76 research outputs found
Inverse problems with Poisson data: Statistical regularization theory, applications and algorithms.
No full textInverse problems with Poisson data arise in many photonic imaging modalities in medicine, engineering and astronomy. The design of regularization methods and estimators for such problems has been studied intensively over the last two decades. In this review we give an overview of statistical regularization theory for such problems, the most important applications, and the most widely used algorithms. The focus is on variational regularization methods in the form of penalized maximum likelihood estimators, which can be analyzed in a general setup. Complementing a number of recent convergence rate results we will establish consistency results. Moreover, we discuss estimators based on a wavelet-vaguelette decomposition of the (necessarily linear) forward operator. As most prominent applications we briefly introduce Positron emission tomography, inverse problems in fluorescence microscopy, and phase retrieval problems. The computation of a penalized maximum likelihood estimator involves the solution of a (typically convex) minimization problem. We also review several efficient algorithms which have been proposed for such problems over the last five years
Learned infinite elements for helioseismology
No full textContext. Acoustic waves in the Sun are affected by the atmospheric layers, but this region is often ignored in forward models because it increases the computational cost.
Aims. The purpose of this work is to take the solar atmosphere into account without significantly increasing the computational cost.
Methods. We solved a scalar-wave equation that describes the propagation of acoustic modes inside the Sun using a finite-element method. The boundary conditions used to truncate the computational domain were learned from the Dirichlet-to-Neumann operator, that is, the relation between the solution and its normal derivative at the computational boundary. These boundary conditions may be applied at any height above which the background medium is assumed to be radially symmetric.
Results. We show that learned infinite elements lead to a numerical accuracy similar to the accuracy that is obtained for a traditional radiation boundary condition in a simple atmospheric model. The main advantage of learned infinite elements is that they reproduce the solution for any radially symmetric atmosphere to a very good accuracy at low computational cost. In particular, when the boundary condition is applied directly at the surface instead of at the end of the photosphere, the computational cost is reduced by 20% in 2D and by 60% in 3D. This reduction reaches 70% in 2D and 200% in 3D when the computational domain includes the atmosphere.
Conclusions. We emphasize the importance of including atmospheric layers in helioseismology and propose a computationally efficient method to do this
On the Well-posedness of the Damped Time-harmonic Galbrun Equation and the Equations of Stellar Oscillations
No full textWe study the time-harmonic Galbrun equation describing the propagation of sound in the presence of a steady background flow. With additional rotational and gravitational terms these equations are also fundamental in helio- and asteroseismology as a model for stellar oscillations. For a simple damping model we prove well-posedness of these equations, i.e., uniqueness, existence, and stability of solutions under mild conditions on the parameters (essentially subsonic flows). The main tool of our analysis is a generalized Helmholtz decomposition
Inverse wave propagation problems without phase information
No full textThis special issue captures recent developments in the classical phase problem in quantum mechanics, optics, and related areas. In particular, Born's rule in quantum mechanics from 1926, for which he was awarded the Nobel Prize in 1954, states that the square of the amplitude of a particle's wave function is proportional to the probability of finding the particle at a given point, whereas the wave function itself has no direct physical interpretation. Similarly, most optical measurement techniques only provide information on the amplitude, but not on the phase of time-harmonic electromagnetic waves. The reason is that in many important cases of monochromatic electromagnetic wave propagation the wave frequency is so great that only the field intensity can be measured by modern technical devices. While classical inverse scattering theory has focused mostly on data with phase information, new exciting results on phaseless data have appeared recently on the theoretical, algorithmic, and experimental side. As two particularly successful experimental techniques which lead to closely related mathematical models involving phase retrieval, we mention phase contrast x-ray imaging and cryo-electron microscopy. The latter was recognized by the Nobel Prize in Chemistry to Dubochet, Frank and Henderson in 2017; see [5, 16]. The twelve papers collected in this special issue focus on different aspects of phaseless inverse problems and provide a good overview on the state-of the-art of this active field of research. They may be grouped into four areas involving three papers each
Generalization of the noise model for time‐distance helioseismology
No full textContext. In time-distance helioseismology, information about the solar
interior is encoded in measurements of travel times between pairs of points on the solar
surface. Travel times are deduced from the cross-covariance of the random wave field.
Here, we consider travel times and also products of travel times as observables. They
contain information about the statistical properties of convection in the Sun.
Aims. We derive analytic formulae for the noise covariance matrix of
travel times and products of travel times.
