The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics.     

Adding Prior Information in FWI through Relative Entropy

Full waveform inversion is an advantageous technique for obtaining high-resolution subsurface information. In the petroleum industry, mainly in reservoir characterisation, it is common to use information from wells as previous information to decrease the ambiguity of the obtained results. For this, we propose adding a relative entropy term to the formalism of the full waveform inversion. In this context, entropy will be just a nomenclature for regularisation and will have the role of helping the converge to the global minimum. The application of entropy in inverse problems usually involves formulating the problem, so that it is possible to use statistical concepts. To avoid this step, we propose a deterministic application to the full waveform inversion. We will discuss some aspects of relative entropy and show three different ways of using them to add prior information through entropy in the inverse problem. We use a dynamic weighting scheme to add prior information through entropy. The idea is that the prior information can help to find the path of the global minimum at the beginning of the inversion process. In all cases, the prior information can be incorporated very quickly into the full waveform inversion and lead the inversion to the desired solution. When we include the logarithmic weighting that constitutes entropy to the inverse problem, we will suppress the low-intensity ripples and sharpen the point events. Thus, the addition of entropy relative to full waveform inversion can provide a result with better resolution. In regions where salt is present in the BP 2004 model, we obtained a significant improvement by adding prior information through the relative entropy for synthetic data. We will show that the prior information added through entropy in full-waveform inversion formalism will prove to be a way to avoid local minimums.     

Regularization inertial proximal point algorithm for common solutions of a finite family of inverse-strongly monotone equations

The aim of the paper is to propose an iterative regularization method of proximal point type for finding a common solution for a finite family of inverse-strongly monotone equations in Hilbert spaces.     

A total variation based approach to correcting surface coil magnetic resonance images

Magnetic resonance images which are corrupted by noise and by smooth modulations are corrected using a variational formulation incorporating a total variation like penalty for the image and a high order penalty for the modulation. The optimality system is derived and numerically discretized. The cost functional used is non-convex, but it possesses a bilinear structure which allows the ambiguity among solutions to be resolved technically by regularization and practically by normalizing the maximum value of the modulation. Since the cost is convex in each single argument, convex analysis is used to formulate the optimality condition for the image in terms of a primal-dual system. To solve the optimality system, a nonlinear Gauss-Seidel outer iteration is used in which the cost is minimized with respect to one variable after the other using an inner generalized Newton iteration. Favorable computational results are shown for artificial phantoms as well as for realistic magnetic resonance images. Reported computational times demonstrate the feasibility of the approach in practice.     

Convergence results for a self-dual regularization of convex problems

We study a one-parameter regularization technique for convex optimization problems whose main feature is self-duality with respect to the Legendre–Fenchel conjugation. The self-dual technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one. For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater's condition.     

Spatially adaptive sparse grids for high-dimensional data-driven problems

Sparse grids allow one to employ grid-based discretization methods in data-driven problems. We present an extension of the classical sparse grid approach that allows us to tackle high-dimensional problems by spatially adaptive refinement, modified ansatz functions, and efficient regularization techniques. The competitiveness of this method is shown for typical benchmark problems with up to 166 dimensions for classification in data mining, pointing out properties of sparse grids in this context. To gain insight into the adaptive refinement and to examine the scope for further improvements, the approximation of non-smooth indicator functions with adaptive sparse grids has been studied as a model problem. As an example for an improved adaptive grid refinement, we present results for an edge-detection strategy.     

Interpolation on Sparse Grids and Tensor Products of Nikol'skij–Besov Spaces

We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of tensor products of Nikol'skij–Besov spaces. To this end, we make use of the uniformity of the considered tensor norms and provide a unified approach to error estimates for the interpolation of univariate periodic functions from Nikol'skij–Besov spaces.     

基于稀疏表示和特征加权的离格双耳声源定位*

基于头相关传递函数数据库的传统双耳声源定位方法的定位角度往往被限定在头相关传递函数数据库的离散测量点上。当头相关传递函数数据库的测量方位角间隔较大时,这类算法的性能会显著下降,这就是典型的离格问题。该文提出了基于加权宽带稀疏贝叶斯学习的离格双耳声源定位算法。首先该算法建立离格双耳信号的稀疏表示模型,然后利用双耳相干与扩散能量比特征对各个频点进行加权以降低噪声和混响的影响,最后通过加权宽带稀疏贝叶斯学习方法估计离格声源的方位角。实验结果表明,该算法在各种复杂的声学环境下都有着较高的定位精度和鲁棒性,特别是提高了离格条件下的声源定位性能。     

Variational assimilation in combination with the regularization method for sea level pressure retrieval from QuikSCAT scatterometer data I: Theoretical frame construction

A new method of constructing a sea level pressure field from satellite microwave scatterometer measurements is presented. It is based on variational assimilation in combination with a regularization method using geostrophic vorticity to construct a sea level pressure field from scatterometer data that are given in this paper, which offers a new idea for the application of scatterometer measurements. Firstly, the geostrophic vorticity from the scatterometer data is computed to construct the observation field, and the vorticity field in an area and the sea level pressure on the borders are assimilated. Secondly, the gradient of sea level pressure (semi-norm) is used as the stable functional to educe the adjoint system, the adjoint boundary condition and the gradient of the cost functional in which a weight parameter is introduced for the harmony of the system and the Tikhonov regularization techniques in inverse problem are used to overcome the ill-posedness of the assimilation. Finally, the iteration method of the sea level pressure field is developed.