Interpolation Of Sequences Research Articles

Algorithm XXXX: The Delaunay Density Diagnostic

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is sufficient to identify the relevant variations in outputs—i.e., the “geometric scale” of the function—despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this paper, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2-10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in \(L^{2}\) norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e. static) datasets of input–output pairs. We include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

Pre-Trained Transformer-Based Parallel Multi-Channel Adaptive Image Sequence Interpolation Network

Pre-Trained Transformer-Based Parallel Multi-Channel Adaptive Image Sequence Interpolation Network

Інтерполяційна задача у вагових просторах Пелі-Вінера

Yu. Lyubarskii and K. Seip K. (Revista Matematica Iberoamericana, 1997 (13), № 2) found the criterion for the existence of a uniqe solution of the interpolation problem f(λk) = bk in the Paley-Wiener spaces in terms of Makenhoupt’s (Ap) condition of entire functions of the exponential type at most π, where p is a real number more than 1, whose restriction to the real line coincides with the class of functions which module degree number p is integrable on R whith p-norm. These results make it possible to obtain the criterion of the unconditional basisness of the system of exponentials in the space of functions which module degree number p is an integrable on (–π; π). At the same time the sequence (λk) with a unique limit point at the infinity, for which mentioned interpolation problem has a uniqe solution is called a complete interpolating sequence in the Paley-Wiener spaces. For p = 2 those descriptions coincides with those given by Pavlov (1979), Nikolsky (1980), and Minkin (1982). We generalize these results to the weighted spaces of entire functions of the exponential type at most σ with the p-norm (there is a power function with exponent ω as a weight), where σ is a nonnegative real number, ω – real number more then –1. That is, we find conditions for the completeness of the interpolating sequence (λk) in the Paley-Wiener weighted space. Different forms of these conditions are considered. Among them there are Mackenhoupt’s conditions (continuous and discrete (Ap) conditions). There is proved that if (λk) is a complete interpolation sequence in the Paley-Wiener weighted space, then it is a relatively dense set in the space C. Also there constructed an example of a complete interpolating sequence for σ = π

Wind pressure field reconstruction using a variance-extended KSI method: Both deterministic and probabilistic applications

Wind tunnel experiment is an essential measure for acquiring wind pressure information on the surface of structures. However, it is hard to acquire the complete wind pressure field information because of the restrictions of the measuring equipment capability or inner space of rigid experimental models. For this reason, this paper proposes a reliable wind pressure field reconstruction method using a variance-extended Kriging sequence interpolation. Besides the commonly deterministic reconstruction generated via conventional methods, this method can achieve the reconstruction of wind pressure coefficient time history or statistical moments at any required instant or location from the probabilistic perspective. More importantly, it can effectively avoid the repetitive procedure in addressing the sequence interpolation problem. Numerical examples are employed to illustrate the performance of the proposed method. The experimental result demonstrates that this method can provide a reliable reconstruction of the wind pressure field, and thus may have a great potential in practice.

Interpolatory Taylor and Lidstone series

We develop Taylor type series expansions for entire functions of order zero using general interpolation sequences that diverge to infinity sufficiently fast. We also derive two-point and multi-point Lidstone type series expansions for entire functions of order zero. The coefficients of these Taylor and Lidstone type series are expressed in terms of values of a certain integral operator. Our results imply as special cases several recent results on [Formula: see text]-Taylor and [Formula: see text]-Lidstone expansions of order zero entire functions, and provide a general technique for deriving such series.

A Hydrogeological Conceptual Model Refines the Behavior of a Mediterranean Coastal Aquifer System: A Key to Sustainable Groundwater Management (Grombalia, NE Tunisia)

The Mediterranean coastal aquifer system of the Grombalia basin (NE Tunisia) offers immense potential as a source of fresh water for agriculture, industry, and drinking water supply. Nonetheless, due to its intricate hydrogeological characteristics and the prevailing issue of groundwater salinity, comprehending its groundwater system behavior becomes crucial for the effective and sustainable management of this aquifer system. Based on the hydrogeological characterization of the Grombalia basin, a novel 3D hydrogeological conceptual model was developed to enhance the understanding of its complex aquifer system. The integration of insights from geological, hydrogeological, hydrodynamic, and hydrochemical components facilitated the construction of the hydrogeological conceptual model. Although the model’s validity faced initial uncertainties due to spatial interpolation of lithological sequences, this study’s thorough and encompassing hydrogeological investigation overcame these limitations. As a result, a more informed comprehension of the aquifer system complexities was achieved. This study reveals that the basin is underlain by an extensive, cohesive Mio–Plio–Quaternary aquifer system. The model demonstrates vertical and lateral hydrogeological continuity between the Quaternary and underlying Mio–Pliocene deposits, enabling groundwater flow and exchange between these layers. Over-abstraction of the Mio–Plio–Quaternary aquifer system has led to a significant drop in piezometric levels and raised the risk of seawater intrusion. These findings emphasize the critical necessity of taking into account the interconnections among hydrogeological units to ensure sustainable groundwater management. The developed conceptual model offers a key tool for understanding the hydrodynamic functioning of the Grombalia aquifer system with a view toward guiding future groundwater management strategies. The application of this approach in the Grombalia basin suggests its potential applicability to other regional aquifers facing comparable challenges.