Methods. The basic assumption of the model is that noise is the result
of the stochastic excitation of solar waves, a random process that is stationary and
Gaussian. We generalize the existing noise model by dropping the assumption of horizontal
spatial homogeneity. Using a recurrence relation, we calculate the noise covariance
matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments
of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of
the travel times.
Results. All noise covariance matrices depend only on the expectation
value of the cross-covariance of the observed wave field. For products of travel times,
the noise covariance matrix consists of three terms proportional to 1 /T,
1
/T2, and 1
/T3, where T is the duration of the
observations. For typical observation times of a few hours, the term proportional to
1
/T2 dominates and
Cov [
τ1τ2,τ3τ4
] ≈ Cov [ τ1,τ3
] Cov [ τ2,τ4 ]
+ Cov [ τ1,τ4 ]
Cov [ τ2,τ3
], where the τi are arbitrary travel
times. This result is confirmed for p1 travel times by Monte Carlo
simulations and comparisons with SDO/HMI observations.
Conclusions. General and accurate formulae have been derived to model
the noise covariance matrix of helioseismic travel times and products of travel times.
These results could easily be generalized to other methods of local helioseismology, such
as helioseismic holography and ring diagram analysis
Convergence rates of general regularization methods for statistical inverse problems and applications
No full textDuring the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as í-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector products. Our results are applied to various problems, in particular we obtain precise convergence rates for satellite gradiometry, L2-boosting, and errors in variable problems
On resonances in open systems
No full textAn open system or open resonator is a domain of wave activity separated from the exterior by a partly open or partly transparent surface. Such open resonators lose energy to infinity through radiation. The numerical computation of the corresponding resonances is complicated by spurious reflections of the outgoing waves at the necessarily finite grid boundaries. These reflections can be reduced to extremely low levels by applying perfectly matched layer (PML) absorbing boundary conditions, which separate the discrete resonances from the continuous spectrum. Using a simple one-dimensional model problem, the influence of the various PML parameters is determined by a numerical error analysis. In addition to one-dimensional open resonators, two-dimensional open resonators as well as various resonating structures in waveguides are considered, and the resonant spectra and selected modes are evaluated. For the first time, leaky modes are computed for several resonating structures in a waveguide in addition to the trapped modes published in the literature. In applications, leaky mode resonances are often more important than trapped mode resonances. Gap tones, observed in a model problem of high-lift configurations, are identified as transversal resonant modes with the lowest radiation losses
Density matrix reconstructions in ultrafast transmission electron microscopy: Uniqueness, stability, and convergence rates
No full textIn the recent paper Priebe et al (2017 Nat. Photon. 11 793–7) the first experimental determination of the density matrix of a free electron beam has been reported. The employed method leads to a linear inverse problem with a positive semidefinite operator as unknown. The purpose of this paper is to complement the experimental and algorithmic results in the work mentioned above by a mathematical analysis of the inverse problem concerning uniqueness, stability, and rates of convergence under different types of a priori information
Learned Infinite Elements
No full textWe study the numerical solution of scalar time-harmonic wave equations on unbounded domains which can be split into a bounded interior domain of primary interest and an exterior domain with separable geometry. To compute the solution in the interior domain, approximations to the Dirichlet-to-Neumann (DtN) map of the exterior domain have to be imposed as transparent boundary conditions on the artificial coupling boundary. Although the DtN map can be computed by separation of variables, it is a nonlocal operator with dense matrix representations, and hence computationally inefficient. Therefore, approximations of DtN maps by sparse matrices, usually involving additional degrees of freedom, have been studied intensively in the literature using a variety of approaches including different types of infinite elements, local nonreflecting boundary conditions, and perfectly matched layers. The entries of these sparse matrices are derived analytically, e.g., from transformations or asymptotic expansions of solutions to the differential equation in the exterior domain. In contrast, in this paper we propose to “learn” the matrix entries from the DtN map in its separated form by solving an optimization problem as a preprocessing step. Theoretical considerations suggest that the approximation quality of learned infinite elements improves exponentially with increasing number of infinite element degrees of freedom, which is confirmed in numerical experiments. These numerical studies also show that learned infinite elements outperform state-of-the-art methods for the Helmholtz equation. At the same time, learned infinite elements are much more flexible than traditional methods as they, e.g., work similarly well for exterior domains involving strong reflections. As the main motivating example we study the atmosphere of the Sun, which is strongly inhomogeneous and exhibits reflections at the corona
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