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Zhong, Y., Ren, K., Runborg, O. & Tsai, R. (2025). Error analysis for the implicit boundary integral method. BIT Numerical Mathematics, 65(1), Article ID 8.
Open this publication in new window or tab >>Error analysis for the implicit boundary integral method
2025 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 65, no 1, article id 8Article in journal (Refereed) Published
Abstract [en]

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 degrees of regularity. First, it is shown that the quadrature error gains an additional order of d-12 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. Then the extension of error estimate to general boundaries under some special circumstances is considered, 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.

Place, publisher, year, edition, pages
Springer Nature, 2025
Keywords
Error analysis, Implicit boundary integral method, Level set, Solvent-excluded surface
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-360580 (URN)10.1007/s10543-024-01051-8 (DOI)001400048000001 ()2-s2.0-85217806551 (Scopus ID)
Note

QC 20250227

Available from: 2025-02-26 Created: 2025-02-26 Last updated: 2025-02-27Bibliographically approved
Lafitte, O. & Runborg, O. (2023). Error estimates for Gaussian beams at a fold caustic. Asymptotic Analysis, 135(1-2), 209-255
Open this publication in new window or tab >>Error estimates for Gaussian beams at a fold caustic
2023 (English)In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 135, no 1-2, p. 209-255Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
IOS Press, 2023
Keywords
caustics, error estimates, Gaussian beams, High frequency wave asymptotics
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:kth:diva-339480 (URN)10.3233/ASY-231852 (DOI)001086995000008 ()2-s2.0-85175196377 (Scopus ID)
Note

QC 20231113

Available from: 2023-11-13 Created: 2023-11-13 Last updated: 2023-11-13Bibliographically approved
Izzo, F., Runborg, O. & Tsai, R. (2023). High-order corrected trapezoidal rules for a class of singular integrals. Advances in Computational Mathematics, 49(4), Article ID 60.
Open this publication in new window or tab >>High-order corrected trapezoidal rules for a class of singular integrals
2023 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 49, no 4, article id 60Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer Nature, 2023
Keywords
Singular integrals, Trapezoidal rules, Level set methods, Closest point projection, Boundary integral formulations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-333754 (URN)10.1007/s10444-023-10060-0 (DOI)001036744400001 ()2-s2.0-85165671333 (Scopus ID)
Note

QC 20230810

Available from: 2023-08-10 Created: 2023-08-10 Last updated: 2023-08-10Bibliographically approved
Izzo, F., Runborg, O. & Tsai, R. (2022). Corrected trapezoidal rules for singular implicit boundary integrals. Journal of Computational Physics, 461, 111193, Article ID 111193.
Open this publication in new window or tab >>Corrected trapezoidal rules for singular implicit boundary integrals
2022 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 461, p. 111193-, article id 111193Article in journal (Refereed) Published
Abstract [en]

We present new higher-order quadratures for a family of boundary integral operators re derived using the approach introduced in Kublik et al. (2013) [7]. 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.

Place, publisher, year, edition, pages
Elsevier BV, 2022
Keywords
Level set methods, Closest point projection, Boundary integral formulations, Singular integrals, Trapezoidal rules
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-313343 (URN)10.1016/j.jcp.2022.111193 (DOI)000793700800008 ()2-s2.0-85128231870 (Scopus ID)
Note

QC 20220602

Available from: 2022-06-02 Created: 2022-06-02 Last updated: 2022-10-07Bibliographically approved
Appelö, D., Garcia, F., Alvarez Loya, A. & Runborg, O. (2022). El-WaveHoltz: A time-domain iterative solver for time-harmonic elastic waves. Computer Methods in Applied Mechanics and Engineering, 401, Article ID 115603.
Open this publication in new window or tab >>El-WaveHoltz: A time-domain iterative solver for time-harmonic elastic waves
2022 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 401, article id 115603Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier BV, 2022
Keywords
Elastic wave equation, Helmholtz equation, Time-harmonic scattering, Acoustics, Convergence of numerical methods, Elastic waves, Harmonic analysis, Iterative methods, Linear systems, Shear flow, Time domain analysis, Elastic wave equations, Energy-conserving, Fixed-point iterations, Helmholtz problems, Helmholtz's equations, Iterative solvers, Time domain, Time-harmonic, Time-stepping schemes, Wave equations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-326975 (URN)10.1016/j.cma.2022.115603 (DOI)000985149900005 ()2-s2.0-85137751557 (Scopus ID)
Note

QC 20230522

Available from: 2023-05-16 Created: 2023-05-16 Last updated: 2023-06-06Bibliographically approved
Leitenmaier, L. & Runborg, O. (2022). Heterogeneous Multiscale Methods for the Landau–Lifshitz Equation. Journal of Scientific Computing, 93(3), Article ID 76.
Open this publication in new window or tab >>Heterogeneous Multiscale Methods for the Landau–Lifshitz Equation
2022 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 93, no 3, article id 76Article in journal (Refereed) Published
Abstract [en]

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. Further important factors that are taken into account are the choice of time integrator and the initial data for the micro problem which has to be set appropriately to get a consistent scheme. Numerical examples in one and two space dimensions and for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.