On an interpolation sequence for a weighted Bergman space on a Hilbert unit ball

The purpose is to provide a generalization of Carleson's Theorem on interpolating sequences when dealing with a sequence in the open unit ball of a Hilbert space. Precisely, we interpolate a sequence by a function belonging to a weighted Bergman space of infinite order on a unit Hilbert ball and we furnish explicitly the upper bound corresponding to the interpolation constant.

Smooth Planning for Manipulator with Multi-dimensional Actuator based on Quintic B-spline

In order to solve the impact problem of uneven velocity during continuous operation for manipulator with multi-dimensional actuator, a smooth planning scheme based on Quintic B-spline algorithm was proposed. Firstly, the physical model of multi-dimensional actuator and its working principle were introduced. At the same time, the motion of the output axis vertex in space was decomposed into three steps, which can make the position relationship between the points more intuitive and convenient for our study. Secondly, the characteristics of Quintic B-spline algorithm were analysed. In this step, the B-spline interpolation function was defined at first, and the basis function of polynomial of degree p was defined according to the relevant information and its characteristics were indicated. Then, we analysed the node sequence parameters. For Quintic B-spline algorithm, when the position sequence of the rotor running track was given, three conditions must be satisfied to uniquely determine the corresponding B-spline function, namely: node vector, control vertex and the degree p of the B-spline curve. On this basis, the boundary constraints and control vertices were solved, which was a very important step. By solving the control vertices of Quintic B-spline curve, a smooth and continuous angular displacement curve can be obtained. The addition of four boundary conditions greatly increases the controllability of the system, making the rotor can start and stop in any way within the constraint conditions, and the connection of each interpolation point is smooth and continuous. The Quintic B-spline algorithm interpolation curve was obtained by simulation based on the time-position sequence of the actuator interpolation points. The results of the interpolation curve verified the effectiveness of the Quintic B-spline algorithm. In addition, three continuous running planning conditions were designed for the same interpolation sequence of the rotor to verify the controllability of the algorithm to the angular velocity and angular acceleration of the rotor during operation. The simulation results show that the Quintic B-spline algorithm has strong controllability for the angular velocity of actuator, and can effectively alleviate the soft impact of the system. Finally, a smooth planning experimental platform with multi-dimensional actuator was built. By setting the same trajectory and different interpolation points, the proposed Quintic B-spline algorithm was compared with the existing S-shaped curve planning algorithm based on the improvement. The results show that the proposed scheme has high control accuracy and strong impact mitigation performance, which can provide a reference for other multi-dimensional actuator trajectory planning.

ON p-ADIC INTERPOLATION IN TWO OF MAHLER’S PROBLEMS

AbstractMotivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers $(u_n)_{n\geq 0}$ . We show that if $(u_n)_{n\geq 0}$ is a sequence of integers with $u_n = O(n)$ which can be p-adically interpolated by an analytic function $f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$ , then $f(x)$ is a polynomial function of degree at most one. The case $u_n=O(n^d)$ with $d>1$ is also considered with additional conditions. Moreover, if X and Y are subsets of $\mathbb {Z}$ dense in $\mathbb {Z}_p$ , we prove that there are uncountably many p-adic analytic injective functions $f:\mathbb {Z}_p\to \mathbb {Q}_p$ , with rational coefficients, such that $f(X)=Y$ .

Robust interpolation of sequences with periodically stationary multiplicative seasonal increments

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

Extremal functional -interpolation on an arbitrary mesh on the real axis

Abstract The Golomb-de Boor problem of extremal interpolation of infinite real sequences with smallest -norm of the th derivative of the interpolant, , on an arbitrary mesh on the real axis is studied under constraints on the norms of the corresponding divided differences. For this smallest norm, lower estimates are obtained for any in terms of -splines. For the second derivative, this quantity is estimated from below and above by constants depending on the parameter . Bibliography: 13 titles.

