arXiv subjects

Olof Runborg

Publications and source records attributed to Olof Runborg.

At least 19 recordsLinked to original sources

arXiv2026

The WaveHoltz Heterogeneous Multiscale Method

We consider the numerical solution of the wave equation in materials with rapidly varying coefficients, and time harmonic sources. For these problems, direct discretization is prohibitively costly, and instead multiscale methods are used. There are several multiscale methods that directly discretize in the frequency domain. In this work we instead start in the time-domain and combine a finite difference Heterogeneous Multiscale Method (HMM) for the wave equation with the WaveHoltz method. Each WaveHoltz iteration marches the wave equation towards the time-periodic Helmholtz solution. The advantages of the WaveHoltz method relative to traditional Helmholtz solvers carry over directly to the multiscale problems considered here. Since, in addition, the time-domain solver does not artificially impose boundary conditions on the micro-scale problems, no boundary errors from the micro-scale problems are present in the homogenized frequency domain solution.

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arXiv2026

Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions

We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including those with jump discontinuities in the function and its derivatives -- and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work.

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arXiv2025

Convergence of the Waveholtz Iteration on $\mathbb{R}^d$

In this paper we analyse the Waveholtz method, a time-domain iterative method for solving the Helmholtz iteration, in the constant-coefficient case in all of $\mathbb{R}^d$. We show that the difference between a Waveholtz iterate and the outgoing Helmholtz solution satisfies a Helmholtz equation with a particular kind of forcing. For this forcing, we prove a frequency-explicit estimate in weighted Sobolev norms, that shows a decrease of the differences as $1/\sqrt{n}$ in terms of the iteration number $n$. This guarantees the convergence of the real parts of the Waveholtz iterates to the real part of the outgoing solution of the Helmholtz equation.

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arXiv2024

Convergence of the Semi-Discrete WaveHoltz Iteration

In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O({\omega})$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results.

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arXiv2023

Error Analysis for the Implicit Boundary Integral Method

The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform Cartesian grids for boundaries with different degree of regularities. We first show that the quadrature error gains an addition order of $\frac{d-1}{2}$ from the curvature for a strongly convex smooth boundary due to the ``randomness'' in the signed distances. This gain is discounted for degenerated convex surfaces. We then extend the error estimate to general boundaries under some special circumstances, including how quadrature error depends on the boundary's local geometry relative to the underlying grid. Bounds on the variance of the quadrature error under random shifts and rotations of the lattices are also derived.

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arXiv2023

Error estimates for Gaussian beams at a fold caustic

In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order $O(k^{-5/6})$ where $k$ is the wave number in Helmholtz.

math.CA↗

arXiv2022

Corrected Trapezoidal Rule-IBIM for linearized Poisson-Boltzmann equation

In this paper, we solve the linearized Poisson-Boltzmann equation, used to model the electric potential of macromolecules in a solvent. We derive a corrected trapezoidal rule with improved accuracy for a boundary integral formulation of the linearized Poisson-Boltzmann equation. More specifically, in contrast to the typical boundary integral formulations, the corrected trapezoidal rule is applied to integrate a system of compacted supported singular integrals using uniform Cartesian grids in $\mathbb{R}^3$, without explicit surface parameterization. A Krylov method, accelerated by a fast multipole method, is used to invert the resulting linear system. We study the efficacy of the proposed method, and compare it to an existing, lower order method. We then apply the method to the computation of electrostatic potential of macromolecules immersed in solvent. The solvent excluded surfaces, defined by a common approach, are merely piecewise smooth, and we study the effectiveness of the method for such surfaces.

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arXiv2022

Convergence of a class of high order corrected trapezoidal rules

We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nodes close to the singularity are judiciously corrected based on these expansions. Then we define the composite corrected trapezoidal rules for a larger family of functions using series expansions around the point singularity and applying corrected trapezoidal rules appropriately. We prove that we can achieve high order accuracy by using a sufficient number of correction nodes around the point singularity and of expansion terms.

