauter, Stefan (2025)
    SAM Research Report
    A novel variational formulation of layer potentials and boundary integral op- erators generalizes their classical construction by Green’s functions, which are not explicitly available for Helmholtz problems with variable coefficients. Wavenumber explicit estimates and properties like jump conditions follow directly from their variational definition and enable a non-local (“integral”) formulation of acoustic transmission problems (TP) with piecewise Lipschitz coefficients. We obtain the well-posedness of the integral equations directly from the stability of the underlying TP. The simultaneous analysis for general dimensions and complex wavenumbers (in this paper) imposes an artificial boundary on the external Helmholtz problem and employs recent insights into the associated Dirichlet-to-Neumann map.
  • Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem
    Item type: Journal Article
    Hiptmair, Ralf; Moiola, Andrea; Spence, Euan A. (2022)
    SIAM Journal on Applied Mathematics
    We consider the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur and propose modified boundary integral equations that are not affected by them.
Publications1 - 10 of 153