Functional differentiation of integral operators of special form and some questions of the inverse interpolation

This article is devoted to the problem of operator interpolation and functional differentiation. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations containing the first variational derivatives of the required functional are given. For functionals defined on sets of functions and square matrices, various interpolating polynomials of the Hermitе type with nodes of the second multiplicity, which contain the first variational derivatives of the interpolated operator, are constructed. The presented solutions of the Hermitе interpolation problems are based on the algebraic Chebyshev system of functions. For analytic functions with an argument from a set of square matrices, explicit formulas for antiderivatives of functionals are obtained. The solution of some differential equations with integral operators of a special form and the first variational derivatives is found. The problem of the inverse interpolation of functions and operators is considered. Explicit schemes for constructing inverse functions and functionals, including the case of functions of a matrix variable, obtained using certain well-known results of interpolation theory, are demonstrated. Data representation is illustrated by a number of examples.

Mathematical Problems of Artificial Intelligence and Artificial Neural Networks

Предложен общий топологический подход для анализа искусственных нейронных сетей на основе симплициальных комплексов и свойств аппроксимации непрерывных отображений их симплициальными приближениями. Выявлены существенные для этого класса задач явления вычислительной неустойчивости, связанной с общими проблемами некорректных задач в гильбертовом пространстве и методами их регуляризации, типичными для обработки Big Data. Сформулированы критерии точности и применимости моделей искусственных нейронных сетей, рассмотрены примеры их реализации на основе теории интерполяции функций. Развитие идей П.Л.Чебышёва о наилучшем приближении служит отправной точкой для широкого класса математических исследований по оптимизации обучающих наборов для построения ИНС. We propose a general topological approach to the analysis of artificial neural networks using simplicial complexes and the approximation of continuous mappings with simplicial ones. The essential properties of numerical instability in such problems were identified. It is associated with ill-posed problems in Hilbert space and regularization methods typically applied to Big Data processing. We formulated the criteria of artificial neural network accuracy and applicability and included some implementation examples based on the interpolation theory. Advancing P.L. Chebyshev’s ideas about the best approximation may be an entry point to various mathematical research on artificial neural network training dataset optimization.  

A minimum curvature algorithm for tomographic reconstruction of atmospheric chemicals based on optical remote sensing

Optical remote sensing (ORS) combined with the computerized tomography (CT) technique is a powerful tool to retrieve a two-dimensional concentration map over an area under investigation. Whereas medical CT usually uses a beam number of hundreds of thousands, ORS-CT usually uses a beam number of dozens, thus severely limiting the spatial resolution and the quality of the reconstructed map. The smoothness a priori information is, therefore, crucial for ORS-CT. Algorithms that produce smooth reconstructions include smooth basis function minimization, grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is promising because of the fast speed. However, its theoretical basis must be clarified to better understand the characteristics of its smoothness constraints. Moreover, the computational efficiency and reconstruction quality need to be improved for practical applications. This paper first treated the LTD algorithm as a special case of the Tikhonov regularization that uses the approximation of the third-order derivative as the regularization term. Then, to seek more flexible smoothness constraints, we successfully incorporated the smoothness seminorm used in variational interpolation theory into the reconstruction problem. Thus, the smoothing effects can be well understood according to the close relationship between the variational approach and the spline functions. Furthermore, other algorithms can be formulated by using different seminorms. On the basis of this idea, we propose a new minimum curvature (MC) algorithm by using a seminorm approximating the sum of the squares of the curvature, which reduces the number of linear equations to half that in the LTD algorithm. The MC algorithm was compared with the non-negative least square (NNLS), GT-MG, and LTD algorithms by using multiple test maps. The MC algorithm, compared with the LTD algorithm, shows similar performance in terms of reconstruction quality but requires only approximately 65 % the computation time. It is also simpler to implement than the GT-MG algorithm because it directly uses high-resolution grids during the reconstruction process. Compared with the traditional NNLS algorithm, it shows better performance in the following three aspects: (1) the nearness of reconstructed maps is improved by more than 50 %, (2) the peak location accuracy is improved by 1–2 m, and (3) the exposure error is improved by 2 to 5 times. Testing results indicated the effectiveness of the new algorithm according to the variational approach. More specific algorithms could be similarly further formulated and evaluated. This study promotes the practical application of ORS-CT mapping of atmospheric chemicals.