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arXiv2022

Extensions and Analysis of an Iterative Solution of the Helmholtz Equation via the Wave Equation

In this paper we extend analysis of the WaveHoltz iteration -- a time-domain iterative method for the solution of the Helmholtz equation. We expand the previous analysis of energy conserving problems and prove convergence of the WaveHoltz iteration for problems with impedance boundary conditions in a single spatial dimension. We then consider interior Dirichlet/Neumann problems with damping in any spatial dimension, and show that for a sufficient level of damping the WaveHoltz iteration converges in a number of iteration independent of the frequency. Finally, we present a discrete analysis of the WaveHoltz iteration for a family of higher order time-stepping schemes. We show that the fixed-point of the discrete WaveHoltz iteration converges to the discrete Helmholtz solution with the order of the time-stepper chosen. We present numerical examples and demonstrate that it is possible to completely remove time discretization error from the WaveHoltz solution through careful analysis of the discrete iteration together with updated quadrature formulas.

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arXiv2022

El-WaveHoltz: A Time-Domain Iterative Solver for Time-Harmonic Elastic Waves

We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.

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arXiv2022

High order corrected trapezoidal rules for a class of singular integrals

We present a family of high order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the Implicit Boundary Integral formulation of surface integrals involving the Laplace layer kernels.

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arXiv2021

Heterogeneous multiscale methods for the Landau-Lifshitz equation

In this paper, we present a finite difference heterogeneous multiscale method for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Numerical examples for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.

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arXiv2021

Corrected trapezoidal rules for singular implicit boundary integrals

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.

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arXiv2021

Upscaling errors in Heterogeneous Multiscale Methods for the Landau-Lifshitz equation

In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau-Lifshitz equation with highly oscillatory coefficient is chosen as the micro problem for all models. We then show that the averaging errors only depend on $\varepsilon$, the size of the microscopic oscillations, as well as the size of the averaging domain in time and space and the choice of averaging kernels.

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arXiv2020

Homogenization of the Landau-Lifshitz equation

In this paper, we consider homogenization of the Landau-Lifshitz equation with a highly oscillatory material coefficient with period $\varepsilon$ modeling a ferromagnetic composite. We derive equations for the homogenized solution to the problem and the corresponding correctors and obtain estimates for the difference between the exact and homogenized solution as well as corrected approximations to the solution. Convergence rates in $\varepsilon$ over times $O(\varepsilon^\sigma)$ with $0\leq \sigma\leq 2$ are given in the Sobolev norm $H^q$, where $q$ is limited by the regularity of the solution to the detailed Landau-Lifshitz equation and the homogenized equation. The rates depend on $q$, $\sigma$ and the the number of correctors.

math.AP↗

arXiv2019

WaveHoltz: Iterative Solution of the Helmholtz Equation via the Wave Equation

A new idea for iterative solution of the Helmholtz equation is presented. We show that the iteration which we denote WaveHoltz and which filters the solution to the wave equation with harmonic data evolved over one period, corresponds to a coercive operator or a positive definite matrix in the discretized case.

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arXiv2019

Stochastic regularity of general quadratic observables of high frequency waves

We consider the wave equation with uncertain initial data and medium, when the wavelength $\varepsilon$ of the solution is short compared to the distance traveled by the wave. We are interested in the statistics for quantities of interest (QoI), defined as functionals of the wave solution, given the probability distributions of the uncertain parameters in the wave equation. Fast methods to compute this statistics require considerable smoothness in the mapping from parameters to the QoI, which is typically not present in the high frequency case, as the oscillations on the $\varepsilon$ scale in the wave field is inherited by the QoIs. The main contribution of this work is to identify certain non-oscillatory quadratic QoIs and show $\varepsilon$-independent estimates for the derivatives of the QoI with respect to the parameters, when the wave solution is replaced by a Gaussian beam approximation.

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arXiv2016

Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media

This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macromodel. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method, that we consider here, was previously addressed only in purely periodic media although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally-periodic medium where slow and fast variations are allowed at the same time. We then prove that HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multi-dimensional setting. The theoretical findings here imply an improved convergence rate in one-dimension, which also justifies the numerical observations from our earlier study.

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