Deep learning based missing data recovery of non-stationary wind velocity

Wireless sensor technology and wireless sensor network are the developing trend of structural health monitoring (SHM) system in the future. Due to working principles, environment and other various factors, random data missing may occur easily during data transmission by mobile wireless sensors. The randomness and fluctuation of non-stationary wind velocity is strong and complex, affected by turbulence effect. Regarding random missing of non-stationary wind velocity in data transmission, a missing data recovery algorithm based on secondary decomposition of signals, convolutional neural network (CNN) and Kriging based sequence interpolation (KSI) is proposed in this paper. Before data transmission, the non-stationary wind velocity is decomposed by wavelet transform (WT) and empirical wavelet transform (EWT) to obtain a set of sub-signals with uniform frequency. Then low-frequency sub-signals are supplemented by cubic spline interpolation (CSI). Due to the randomness and complexity of high-frequency sub-signals, CNN is used to establish the recovery model to achieve preliminary recovery results. Generally, there exists lots of extreme points in high-frequency sub-signals. The recovery error may be large due to strong randomness of extreme points. Therefore, KSI based extreme value estimation is proposed for high-frequency sub-signals to estimate extreme curves, which are subsequently used to correct the preliminary recovery results. Numerical examples of field monitoring downburst data testify the superiority of the proposed algorithm. In addition, structural dynamic analysis fully verifies the effectiveness of extreme value correction.

Dexamethasone does not affect endolymphatic hydrops (EH) in patients with Meniere's disease within 24 h, and intratympanic administration of gadolinium plus dexamethasone simplifies high-quality imaging of EH using a novel protocol of 7 min

BackgroundIntratympanic administration of gadolinium chelate allows for a better visualization of endolymphatic hydrops (EH) using MRI than intravenous injection and was recently further improved to facilitate high-quality imaging of EH in 7 min. The aim of the present study was to simplify the intratympanic administration protocol by mixing gadolinium chelate with therapeutic dexamethasone and to evaluate the effects of this mixture on the visualization of EH in MRI. Materials and methodsIn an in vitro study, the potential impact of gadolinium-diethylenetriamine pentaacetic acid (Gd-DTPA) on the stability of dexamethasone was evaluated by analyzing dynamic changes in dexamethasone with high-performance liquid chromatography (HPLC) after mixing with Gd-DTPA. Ten patients with definite Meniere's disease (MD) were recruited to study the potential interference of dexamethasone on MRI visualization of EH, and 49 patients with MD were recruited to evaluate the effect of intratympanic injection of Gd-DTPA mixed with dexamethasone on MRI of EH using a 3T MR machine and a novel heavily T2-weighted 3-dimensional fluid-attenuated inversion recovery reconstructed using a magnitude plus zero-filled interpolation (hT2FLAIR-MZFI) sequence. ResultsThe retention times and peak area of dexamethasone in HPLC were not modified by the addition of Gd-DTPA. EH grading in the cochlea and vestibule was not influenced in any ear by intratympanic injection of dexamethasone. Excellent inner ear images were obtained from all patients, and EHs with various grades were displayed. There were significant correlations between diagnosis and cochlear EH (p < 0.01, Spearman's Rho), between diagnosis and vestibular EH (p < 0.01, Spearman's Rho), and between cochlear and vestibular EH (p < 0.01, Spearman's Rho). The distribution of Gd-DTPA plus dexamethasone negatively correlated with the grade of vestibular EH. Injury of the endolymph-perilymph barrier was detected in one cochlea and three vestibules of 59 inner ears with MD. ConclusionsIntratympanic administration of Gd-DTPA plus dexamethasone yielded high-quality MRI images of EH in patients with MD using a novel 7-min protocol and simplified the clinical application. Intratympanic administration of Gd-DTPA plus dexamethasone might be used to test its therapeutic effect in future work. Level of evidence3.

Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local Lipschitz conditions of state variables

AbstractThis paper develops strong convergence of the Euler–Maruyama (EM) schemes for approximating McKean–Vlasov stochastic differential equations (SDEs). In contrast to the existing work, a novel feature is the use of a much weaker condition—local Lipschitzian in the state variable, but under uniform linear growth assumption. To obtain the desired approximation, the paper first establishes the existence and uniqueness of solutions of the original McKean–Vlasov SDE using a Euler-like sequence of interpolations and partition of the sample space. Then, the paper returns to the analysis of the EM scheme for approximating solutions of McKean–Vlasov SDEs. A strong convergence theorem is established. Moreover, the convergence rates under global conditions are obtained.

100 kHz CH2O imaging realized by lower speed planar laser-induced fluorescence and deep learning.

