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Publications (10 of 59) Show all publications
Popovic, J. & Runborg, O. (2017). Adaptive fast interface tracking methods. Journal of Computational Physics, 337, 42-61
Open this publication in new window or tab >>Adaptive fast interface tracking methods
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 337, p. 42-61Article in journal (Refereed) Published
Abstract [en]

In this paper, we present a fast time adaptive numerical method for interface tracking. The method uses an explicit multiresolution description of the interface, which is represented by wavelet vectors that correspond to the details of the interface on different scale levels. The complexity of standard numerical methods for interface tracking, where the interface is described by N marker points, is 0 (N/Delta t),when a time step At is used. The methods that we propose in this paper have 0 (TOL-1/P log N + N log N) computational cost, at least for uniformly smooth problems, where TOL is some given tolerance and p is the order of the time stepping method that is used for time advection of the interface. The adaptive method is robust in the sense that it can handle problems with both smooth and piecewise smooth interfaces (e.g. interfaces with corners) while keeping a low computational cost. We show numerical examples that verify these properties.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Interface tracking, Multiresolution analysis, Fast algorithms, Adaptive methods
National Category
Computer and Information Sciences
Identifiers
urn:nbn:se:kth:diva-206672 (URN)10.1016/j.jcp.2017.02.017 (DOI)000398874700003 ()2-s2.0-85013831555 (Scopus ID)
Note

QC 20170510

Available from: 2017-05-10 Created: 2017-05-10 Last updated: 2018-01-13Bibliographically approved
Malenova, G., Motamed, M. & Runborg, O. (2017). Regularity of stochastic observables in Gaussian beam superposition of high-frequency waves. Research in mathematical sciences
Open this publication in new window or tab >>Regularity of stochastic observables in Gaussian beam superposition of high-frequency waves
2017 (English)In: Research in mathematical sciences, ISSN 2197-9847Article in journal (Other academic) Published
Abstract [en]

We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-238869 (URN)10.1186/s40687-016-0091-8 (DOI)000412664800001 ()2-s2.0-85050381408 (Scopus ID)
Note

QC 20181114

Available from: 2018-11-13 Created: 2018-11-13 Last updated: 2018-11-14Bibliographically approved
Malenova, G., Motamed, M. & Runborg, O. (2017). Stochastic regularity of a quadratic observable of high-frequency waves. RESEARCH IN THE MATHEMATICAL SCIENCES, 4, Article ID UNSP 3.
Open this publication in new window or tab >>Stochastic regularity of a quadratic observable of high-frequency waves
2017 (English)In: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 4, article id UNSP 3Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2017
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-216645 (URN)10.1186/s40687-016-0091-8 (DOI)000412664800001 ()2-s2.0-85050381408 (Scopus ID)
Note

QC 20171031

QC 20181017

Available from: 2017-10-31 Created: 2017-10-31 Last updated: 2018-10-17Bibliographically approved
Malenova, G., Motamed, M., Runborg, O. & Tempone, R. (2016). A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty. SIAM/ASA Journal on Uncertainty Quantification, 4(1), 1084-1110
Open this publication in new window or tab >>A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty
2016 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, no 1, p. 1084-1110Article in journal (Refereed) Published
Abstract [en]

We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase, and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, u(epsilon), is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments for simplified problems and numerical evidence that quantities of interest based on local averages of |u(epsilon)|(2) are smooth, with derivatives in the stochastic space uniformly bounded in epsilon, where epsilon denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics Publications, 2016
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-214529 (URN)10.1137/15M1029230 (DOI)000407996700042 ()2-s2.0-85033578756 (Scopus ID)
Note

QC 20170929

Available from: 2017-09-29 Created: 2017-09-29 Last updated: 2018-11-14Bibliographically approved
Arjmand, D. & Runborg, O. (2016). A time dependent approach for removing the cell boundary error in elliptic homogenization problems. Journal of Computational Physics, 314, 206-227
Open this publication in new window or tab >>A time dependent approach for removing the cell boundary error in elliptic homogenization problems
2016 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 314, p. 206-227Article in journal (Refereed) Published
Abstract [en]

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(epsilon/eta) error in the computation, where epsilon is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of epsilon/eta in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(epsilon/eta) error to O(epsilon) in general non-periodic media.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Multiscale problems, Homogenization, Elliptic PDEs
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-186597 (URN)10.1016/j.jcp.2016.03.009 (DOI)000374122100013 ()2-s2.0-84961194378 (Scopus ID)
Note

QC 20160601

Available from: 2016-06-01 Created: 2016-05-13 Last updated: 2017-11-30Bibliographically approved
Liu, H., Runborg, O. & Tanushev, N. M. (2016). Sobolev and max norm error estimates for Gaussian beam superpositions. Communications in Mathematical Sciences, 14(7), 2037-2072
Open this publication in new window or tab >>Sobolev and max norm error estimates for Gaussian beam superpositions
2016 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 14, no 7, p. 2037-2072Article in journal (Refereed) Published
Abstract [en]

This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schrödinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength e. The estimates are performed for the scalar wave equation and the Schrödinger equation. Our result demonstrates that a Gaussian beam superposition with kth order beams converges to the exact solution as O(εk/2-s) in order s Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is O(ε⌈k/2⌉) and away from the essential support of the solution, the convergence is spectral in ε. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate O(ε(k-n)/2) in n spatial dimensions.

