| 1. |
- Carlsson, Marcus, et al.
(författare)
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On phase retrieval via matrix completion and the estimation of low rank PSD matrices
- 2020
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 36:1
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Tidskriftsartikel (refereegranskat)abstract
- Given underdetermined measurements of a positive semi-definite (PSD) matrix X of known low rank K, we present a new algorithm to estimate X based on recent advances in non-convex optimization schemes. We apply this in particular to the phase retrieval problem for Fourier data, which can be formulated as a rank 1 PSD matrix recovery problem. Moreover, we provide a theory for how oversampling affects the stability of the lifted inverse problem.
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| 2. |
- Shestopalov, Yury, 1953-, et al.
(författare)
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Determination of permittivity of an inhomogeneous dielectric body in a waveguide
- 2011
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Ingår i: Inverse Problems. - Philadelphia : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 27, s. 095010-1-095010-12
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Tidskriftsartikel (refereegranskat)abstract
- The determination of permittivity of an inhomogeneous dielectric body located in a rectangular waveguide is considered. An iteration method for the numerical solution of the problem is proposed. Convergence of the method is proved. Numerical results for the determination of permittivity of a dielectric body are presented
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| 3. |
- Andersson, Fredrik
(författare)
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The Doppler moment transform in Doppler tomography
- 2005
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 21:4, s. 1249-1274
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Tidskriftsartikel (refereegranskat)abstract
- The problem of reconstructing two-dimensional vector fields from tomographic measurements based on Doppler techniques is addressed. Measurements are made along lines, from which distributions of the field component parallel to the line are obtained. A new transform, the Doppler moment transform, which extracts local information from such data, is introduced. In contrast to most previous approaches in vector tomography that are only able to treat solenoidal fields, the Doppler moment transform provides information also of the irrotational part of the field. Specifically, it gives rise to partial differential equations in the field components. Properties of the new transform and the corresponding equations are discussed, shedding light on questions of uniqueness. It is also shown that the differential equations can be combined to form algebraic equations in the field components. Finally, a reconstruction scheme is constructed and some numerical experiments are provided.
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| 4. |
- Carlsson, Marcus, et al.
(författare)
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An unbiased approach to compressed sensing
- 2020
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Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 36:11
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Tidskriftsartikel (refereegranskat)abstract
- In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system Ax = b. Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For b of the form Ax(0) + epsilon, where x(0) is the ground truth and epsilon is noise, the 'oracle solution' is the one you get if you a priori know the support of x(0), and is the best solution one could hope for. We provide a non-convex functional whose global minimum is the oracle solution, with the property that any other local minimizer necessarily has high cardinality. We provide estimates of the type parallel to(x) over cap - x(0)parallel to(2) <= C parallel to epsilon parallel to(2) that are significantly lower than for competing methods or theorems, and our theory relies on soft assumptions on the matrix A, in comparison with standard results in the field. The framework also allows to incorporate a priori information on the cardinality of the sought vector. In this case we show that despite being non-convex, our cost functional has no spurious local minima and the global minima is again the oracle solution, thereby providing the first method which is guaranteed to find this point for reasonable levels of noise, without resorting to combinatorial methods.
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| 5. |
- Kurasov, Pavel
(författare)
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Scattering matrices with finite phase shift and the inverse scattering problem
- 1996
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 12:3, s. 295-307
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Tidskriftsartikel (refereegranskat)abstract
- The inverse scattering problem for the Schrödinger operator on the half-axis is studied. it is shown that this problem can be solved for the scattering matrices with arbitrary finite phase shift on the real axis if one admits potentials with long-range oscillating tails at infinity. The solution of the problem is constructed with the help of the Gelfand-Levitan-Marchenko procedure. The inverse problem has no unique solution for the standared set of scattering data which includes the scattering matrix, energies of the bound states and corresponding normalizing constants. This fact is related to zeros of the spectral density on the real axis. It is proven that the inverse problem has a unique solution in the defined class of potentials if the zeros of the spectral density are added to the set of scattering data.
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| 6. |
- Kohlenberg, Jennifer, et al.
(författare)
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The inverse spectral problem for the discrete cubic string
- 2007
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 23:1, s. 99-121
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Tidskriftsartikel (refereegranskat)abstract
- Given a measure m on the real line or a finite interval, the cubic string is the third-order ODE - ′′′ ≤ zm where z is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a non-self-adjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper, we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis-Procesi equation. We solve the spectral and inverse spectral problem for the case of m being a finite positive discrete measure. In particular, explicit determinantal formulae for the measure m are given. These formulae generalize Stieltjes' formulae used by Krein in his study of the corresponding second-order ODE -″ ≤ zm. © 2007 IOP Publishing Ltd.
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| 7. |
- Shestopalov, Youri, 1953-, et al.
(författare)
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Existence and uniqueness of solution to the inverse problem of complex permittivity reconstruction of a dielectric body in a waveguide
- 2010
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Ingår i: Inverse Problems. - Bristol : IoP. - 0266-5611 .- 1361-6420. ; 26:10
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents a statement, a proof of uniqueness, and a method of solution to the inverse problem of the determination of permittivity of a lossy dielectric inclusion in a three-dimensional waveguide of rectangular cross-section fromthe transmission characteristics. The approach is based on the solution to a volume singular integral equation (VSIE). The examination of this equationis based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwell equations and the equivalence of this BVP andVSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. The permittivity reconstruction employs a method ofiterations applied to the solution of VSIE
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| 8. |
- Andersson, Fredrik, et al.
(författare)
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A variational formulation for interpolation of seismic traces with derivative information
- 2015
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 31:5
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Tidskriftsartikel (refereegranskat)abstract
- We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvalue of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. The variational problem is used to justify the solutions of a non-standard anisotropic diffusion problem as reasonable interpolated images. We show existence and uniqueness for this type of anisotropic diffusion. In particular, the uniqueness property is important as it guarantees that the solution can be obtained by the numerical schemes we propose.
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| 9. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem
- 2010
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:11
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Tidskriftsartikel (refereegranskat)abstract
- This study concerns a posteriori error estimates in a globally convergent numerical method for a hyperbolic coefficient inverse problem. Using the Laplace transform the model problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term in both a priori and a posteriori settings and derive optimal a posteriori error estimates for a finite-element approximation of this problem. Numerical experiments justify the efficiency of a posteriori estimates in the globally convergent approach.
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| 10. |
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