arXiv subjects

Ralf Hiptmair

Publications and source records attributed to Ralf Hiptmair.

30 records · Page 2Linked to original sources

arXiv2021

First-Kind Boundary Integral Equations for the Dirac Operator in 3D Lipschitz Domains

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.

math.AP↗

arXiv2021

Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation

We present a quantum algorithm to solve systems of linear equations of the form $A\mathbf{x}=\mathbf{b}$, where $A$ is a tridiagonal Toeplitz matrix and $\mathbf{b}$ results from discretizing an analytic function, with a circuit complexity of $poly(\log(κ), 1/\sqrtε, \log(N))$, where $N$ denotes the number of equations, $ε$ is the accuracy, and $κ$ the condition number. The \emph{repeat-until-success} algorithm has to be run $\mathcal{O}\left(κ/(1-ε)\right)$ times to succeed, leveraging amplitude amplification, and sampled $\mathcal{O}(1/ε^2)$ 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 $1/\sqrtε$ circuit complexity instead of $1/ε$ and which can be parallelized. 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.

quant-ph↗

arXiv2020

A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. The transmission interfaces may join the domain boundary in a general way independent of the location of the boundary conditions. We will derive a formulation as a \textit{direct}, \textit{space-time retarded boundary integral equation}, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary. We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain.

math.NA↗

arXiv2019

Elastic energy regularization for inverse obstacle scattering problems

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how self-intersections can be avoided by penalization with the Möbius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions. In the second part of the paper the discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on a shape-manifold. Numerical examples demonstrate the feasibility of reconstructing non-star-shaped obstacles.

math.NA↗

arXiv2017

Closed-Form Exact Inverses of the Weakly Singular and Hypersingular Operators On Disks

We introduce new boundary integral operators which are the exact inverses of the weakly singular and hypersingular operators for the Laplacian on flat disks. Moreover, we provide explicit closed forms for them and prove the continuity and ellipticity of their corresponding bilinear forms in the natural Sobolev trace spaces. This permit us to derive new Calderón-type identities that can provide the foundation for optimal operator preconditioning in Galerkin boundary element methods.

math.AP↗

arXiv2015

Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation

We describe and analyse a space-time Trefftz discontinuous Galerkin method for the wave equation. The method is defined for unstructured meshes whose internal faces need not be aligned to the space-time axes. We show that the scheme is well-posed and dissipative, and we prove a priori error bounds for general Trefftz discrete spaces. A concrete discretisation can be obtained using piecewise polynomials that satisfy the wave equation elementwise.

math.NA↗

arXiv2010

Discrete compactness for the p-version of discrete differential forms

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.

math.NA↗

arXiv2010

Eulerian and Semi-Lagrangian Methods for Convection-Diffusion for Differential Forms

We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Details of implementation are discussed as well as an application to the discretization of eddy current equations in moving media.

math.NA↗

arXiv2009

Convergence of the Natural hp-BEM for the Electric Field Integral Equation on Polyhedral Surfaces

We consider the variational formulation of the electric field integral equation (EFIE) on bounded polyhedral open or closed surfaces. We employ a conforming Galerkin discretization based on div-conforming Raviart-Thomas boundary elements (BEM) of locally variable polynomial degree on shape-regular surface meshes. We establish asymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or for sufficiently high polynomial degree.

math.NA↗

arXiv2009

Discrete Compactness for p-Version of Tetrahedral Edge Elements

We consider the first family of $\Hcurl$-conforming Nedéléc finite elements on tetrahedral meshes. Spectral approximation ($p$-version) is achieved by keeping the mesh fixed and raising the polynomial degree $p$ uniformly in all mesh cells. We prove that the associated subspaces of discretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discrete compactness property as $p\to\infty$. This permits us to conclude asymptotic spectral correctness of spectral Galerkin finite element approximations of Maxwell eigenvalue problems.

math.NA↗

arXiv2009

Local Multigrid in H(curl)

We consider H(curl)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H1-context along with local discrete Helmholtz-type decompositions of the edge element space.

math.NA↗

arXiv2006

Conservative discretization of the Einstein-Dirac equations in spherically symmetric spacetime

In computational relativity, critical behaviour near the black hole threshold has been studied numerically for several models in the last decade. In this paper we present a spatial Galerkin method, suitable for finding numerical solutions of the Einstein-Dirac equations in spherically symmetric spacetime (in polar/areal coordinates). The method features exact conservation of the total electric charge and allows for a spatial mesh adaption based on physical arclength. Numerical experiments confirm excellent robustness and convergence properties of our approach. Hence, this new algorithm is well suited for studying critical behaviour.

gr-qc↗