Place, publisher, year, edition, pages
International Press of Boston, 2016
Keywords
Error estimates, Gaussian beams, High-frequency wave propagation, Max norm, Sobolev norm
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-195136 (URN)10.4310/CMS.2016.v14.n7.a12 (DOI)000385990500012 ()2-s2.0-84990954990 (Scopus ID)
Funder
Swedish e‐Science Research Center
Note

QC 20161107

Available from: 2016-11-07 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approved
Motamed, M. & Runborg, O. (2015). A wavefront-based Gaussian beam method for computing high frequency wave propagation problems. Computers and Mathematics with Applications, 69(9), 949-963
Open this publication in new window or tab >>A wavefront-based Gaussian beam method for computing high frequency wave propagation problems
2015 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 69, no 9, p. 949-963Article in journal (Refereed) Published
Abstract [en]

We present a novel wavefront method based on Gaussian beams for computing high frequency wave propagation problems. Unlike standard geometrical optics, Gaussian beams compute the correct solution of the wave field also at caustics. The method tracks a front of two canonical beams with two particular initial values for width and curvature. In a fast post-processing step, from the canonical solutions we recreate any other Gaussian beam with arbitrary initial data on the initial front. This provides a simple mechanism to include a variety of optimization processes, including error minimization and beam width minimization, for a posteriori selection of optimal beam initial parameters. The performance of the method is illustrated with two numerical examples.

Keywords
Asymptotic approximation, High frequency, Summation of Gaussian beams, Wave propagation, Wavefront methods
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-166945 (URN)10.1016/j.camwa.2015.02.021 (DOI)000353613500004 ()2-s2.0-84926422117 (Scopus ID)
Funder
Swedish e‐Science Research Center
Note

QC 20150529

Available from: 2015-05-29 Created: 2015-05-21 Last updated: 2017-12-04Bibliographically approved
Di, Y., Popovic, J. & Runborg, O. (2015). AN ADAPTIVE FAST INTERFACE TRACKING METHOD. Journal of Computational Mathematics, 33(6), 576-586
Open this publication in new window or tab >>AN ADAPTIVE FAST INTERFACE TRACKING METHOD
2015 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 33, no 6, p. 576-586Article in journal (Refereed) Published
Abstract [en]

An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiscale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.

Place, publisher, year, edition, pages
Global Science Press, 2015
Keywords
Interface tracking, Multiresolution, adaptivity, Fast algorithms
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-180181 (URN)10.4208/jcm.1503-m4532 (DOI)000365818500002 ()2-s2.0-84954535063 (Scopus ID)
Note

QC 20150111

Available from: 2016-01-11 Created: 2016-01-07 Last updated: 2017-12-01Bibliographically approved
Kieri, E., Kreiss, G. & Runborg, O. (2015). Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrodinger Equations. Advances in Applied Mathematics and Mechanics, 7(6), 687-714
Open this publication in new window or tab >>Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrodinger Equations
2015 (English)In: Advances in Applied Mathematics and Mechanics, ISSN 2070-0733, E-ISSN 2075-1354, Vol. 7, no 6, p. 687-714Article in journal (Refereed) Published
Abstract [en]

In the semiclassical regime, solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

Place, publisher, year, edition, pages
[Kieri, Emil; Kreiss, Gunilla] Uppsala Univ, Dept Informat Technol, Div Comp Sci, Uppsala, Sweden. [Runborg, Olof] KTH, Dept Math, Uppsala, Sweden. [Runborg, Olof] KTH, SeRC, Uppsala, Sweden.: , 2015
Keywords
Gaussian beams, semiclassical Schrodinger equation, hybrid methods
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-174200 (URN)10.4208/aamm.2013.m411 (DOI)000361055400001 ()2-s2.0-84960871763 (Scopus ID)
Note

QC 20151014

Available from: 2015-10-14 Created: 2015-10-02 Last updated: 2017-12-01Bibliographically approved
Häggblad, J. & Runborg, O. (2014). Accuracy of staircase approximations in finite-difference methods for wave propagation. Numerische Mathematik, 128(4), 741-771
Open this publication in new window or tab >>Accuracy of staircase approximations in finite-difference methods for wave propagation
2014 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, no 4, p. 741-771Article in journal (Refereed) Published
Abstract [en]

While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

Keywords
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95502 (URN)10.1007/s00211-014-0625-1 (DOI)000344752100005 ()
Funder
Swedish e‐Science Research Center
Note

QC 20141212

Updated from manuscript to article in journal.

Available from: 2012-05-28 Created: 2012-05-28 Last updated: 2017-12-07Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0002-6321-8619

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