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Alexander Rieder

Publications and source records attributed to Alexander Rieder.

At least 19 recordsLinked to original sources
arXiv2024

Towards optimal hierarchical training of neural networks

We propose a hierarchical training algorithm for standard feed-forward neural networks that adaptively extends the network architecture as soon as the optimization reaches a stationary point. By solving small (low-dimensional) optimization problems, the extended network provably escapes any local minimum or stationary point. Under some assumptions on the approximability of the data with stable neural networks, we show that the algorithm achieves an optimal convergence rate s in the sense that loss is bounded by the number of parameters to the -s. As a byproduct, we obtain computable indicators which judge the optimality of the training state of a given network and derive a new notion of generalization error.

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arXiv2024

A $p$-version of convolution quadrature in wave propagation

We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the timestep size ($h$-method), we achieve accuracy by increasing the order of the method ($p$-method). We base this method on discontinuous Galerkin timestepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observes(root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the findings.

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arXiv2024

FEM-BEM coupling for the high-frequency Helmholtz problem

We present a wavenumber-explicit analysis of FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in arXiv:2004.03523 for conforming discretizations and in arXiv:2105.06173 for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that $kh/p$ be sufficiently small and that $\log(k) / p$ be bounded imply quasi-optimality of both conforming and DG-method, where $k$ is the wavenumber, $h$ the mesh size, and $p$ the approximation order. The analysis relies on a $k$-explicit regularity theory for a three-field coupling formulation.

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arXiv2023

On interpolation spaces of piecewise polynomials on mixed meshes

We consider fractional Sobolev spaces $H^θ$, $θ\in (0,1)$, on 2D domains and $H^1$-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shape-regular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the $L^2$-norm on the one hand and the $H^1$-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between $H^1$ and $H^θ$.

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arXiv2023

FEM-BEM coupling in Fractional Diffusion

We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space $\mathbb{R}^d$ . Based on a reformulation using the well-known Caffarelli-Silvestre extension, we study a modified variational formulation to obtain well-posedness of the discrete problem. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.

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arXiv2023

Fractional Diffusion in the full space: decay and regularity

We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $\R^d \times (0,\YY)$ converge to the solution of the original problem as $\YY \rightarrow \infty$. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.

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arXiv2022

An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

We consider a space-time fractional parabolic problem. Combining a sinc-quadrature based method for discretizing the Riesz-Dunford integral with $hp$-FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an $hp$-quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times $t$, proving robustness with respect to startup singularities due to data incompatibilities.

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arXiv2021

Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.

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arXiv2020

Double exponential quadrature for fractional diffusion

We introduce a novel discretization technique for both elliptic and parabolic fractional diffusion problems based on double exponential quadrature formulas and the Riesz-Dunford functional calculus. Compared to related schemes, the new method provides faster convergence with fewer parameters that need to be adjusted to the problem. The scheme takes advantage of any additional smoothness in the problem without requiring a-priori knowledge to tune parameters appropriately. We prove rigorous convergence results for both, the case of finite regularity data as well as for data in certain Gevrey-type classes. We confirm our findings with numerical tests.

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arXiv2020

On superconvergence of Runge-Kutta convolution quadrature for the wave equation

The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $\abs{s}$(up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.

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arXiv2020

Metastable Speeds in the Fractional Allen-Cahn Equation

We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-Δ)^{α/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $α$ of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of $α$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for $α\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

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arXiv2020

FEM-BEM mortar coupling for the Helmholtz problem in three dimensions

We present a FEM-BEM coupling strategy for time-harmonic acoustic scattering in media with variable sound speed. The coupling is realized with the aid of a mortar variable that is an impedance trace on the coupling boundary. The resulting sesquilinear form is shown to satisfy a Garding inequality. Quasi-optimal convergence is shown for sufficiently fine meshes. Numerical examples confirm the theoretical convergence results.

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arXiv2020

Runge-Kutta approximation for $C_0$-semigroups in the graph norm with applications to time domain boundary integral equations

We consider the approximation to an abstract evolution problem with inhomogeneous side constraint using $A$-stable Runge-Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator. These results are used to study convolution quadrature based discretizations of a wave scattering and a heat conduction problem.

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arXiv2019

Stable decompositions of $hp$-BEM spaces and an optimal Schwarz preconditioner for the hypersingular integral operator in 3D

We consider fractional Sobolev spaces $H^θ(Γ)$, $θ\in [0,1]$, on a 2D surface $Γ$. We show that functions in $H^θ(Γ)$ can be decomposed into contributions with local support in a stable way. Stability of the decomposition is inherited by piecewise polynomial subspaces. Applications include the analysis of additive Schwarz preconditioners for discretizations of the hypersingular integral operator by the $p$-version of the boundary element method with condition number bounds that are uniform in the polynomial degree $p$.

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arXiv2019

$hp$-FEM for the fractional heat equation

We consider a time dependent problem generated by a nonlocal operator in space. Applying a discretization scheme based on $hp$-Finite Elements and a Caffarelli-Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-Discontinuous Galerkin based timestepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the original domain $Ω$.

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arXiv2017

Time-domain boundary integral equation modeling of heat transmission problems

This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in spatial semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. In comparison to the traditional Laplace domain approach, we show our approach gives better estimates.

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arXiv2016

Runge-Kutta convolution quadrature and FEM-BEM coupling for the time dependent linear Schr\"odinger equation

We propose a numerical scheme to solve the time dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element discretization in space. Since the Schr\"odinger equation is posed on the whole space $\R^d$ we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

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