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| 2. |
- Andersson, Fredrik, et al.
(författare)
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A multi-scale approach to hyperbolic evolution equations with limited smoothness
- 2008
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302. ; 33:6, s. 988-1017
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Tidskriftsartikel (refereegranskat)abstract
- We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder regularity s >= 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursionas in the computation of certain multiple scattering seriesrevealing a curvelet-curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities.
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| 3. |
- Andersson, Fredrik, et al.
(författare)
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Frequency Extrapolation through Sparse Sums of Lorentzians
- 2014
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Ingår i: Journal of Earth Science. - : Springer Science and Business Media LLC. - 1867-111X .- 1674-487X. ; 25:1, s. 117-125
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Tidskriftsartikel (refereegranskat)abstract
- Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are assigned: translation and scale. These parameters can be found by using a method for complex frequency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.
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| 4. |
- Andersson, Fredrik, et al.
(författare)
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Multi-scale discrete approximation of Fourier integral operators
- 2012
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Ingår i: Multiscale Modeling & Simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 10:1, s. 111-135
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Tidskriftsartikel (refereegranskat)abstract
- Abstract in UndeterminedWe develop a discretization and computational procedures for approximation of the action of Fourier integral operators the canonical relations of which are graphs. Such operators appear, for instance, in the formulation of imaging and inverse scattering of seismic reflection data. Our discretization and algorithms are based on a multiscale low-rank expansion of the action of Fourier integral operators using the dyadic parabolic decomposition of phase space and on explicit constructions of low-rank separated representations using prolate spheroidal wave functions, which directly reflect the geometry of such operators. The discretization and computational procedures connect to the discrete almost symmetric wave packet transform. Numerical wave propagation and imaging examples illustrate our computational procedures.
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| 5. |
- Andersson, Fredrik, et al.
(författare)
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Multiscale Reverse-Time-Migration-Type Imaging Using the Dyadic Parabolic Decomposition of Phase Space
- 2015
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Ingår i: SIAM Journal of Imaging Sciences. - 1936-4954. ; 8:4, s. 2383-2411
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Tidskriftsartikel (refereegranskat)abstract
- We develop a representation of reverse-time migration (RTM) in terms of Fourier integral operators, the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse-time continuation from the boundary of scattering data and for RTM. The algorithms are derived from those we recently developed for the discrete approximate evaluation of the action of Fourier integral operators and inherit their conceptual and numerical properties.
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| 6. |
- Andersson, Fredrik, et al.
(författare)
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Nonlinear approximation of functions in two dimensions by sums of exponential functions
- 2010
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Ingår i: Applied and Computational Harmonic Analysis. - : Elsevier BV. - 1096-603X .- 1063-5203. ; 29:2, s. 156-181
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of approximating a given function in two dimensions by a sum of exponential functions, with complex-valued exponents and coefficients. In contrast to Fourier representations where the exponentials are fixed, we consider the nonlinear problem of choosing both the exponents and coefficients. In this way we obtain accurate approximations with only few terms. Our approach is built on recent work done by G. Beylkin and L Monzon in the one-dimensional case. We provide constructive methods for how to find the exponents and the coefficients, and provide error estimates. We also provide numerical simulations where the method produces sparse approximations with substantially fewer terms than what a Fourier representation produces for the same accuracy. (c) 2009 Elsevier Inc. All rights reserved.
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| 7. |
- Andersson, Fredrik, et al.
(författare)
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Nonlinear approximation of functions in two dimensions by sums of wave packets
- 2010
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Ingår i: Applied and Computational Harmonic Analysis. - : Elsevier BV. - 1096-603X .- 1063-5203. ; 29:2, s. 198-213
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of approximating functions that arise in wave-equation imaging by sums of wave packets. Our objective is to find sparse decompositions of image functions, over a finite range of scales. We also address the naturally connected task of numerically approximating the wavefront set. We present an approximation where we use the dyadic parabolic decomposition, but the approach is not limited to only this type. The approach makes use of expansions in terms of exponentials, while developing an algebraic structure associated with the decomposition of functions into wave packets. (c) 2009 Elsevier Inc. All rights reserved.
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| 8. |
- Andersson, Fredrik, et al.
(författare)
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Sparse approximation of functions using sums of exponentials and AAK theory
- 2011
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Ingår i: Journal of Approximation Theory. - : Elsevier BV. - 0021-9045. ; 163:2, s. 213-248
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of approximating functions by sums of few exponentials functions, either on an interval or on the positive half-axis. We study both continuous and discrete cases, i.e. when the function is replaced by a number of equidistant samples. Recently, an algorithm has been constructed by Beylkin and Monzón for the discrete case. We provide a theoretical framework for understanding how this algorithm relates to the continuous case.
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| 9. |
- de Hoop, Maarten V., et al.
(författare)
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Analysis of wavenumber resonances for the Rayleigh system in a half space
- 2023
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Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : Royal Society. - 1364-5021 .- 1471-2946. ; 479:2277
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Tidskriftsartikel (refereegranskat)abstract
- We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyse their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the Riemann surface as expected.
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| 10. |
- De Hoop, Maarten V, et al.
(författare)
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Inverse problem for Love waves in a layered, elastic half-space
- 2024
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Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; :045013, s. 1-44
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.
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