Security-Oriented Indoor Robots Tracking: An Object Recognition Viewpoint

Indoor robots, in particular AI-enhanced robots, are enabling a wide range of beneficial applications. However, great cyber or physical damages could be resulted if the robots’ vulnerabilities are exploited for malicious purposes. Therefore, a continuous active tracking of multiple robots’ positions is necessary. From the perspective of wireless communication, indoor robots are treated as radio sources. Existing radio tracking methods are sensitive to indoor multipath effects and error-prone with great cost. In this backdrop, this paper presents an indoor radio sources tracking algorithm. Firstly, an RSSI (received signal strength indicator) map is constructed based on the interpolation theory. Secondly, a YOLO v3 (You Only Look Once Version 3) detector is applied on the map to identify and locate multiple radio sources. Combining a source’s locations at different times, we can reconstruct its moving path and track its movement. Experimental results have shown that in the typical parameter settings, our algorithm’s average positioning error is lower than 0.39 m, and the average identification precision is larger than 93.18% in case of 6 radio sources.

A minimum curvature algorithm for tomographic reconstruction of atmospheric chemicals based on optical remote sensing

Optical remote sensing (ORS) combined with computerized tomography (CT) technique is a powerful tool to retrieve a two-dimensional concentration map over the area under investigation. But unlike the medical CT, the beam number used in ORS-CT is usually dozens comparing to up to hundreds of thousands in the former, which severely limits the spatial resolution and the quality of the reconstructed map. This situation makes the “smoothness” a priori information especially necessary for ORS-CT. Algorithms which produce smooth reconstructions include smooth basis function minimization (SBFM), grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is a promising one with fast speed and simple realization. But its characteristics and the theory basis are not clear. Moreover, the computation efficiency and the reconstruction quality need to be improved for practical applications. This paper employs two theories, i.e., Tikhonov regularization and spatial interpolation, to produce a smooth reconstruction by ORS-CT. Within the two theories’ frameworks, new algorithms can be explored in order to improve the performance. For example, we propose a new minimum curvature (MC) algorithm based on the variational approach in the theory of the spatial interpolation, which reduces the number of linear equations by half comparing to that in the LTD algorithm using the biharmonic equation instead of the smoothness seminorm. We compared our MC algorithm with the non-negative least square (NNLS), GT-MG, and LTD algorithms using multiple test maps. The MC and the LTD algorithms have similar performance on the reconstruction quality. But the MC algorithm needs only about 65 % computation time of the LTD algorithm. It is much simpler in realization than the GT-MG algorithm by using high-resolution grids directly during the reconstruction process to generate a high-resolution map immediately after one reconstruction process is done. Comparing to the traditional NNLS algorithm, it shows better performance in three aspects: (1) the nearness of reconstructed maps is improved by more than 50 %; (2) the peak location accuracy is improved by 1–2 m; and (3) the exposure error is improved by more than 10 times. The testing results show the effectiveness of the new algorithm based on the spatial interpolation theory. Similarly, other algorithms may also be formulated to address problems such as the over-smooth issue in order to further improve the reconstruction equality. The studies will promote the practical application of the ORS-CT mapping of atmospheric chemicals.

CARLESON INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS

This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result for H ∞ can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spaces H p . Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

Direct and Inverse Fractional Abstract Cauchy Problems

We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p, p ∈ ( 0 , ∞ ] instead of the indices θ , ∞. As a possible application of the abstract theorems, some examples of partial differential equations are given.