Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. 10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gˆateaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (Numer. Math. 133:655–684, 2016) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the nonexisting Gˆateaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

Photonic-dispersion neural networks for inverse scattering problems

Inferring the properties of a scattering objective by analyzing the optical far-field responses within the framework of inverse problems is of great practical significance. However, it still faces major challenges when the parameter range is growing and involves inevitable experimental noises. Here, we propose a solving strategy containing robust neural-networks-based algorithms and informative photonic dispersions to overcome such challenges for a sort of inverse scattering problem—reconstructing grating profiles. Using two typical neural networks, forward-mapping type and inverse-mapping type, we reconstruct grating profiles whose geometric features span hundreds of nanometers with nanometric sensitivity and several seconds of time consumption. A forward-mapping neural network with a parameters-to-point architecture especially stands out in generating analytical photonic dispersions accurately, featured by sharp Fano-shaped spectra. Meanwhile, to implement the strategy experimentally, a Fourier-optics-based angle-resolved imaging spectroscopy with an all-fixed light path is developed to measure the dispersions by a single shot, acquiring adequate information. Our forward-mapping algorithm can enable real-time comparisons between robust predictions and experimental data with actual noises, showing an excellent linear correlation (R2 > 0.982) with the measurements of atomic force microscopy. Our work provides a new strategy for reconstructing grating profiles in inverse scattering problems.

Using a Heliospheric Upwinding eXtrapolation Technique to Magnetically Connect Different Regions of the Heliosphere

Understanding how coronal structure propagates and evolves from the Sun and into the heliosphere has been thoroughly explored using sophisticated MHD models. From these, we have a reasonably good working understanding of the dynamical processes that shape the formation and evolution of stream interaction regions and rarefactions, including their locations, orientations, and structure. However, given the technical expertize required to produce, maintain, and run global MHD models, their use has been relatively restricted. In this study, we refine a simple Heliospheric eXtrapolation Technique (HUX) to include not only forward mapping from the Sun to 1 AU (or elsewhere), but backward mapping toward the Sun. We demonstrate that this technique can provide substantially more accurate mappings than the standard, and often applied “ballistic” approximation. We also use machine learning (ML) methods to explore whether the HUX approximation to the momentum equation can be refined without loss of simplicity, finding that it likely provides the optimum balance. We suggest that HUX can be used, in conjunction with coronal models (PFSS or MHD) to more accurately connect measurements made at 1 AU, Stereo-A, Parker Solar Probe, and Solar Orbiter with their solar sources. In particular, the HUX technique: 1) provides a substantial improvement over the “ballistic” approximation for connecting to the source longitude of streams; 2) is almost as accurate, but considerably easier to implement than MHD models; and 3) can be applied as a general tool to magnetically connect different regions of the inner heliosphere together, as well as providing a simple 3-D reconstruction.

The digital great leap forward mapping China's 21st century attempt to create a new growth model

The recent successes of the Chinese modernisation strategy are substantiated by an array of indicators showing an impressive improvement. Irrespective of China's current growth deceleration, these indicators suggest a highly effective implementation of an ambitious roadmap that can ultimately help China to catch up and achieve a global technological leadership. Still, some scholars point to deep structural deficiencies, and maintain that these indicators – however impressive they are – merely scratch the surface, while much deeper change is required in order to maintain economic growth. Therefore, the purpose of this paper (finalized before the ongoing COVID-19 crisis) is to contribute to this burgeoning literature – documenting the outcome and analysing the implications of China's efforts to embrace a new growth model – and analyse the chances of the Chinese digital great leap forward, that is the radical transformation of its prior modernisation trajectory. Drawing on a systematic review of the literature, the author maps, presents and analyses existing indicators quantifying China's progress in shifting to this new development trajectory, identifying also the gaps in the conventional measurement approaches. According to the findings of this paper, there are several easy-to-measure indicators, often used in international comparisons, that indeed confirm the optimistic scenario of China's development prospects in the near future. On the other hand, some hard-to-quantify factors, such as the localization of knowledge and the spreading of innovation, need to be also considered. These latter show a closer association with countries' development level as well as development potential. With regards to these latter particularities, China still has a long way to go.

Γ-stochastic neighbour embedding for feed-forward data visualization

t-Distributed stochastic neighbour embedding is one of the most popular non-linear dimension-reduction techniques used in multiple application domains. In this article, we propose a variation on the embedding neighbourhood distribution, resulting in Γ-stochastic neighbour embedding, which can construct a feed-forward mapping using a radial basis function network. We compare the visualizations generated by Γ-stochastic neighbour embedding with those of t-distributed stochastic neighbour embedding and provide empirical evidence suggesting the network is capable of robust interpolation and automatic weight regularization.

Non-singular acoustic cloak derived by the ray tracing method with rotationally symmetric transformations

Recently, the ray tracing method has been used to derive the non-singular cylindrical invisibility cloaks for out-of-plane shear waves, which is impossible via the transformation method directly owing to the singular push-forward mapping. In this paper, the method is adopted to design a kind of non-singular acoustic cloak. Based on Hamilton's equations of motion, eikonal equation and pre-designed ray equations, we derive several constraint equations for bulk modulus and density tensor. On the premise that the perfect matching conditions are satisfied, a series of non-singular physical profiles can be obtained by arranging the singular terms reasonably. The physical profiles derived by the ray tracing method will degenerate to the transformation-based solutions when taking the transport equation into consideration. This illuminates the essence of the newly designed cloaks that they are actually the so-called eikonal cloaks that can accurately control the paths of energy flux but with small disturbance in energy distribution along the paths. The near-perfect invisible performance has been demonstrated by the numerical ray tracing results and the pressure distribution snapshots. Finally, a kind of reduced cloak is conceived, and the good invisible performance has been measured quantitatively by the normalized scattering width.

Modeling of Cement-Bonded Sand Mould System

The present paper deals with the forward mapping problem of cement bonded sand mould system using fuzzy logic (FL)-based approaches. It is important to note that the performance of an FL-based approach depends on its knowledge base (KB) that is, rule base and data base. Here, three approaches have been proposed to solve the said problem. The first Approach deals with the development of manually constructed Mamdani-based FL system, and the second Approach deals with the optimization of the rule base and data base of the FL system constructed in Approach 1, whereas the third Approach deals with automatic evolution of the FL system, in which the consequent part has also been optimized. A binary coded genetic algorithm (GA) has been used for the said purpose. The performances of the developed approaches are tested in forward mapping of a cement bonded sand mould system. It is to be noted that all the approaches can be effectively used to model the cement-bonded moulding sand system.

A Navigation Safety Support Model for the Strait of Istanbul

In this study, a real time maritime traffic support model is developed for safe navigation in the Strait of Istanbul, also known as the Bosporus. The present model simulates vessel trajectories corresponding to possible headings, using channel geometry, counter traffic, and surface currents as input. A new MATLAB code is developed for the simulation and the Marine GNC Toolbox (Fossen and Perez, 2004) is used for the vessel hydrodynamics and the auto-pilot model. After computing the trajectory tree of the vessel by forward-mapping its position distribution with respect to the initial position vector, the casualty probabilities of each trajectory are found. Within certain restrictions on vessel geometry, the proposed model predicts the safest possible intended course for the transit vessels based on the navigational parameters including position, speed, and course of the vessel. The model is tested for the Strait of Istanbul for validation. Without loss of generality, the model can be used for any narrow channel with a vessel traffic system providing the necessary input.