Browsing by Subject "inverse problem"

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  • Niu, Pingping; Helin, Tapio; Zhang, Zhidong (2020)
    In this work the authors consider an inverse source problem the stochastic fractional diffusion equation. The interested inverse problem is to reconstruct the unknown spatial functions f and g (the latter up to the sign) in the source by the statistics of the final time data u(x, T). Some direct problem results are proved at first, such as the existence, uniqueness, representation and regularity of the solution. Then a reconstruction scheme for f and g up to the sign is given. To tackle the ill-posedness, Tikhonov regularization is adopted and some numerical results are displayed.
  • Helin, Tapio; Kindermann, Stefan; Lehtonen, Jonatan; Ramlau, Ronny (2018)
    Adaptive optics (AO) is a technology in modern ground-based optical telescopes to compensate for the wavefront distortions caused by atmospheric turbulence. One method that allows to retrieve information about the atmosphere from telescope data is so-called SLODAR, where the atmospheric turbulence profile is estimated based on correlation data of Shack-Hartmann wavefront measurements. This approach relies on a layered Kolmogorov turbulence model. In this article, we propose a novel extension of the SLODAR concept by including a general non-Kolmogorov turbulence layer close to the ground with an unknown power spectral density. We prove that the joint estimation problem of the turbulence profile above ground simultaneously with the unknown power spectral density at the ground is ill-posed and propose three numerical reconstruction methods. We demonstrate by numerical simulations that our methods lead to substantial improvements in the turbulence profile reconstruction compared to the standard SLODAR-type approach. Also, our methods can accurately locate local perturbations in non-Kolmogorov power spectral densities.
  • Toivanen, Jussi; Meaney, Alexander; Siltanen, Samuli; Kolehmainen, Ville (2020)
    Multi-energy CT takes advantage of the non-linearly varying attenuation properties of elemental media with respect to energy, enabling more precise material identification than single-energy CT. The increased precision comes with the cost of a higher radiation dose. A straightforward way to lower the dose is to reduce the number of projections per energy, but this makes tomographic reconstruction more ill-posed. In this paper, we propose how this problem can be overcome with a combination of a regularization method that promotes structural similarity between images at different energies and a suitably selected low-dose data acquisition protocol using non-overlapping projections. The performance of various joint regularization models is assessed with both simulated and experimental data, using the novel low-dose data acquisition protocol. Three of the models are well-established, namely the joint total variation, the linear parallel level sets and the spectral smoothness promoting regularization models. Furthermore, one new joint regularization model is introduced for multi-energy CT: a regularization based on the structure function from the structural similarity index. The findings show that joint regularization outperforms individual channel-by-channel reconstruction. Furthermore, the proposed combination of joint reconstruction and non-overlapping projection geometry enables significant reduction of radiation dose.
  • Beretta, Elena; Cerutti, M. Cristina; Ratti, Luca (2021)
    We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.
  • Rundell, William; Zhang, Zhidong (2020)
    We consider the recovery of a source term f (x, t) = p(x)q(t) for the nonhomogeneous heat equation in Omega x (0, infinity) where Omega is a bounded domain in R-2 with smooth boundary partial derivative Omega from overposed lateral data on a sparse subset of partial derivative Omega x (0, infinity). Specifically, we shall require a small finite number N of measurement points on partial derivative Omega and prove a uniqueness result, namely, the recovery of the pair (p, q) within a given class, by a judicious choice of N = 2 points. Naturally, with this paucity of overposed data, the problem is severely ill-posed. Nevertheless we shall show that, provided the data noise level is low, effective numerical reconstructions may be obtained.
  • Kiczko, Adam; Västilä, Kaisa; Kozioł, Adam; Kubrak, Janusz; Kubrak, Elzbieta; Krukowski, Marcin (EGU, 2020)
    Hydrology and Earth System Sciences 24 8 (2020)
    Despite the development of advanced process-based methods for estimating the discharge capacity of vegetated river channels, most of the practical one-dimensional modeling is based on a relatively simple divided channel method (DCM) with the Manning flow resistance formula. This study is motivated by the need to improve the reliability of modeling in practical applications while acknowledging the limitations on the availability of data on vegetation properties and related parameters required by the process-based methods. We investigate whether the advanced methods can be applied to modeling of vegetated compound channels by identifying the missing characteristics as parameters through the formulation of an inverse problem. Six models of channel discharge capacity are compared in respect of their uncertainty using a probabilistic approach. The model with the lowest estimated uncertainty in explaining differences between computed and observed values is considered the most favorable. Calculations were performed for flume and field settings varying in floodplain vegetation submergence, density, and flexibility, and in hydraulic conditions. The output uncertainty, estimated on the basis of a Bayes approach, was analyzed for a varying number of observation points, demonstrating the significance of the parameter equifinality. The results showed that very reliable predictions with low uncertainties can be obtained for process-based methods with a large number of parameters. The equifinality affects the parameter identification but not the uncertainty of a model. The best performance for sparse, emergent, rigid vegetation was obtained with the Mertens method and for dense, flexible vegetation with a simplified two-layer method, while a generalized two-layer model with a description of the plant flexibility was the most universally applicable to different vegetative conditions. In many cases, the Manning-based DCM performed satisfactorily but could not be reliably extrapolated to higher flows.
  • Palacios, Benjamin; Uhlmann, Gunther; Wang, Yiran (2017)
    It is well known that reconstruction algorithms in quantitative susceptibility mapping often contain streaking artifacts. These are nondesirable objects that contaminate the image, and the possibility of removing or at least reducing them has a great practical interest. In [J. K. Choi, H. S. Park, S. Wang, Y. Wang, and J. K. Seo, SIAM J. Imaging Sci., 7 (2014), pp. 1669-1689], the cause of the artifacts is identified as propagation of singularities for a wave-type operator. In this work, we analyze such singularities using microlocal techniques and propose some strategies to reduce the artifacts.
  • Harju, Markus; Kultima, Jaakko; Serov, Valery; Tyni, Teemu (2021)
    The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space W-infinity(1)(R-2). Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.