This paper reports an approach to interpolate planar laser-induced fluorescence (PLIF) images of CH2O between consecutive experimental data by means of computational imaging realized with convolutional neural network (CNN). Such a deep learning based method can achieve higher temporal resolution for 2D visualization of intermediate species in combustion based on high-speed experimental images. The capability of the model was tested for generating 100 kHz PLIF images by interpolating single and multiple PLIF frames into the sequences of experimental images of lower frequencies (50, 33, 25 and 20 kHz). Results show that the prediction indices, including intersection over union (IoU), peak signal to noise ratio (PSNR), structural similarity index (SSIM), and time averaged correlation coefficient at various axial positions could achieve acceptable accuracy. This work sheds light on the utilization of CNN-based models to achieve optical flow computation and image sequence interpolation, also providing an efficient off-line model as an alternative pathway to overcome the experimental challenges of the state-of-the-art ultra-high speed PLIF techniques, e.g., to further increase repetition rate and save data transfer time.

On Interpolation of Some Recurrent Sequences

On Interpolation of Some Recurrent Sequences

Nonstationary wind speed data reconstruction based on secondary correction of statistical characteristics

Data reconstruction is usually used for lost data recovery and data supplement in structural health monitoring (SHM) system. In this paper, a data reconstruction algorithm is proposed for the entire time series data missing at dispersed and consecutive multiple observation points. Kriging-based sequence interpolation (KSI) is adopted for primary data reconstruction and probability density function (PDF) reconstruction. In terms of dispersed data missing, the reconstructed PDF is of great accuracy, which is used to calculate the exact standard deviation to correct the primary reconstructed data and obtain satisfactory performance. Regarding the consecutive data missing, the first correction is insufficient due to the missing of too much information. In addition, it is found to exist a fuzzy mapping relation between the exact standard deviation and extreme values of wind speed series. Thus, the quantile regression combined with deep neural network is proposed for probability reconstruction of the extreme values. The point estimations are adopted to the secondary correction for the reconstructed data with first correction. The effectiveness and validity of the secondary correction strategy have been testified by structural dynamic response analysis. The extreme value of the displacement response under the reconstructed data with the secondary correction is closer to that under the actual wind data, compared with the reconstructed data with first correction.

CARLESON INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS

AbstractThis 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 forH∞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 spacesHp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

Interpolation sequences and nonspanning systems of exponentials on curves

Interpolation sequences of the form are investigated, and also the problem of when the system of exponentials is nonspanning on the family of arbitrary rectifiable curves in the uniform norm. In terms of the interpolation nodes (or equivalently, the exponents of the system of exponentials) a criterion for the interpolation problem to be solvable is established and the strong nonspanning property of is proved. This significantly improves some known results, in particular, results due to Korevaar, Dixon and Berndtsson. Bibliography: 23 titles.

Alongshore Classification and Super Domain Delineation of Coastal Belts Based on Interpretation of Biophysical Catenary Sequences Observed from Satellite Images

Finkl, C.W. and Makowski, C., 2021. Alongshore classification and super domain delineation of coastal belts based on interpretation of biophysical catenary sequences observed from satellite images. Journal of Coastal Research, 37(1), 1–25. Coconut Creek (Florida), ISSN 0749-0208.The cross-shore classification of coastal belts, facilitated by the interpretation of biophysical features from satellite imagery, is an adjunctory approach to traditional alongshore classification procedures. Methods such as the Biophysical Cross-shore Classification System (BCCS) (Finkl and Makowski, 2020a) and the Coastal Belt Linked Classification (CBLC) (Finkl and Makowski, 2020d) not only provide insight into cross-shore natural ecological variability and zonation, but also emphasize eco-geomorphological linkages so that shore-normal transitions and successions can be delineated and understood. The discernment of numerous cross-shore biophysical transect codifications in a satellite image seascape scene (referred to as either BCCS or CBLC code sequences) results in coastal belt segmenting that identifies offshore, nearshore, inshore, and onshore similarities and differences. This, in turn, can be correlated alongshore by establishing the swath widths of each cross-shore transect as an individual domain. Data integration and interpretation processes are similarly applied as procedural management of geological borehole (well) logs to determine drilling and petrophysical parameters. This study demonstrates that once all individual alongshore domains are interpreted for a specific coastal area, they can be collectively amalgamated and concised to create a universal alongshore super domain for the entire coastal belt. The advantage of constructing a Concised Alongshore Super Domain is that the procedure provides a sagacious and rational basis for swath alongshore classification of coastal belts. This study shows the effective use of this methodology across coastal belts of varying latitudes, as the interpolation of shore-normal ecological sequences validates the cross-shore correlation between transects and can result in a universal alongshore classification process in the form of super domains for coastal belts worldwide.