| 1. |
- Flodén, Liselott, 1967-, et al.
(författare)
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A myriad shades of green
- 2009
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Ingår i: Proceedings of Bridges 2009, Banff, Alberta, Canada.
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Konferensbidrag (refereegranskat)abstract
- We discuss the possible application of techniques inspired by the theories of G-convergence and homogenization to understand mixtures of colors and how they appear as observed by the human eye. The ideas are illustrated by pictures describing the equivalent of a convergence process for different kinds of mixtures of colors.
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| 2. |
- Flodén, Liselott, 1967-, et al.
(författare)
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Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence
- 2011
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Ingår i: Annals of Functional Analysis. - Mashdad, Iran : TSMG. - 2008-8752 .- 2008-8752. ; 2:1, s. 84-99
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Tidskriftsartikel (refereegranskat)abstract
- We apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coefficient of a diffeential equation from its solution. We also use thís notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales
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| 3. |
- Flodén, Liselott, et al.
(författare)
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Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales
- 2017
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Ingår i: Progress in Industrial Mathematics at ECMI 2016. - Cham : Springer. - 9783319630816 ; , s. 617-623
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Konferensbidrag (refereegranskat)abstract
- We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε2}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.
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| 4. |
- Flodén, Liselott, 1967-, et al.
(författare)
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Homogenization of parabolic equations with an arbitrary number of scales in both space and time
- 2014
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Ingår i: Journal of Applied Mathematics. - Boston : Hindawi Publishing Corporation. - 1110-757X .- 1687-0042. ; , s. Art. no. 101685-
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Tidskriftsartikel (refereegranskat)abstract
- The main contribution of this paper is the homogenization of the linearparabolic equationtu (x, t) − ·axq1, ...,xqn,tr1, ...,trmu (x, t)= f(x, t)exhibiting an arbitrary finite number of both spatial and temporal scales.We briefly recall some fundamentals of multiscale convergence and providea characterization of multiscale limits for gradients in an evolution settingadapted to a quite general class of well-separated scales, which we nameby jointly well-separated scales (see Appendix for the proof). We proceedwith a weaker version of this concept called very weak multiscale convergence.We prove a compactness result with respect to this latter typefor jointly well-separated scales. This is a key result for performing thehomogenization of parabolic problems combining rapid spatial and temporaloscillations such as the problem above. Applying this compactnessresult together with a characterization of multiscale limits of sequences ofgradients we carry out the homogenization procedure, where we togetherwith the homogenized problem obtain n local problems, i.e. one for eachspatial microscale. To illustrate the use of the obtained result we apply itto a case with three spatial and three temporal scales with q1 = 1, q2 = 2and 0 < r1 < r2.MSC: 35B27; 35K10
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| 5. |
- Flodén, Liselott, 1967-, et al.
(författare)
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On the determination of effective properties of certain structures with non-periodic temporal oscillations
- 2009
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Ingår i: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling. - Wien : Vienna University Press (WUV). - 9783901608353 ; , s. 2627-2630
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Konferensbidrag (refereegranskat)abstract
- We investigate the homogenization of an evolution problem modelled by a parabolic equation, where the coefficient describing the structure is periodic in space but may vary in time in a non-periodic way. This is performed applying a generalization of two-scale convergence called λ-scale convergence. We give a result on the characterization of the λ-scale limit of gradients under certain boundedness assumptions. This is then applied to perform the homogenization procedure. It turns out that, under a certain condition on the rate of change of the temporal variations, the effective property of the given structure can be determined the same way as in periodic cases.
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| 6. |
- Flodén, Liselott, 1967-, et al.
(författare)
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Two-scale convergence: Some remarks and extensions
- 2013
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Ingår i: Pure and Applied Mathematics Quarterly. - : International press of Boston. - 1558-8599 .- 1558-8602. ; 9:3, s. 461-486
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Tidskriftsartikel (refereegranskat)abstract
- We first study the fundamental ideas behind two-scale conver-gence to enhance an intuitive understanding of this notion. The classicaldefinitions and ideas are motivated with geometrical arguments illustratedby illuminating figures. Then a version of this concept, very weak two-scaleconvergence, is discussed both independently and brie°y in the context ofhomogenization. The main features of this variant are that it works alsofor certain sequences of functions which are not bounded inL2 and atthe same time is suited to detect rapid oscillations in some sequences whichare strongly convergent inL2 . In particular, we show how very weaktwo-scale convergence explains in a more transparent way how the oscilla-tions of the governing coe±cient of the PDE to be homogenized causes thedeviation of theG-limit from the weak L2 NxN-limit for the sequence ofcoe±cients. Finally, we investigate very weak multiscale convergence andprove a compactness result for separated scales which extends a previousresult which required well-separated scales.
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| 7. |
- Flodén, Liselott, et al.
(författare)
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Very weak multiscale convergence
- 2010
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Ingår i: Applied Mathematics Letters. - : Elsevier BV. - 0893-9659 .- 1873-5452. ; 23:10, s. 1170-1173
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Tidskriftsartikel (refereegranskat)abstract
- We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.
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| 8. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems
- 2016
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Ingår i: Inverse Problems. - Bristol, United Kingdom : IOP Publishing. - 0266-5611 .- 1361-6420. ; 32:2
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Tidskriftsartikel (refereegranskat)abstract
- Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak L-2-limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.
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| 9. |
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| 10. |
- Holmbom, Anders, 1958-, et al.
(författare)
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A note on parabolic homogenization with a mismatch between the spatial scales
- 2013
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Ingår i: Abstract and Applied Analysis. - : Hindawi Publishing Corporation. - 1085-3375 .- 1687-0409. ; , s. Art. no. 329704-
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Tidskriftsartikel (refereegranskat)abstract
- We consider the homogenization of the linear parabolic problem rho(x/epsilon(2))partial derivative(t)u(epsilon)(x,t) - del . (a(x/epsilon(1), t/epsilon(2)(1))del u(epsilon) (x,t)) = f(x,t) which exhibits a mismatch between the spatial scales in the sense that the coefficient a(x/epsilon(1), t/epsilon(2)(1)) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient rho(x/epsilon(2)) of the time derivative contains a faster spatial scale. It is shown that the faster spatialmicroscale does not give rise to any corrector termand that there is only one local problemneeded to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.
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