Place, publisher, year, edition, pages
Springer Nature, 2022
Keywords
Finite differences, Heterogeneous Multiscale Methods, Landau–Lifshitz, Micromagnetics, Nonlinear equations, Partial differential equations, Approximation errors, Artificial damping, Efficient implementation, Finite difference, Finite difference heterogeneous multiscale methods, Heterogeneous multiscale method, Higher order discretization, Landau Lifshitz equation, Landau-Lifshitz, Numerical methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-328919 (URN)10.1007/s10915-022-01992-8 (DOI)000880353700003 ()2-s2.0-85141479487 (Scopus ID)
Note

QC 20230613

Available from: 2023-06-13 Created: 2023-06-13 Last updated: 2023-06-13Bibliographically approved
Leitenmaier, L. & Runborg, O. (2022). On homogenization of the Landau-Lifshitz equation with rapidly oscillating material coefficient. Communications in Mathematical Sciences, 20(3), 653-694
Open this publication in new window or tab >>On homogenization of the Landau-Lifshitz equation with rapidly oscillating material coefficient
2022 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 20, no 3, p. 653-694Article in journal (Refereed) Published
Abstract [en]

In this paper, we consider homogenization of the Landau-Lifshitz equation with a highly oscillatory material coefficient with period epsilon 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 epsilon over times O(epsilon(sigma)) with 0 <= sigma <= 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 number of correctors.

Place, publisher, year, edition, pages
International Press of Boston, Inc., 2022
Keywords
Homogenization, micromagnetics, magnetization dynamics, multiscale
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-311035 (URN)10.4310/CMS.2022.v20.n3.a3 (DOI)000776355400003 ()2-s2.0-85128299475 (Scopus ID)
Note

Not duplicate with DiVA 1611059

QC 20220421

Available from: 2022-04-21 Created: 2022-04-21 Last updated: 2022-06-25Bibliographically approved
Leitenmaier, L. & Runborg, O. (2022). Upscaling errors in Heterogeneous Multiscale Methods for the Landau-Lifshitz equation. Multiscale Modeling & simulation, 20(1), 1-35
Open this publication in new window or tab >>Upscaling errors in Heterogeneous Multiscale Methods for the Landau-Lifshitz equation
2022 (English)In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 20, no 1, p. 1-35Article in journal (Refereed) Published
Abstract [en]

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 a highly oscillatory coefficient is chosen as the micro problem for all models. We then show that the averaging errors only depend on \varepsilon and 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.

Place, publisher, year, edition, pages
Society for Industrial & Applied Mathematics (SIAM), 2022
Keywords
&nbsp, heterogeneous multiscale methods, micromagnetics, magnetization dynamics
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-310029 (URN)10.1137/21M1409408 (DOI)000760284700001 ()2-s2.0-85130574540 (Scopus ID)
Note

QC 20220323

No duplicate with DiVA 1611064

Available from: 2022-03-23 Created: 2022-03-23 Last updated: 2023-02-21Bibliographically approved
Leitenmaier, L. & Runborg, O. (2021). On homogenization of the Landau-Lifshitz equation with rapidly oscillating material coefficient. Communications in Mathematical Sciences
Open this publication in new window or tab >>On homogenization of the Landau-Lifshitz equation with rapidly oscillating material coefficient
2021 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796Article in journal (Refereed) Accepted
Abstract [en]

In this paper, we consider homogenization of the Landau-Lifshitz equation with a highly oscillatory material coefficient with period ε 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 ε over times O(εσ) with 0 ≤ σ ≤ 2 are given in the Sobolev norm Hq , 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, σ and the number of correctors.

Keywords
Homogenization; Micromagnetics; Magnetization Dynamics; Multiscale
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-304805 (URN)
Note

QC 20211123

Available from: 2021-11-12 Created: 2021-11-12 Last updated: 2022-06-25Bibliographically approved
Appelo, D., Garcia, F. & Runborg, O. (2020). Waveholtz: Iterative solution of the helmholtz equation via the wave equation. SIAM Journal on Scientific Computing, 42(4), A1950-A1983
Open this publication in new window or tab >>Waveholtz: Iterative solution of the helmholtz equation via the wave equation
2020 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 42, no 4, p. A1950-A1983Article in journal (Refereed) Published
Abstract [en]

A new iterative method, the WaveHoltz iteration, for solution of the Helmholtz equation is presented. WaveHoltz is a fixed point iteration that filters the solution to the solution of a wave equation with time periodic forcing and boundary data. The WaveHoltz iteration corresponds to a linear and coercive operator which, after discretization, can be recast as a positive definite linear system of equations. The solution to this system of equations approximates the Helmholtz solution and can be accelerated by Krylov subspace techniques. Analysis of the continuous and discrete cases is presented, as are numerical experiments.

Place, publisher, year, edition, pages
Society for Industrial & Applied Mathematics (SIAM), 2020
Keywords
Helmholtz, wave equation, positive definite
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-283037 (URN)10.1137/19M1299062 (DOI)000568184400002 ()2-s2.0-85093120720 (Scopus ID)
Note

QC 20201005

Available from: 2020-10-05 Created: 2020-10-05 Last updated: 2022-06-25Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-6321-8619

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