Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

Reflection angle/azimuth-dependent least-squares reverse-time migration

Seismic images under complex overburdens like salt are strongly affected by illumination variations due to overburden velocity variations and imperfect acquisition geometries, making it difficult to obtain reliable image amplitudes. Least-squares reverse-time migration (LSRTM) addresses these issues by formulating full wave-equation imaging as a linear inverse problem and solving for a reflectivity model that explains the recorded seismic data. Because subsurface reflection coefficients depend on the incident angle, and possibly on azimuth, quantitative interpretation under complex overburdens requires LSRTM with output in terms of image gathers, e.g., as a function of reflection angle or angle and azimuth. We present a reflection angle- or angle/azimuth-dependent LSRTM method aimed at obtaining physically meaningful image amplitudes interpretable in terms of angle- or angle/azimuth-dependent reflection coefficients. The method is formulated as a linear inverse problem solved iteratively with the conjugate gradient method. It requires an adjoint pair of linear operators for reflection angle/azimuth-dependent migration and demigration based on full wave-equation propagation. We implement these operators in an efficient way by using a mapping approach between migrated shot gathers and subsurface reflection angle/azimuth gathers. To accelerate convergence of the iterative inversion, we apply image-domain preconditioning operators computed from a single de-remigration step. An angle continuity constraint and a structural dip constraint, implemented via shaping regularization, are used to stabilize the solution in the presence of limited illumination and to control the effects of coherent noise. We demonstrate the method on a synthetic data example and on a wide-azimuth streamer dataset from the Gulf of Mexico, where we show that angle/azimuth-dependent LSRTM can achieve significant uplift in subsalt image quality, with overburden- and acquisition-related illumination variation effects on angle/azimuth-dependent image amplitudes largely removed.

Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem

<p style='text-indent:20px;'>Sparse regression plays a very important role in statistics, machine learning, image and signal processing. In this paper, we consider a high-dimensional linear inverse problem with <inline-formula><tex-math id="M3">\begin{document}$ \ell^0 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> penalty to stably reconstruct the sparse signals. Based on the sufficient and necessary condition of the coordinate-wise minimizers, we design a smoothing Newton method with continuation strategy on the regularization parameter. We prove the global convergence of the proposed algorithm. Several numerical examples are provided, and the comparisons with the state-of-the-art algorithms verify the effectiveness and superiority of the proposed method.</p>

Analysis of heating time and of temperature distributions for cylindrical geometry with the use of solution to the inverse problem

Changes in heating time of a cylinder in the furnace for thermal and thermochemical treatments depending on the given heating rate is analysed in this paper. Temperature distributions from the axis to the boundary of the cylinder were determined based on solving non-stationary and non-linear inverse problem for the heat equation. Differences between the temperature on the boundary and along the cylinder axis for processes with the given heating rates from 5 to 10ᵒC/min were calculated. Twofold increase in the heating rate allowed the heating time to be reduced significantly. Increase in the heating rate had no impact on the difference between the temperature on the boundary and on the axis of the cylinder and on the quantity of energy being consumed by heating elements.

Revisiting Vertical Land Motion and Sea Level Trends in the Northeastern Adriatic Sea Using Satellite Altimetry and Tide Gauge Data

We propose a revisited approach to estimating sea level change trends based on the integration of two measuring systems: satellite altimetry and tide gauge (TG) time series of absolute and relative sea level height. Quantitative information on vertical crustal motion trends at six TG stations of the Adriatic Sea are derived by solving a constrained linear inverse problem. The results are verified against Global Positioning System (GPS) estimates at some locations. Constraints on the linear problem are represented by estimates of relative vertical land motion between TG couples. The solution of the linear inverse problem is valid as long as the same rates of absolute sea level rise are observed at the TG stations used to constrain the system. This requirement limits the applicability of the method with variable absolute sea level trends. The novelty of this study is that we tried to overcome such limitations, subtracting the absolute sea level change estimates observed by the altimeter from all relevant time series, but retaining the original short-term variability and associated errors. The vertical land motion (VLM) solution is compared to GPS estimates at three of the six TGs. The results show that there is reasonable agreement between the VLM rates derived from altimetry and TGs, and from GPS, considering the different periods used for the processing of VLM estimates from GPS. The solution found for the VLM rates is optimal in the least square sense, and no longer depends on the altimetric absolute sea level trend at the TGs. Values for the six TGs’ location in the Adriatic Sea during the period 1993–2018 vary from −1.41 ± 0.47 mm y−1 (National Research Council offshore oceanographic tower in Venice) to 0.93 ± 0.37 mm y−1 (Rovinj), while GPS solutions range from −1.59 ± 0.65 (Venice) to 0.10 ± 0.64 (Split) mm y−1. The absolute sea level rise, calculated as the sum of relative sea level change rate at the TGs and the VLM values estimated in this study, has a mean of 2.43 mm y−1 in the period 1974–2018 across the six TGs, a mean standard error of 0.80 mm y−1, and a sample dispersion of 0.18 mm y−1.