Interpolation Series Research Articles

Dimension reduction analysis with mapping and direct integration algorithm

Based on the Lagrange series interpolation, the semi-analytical and direct integration method (DIM) for solving the multi-dimensional partial differential equations are presented in this paper. A multidimensional problem can be simplified to a set of the ordinary differential equation by the mapping technique and Lagrange series interpolation (dimension reduction). The semi-analytical and numerical solutions by DIM are derived for the partial differential equation with variable coefficients. Comparisons with analytical solutions are performed in order to demonstrate the accuracy and efficiency of the proposed method.

Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

• A novel finite and infinite block Petrov–Galerkin method is applied for crack problems in functionally graded materials . • The problem domain is discretized by a few blocks in the computation which saves tedious mesh generation . • The displacements and stresses are continuous at interfaces of neighboring blocks. • Due to the usage of Petrov–Galerkin formulation, the BPGM is a weak-form method. • Numerical results show that the proposed approach is accurate and efficient. This paper presents a novel weak-form block Petrov–Galerkin method (BPGM) for linear elastic and crack problems in functionally graded materials with bounded and unbounded problem domains. The main idea of this approach is to combine the meshless local Petrov–Galerkin method with block method. Once the problem domain is discretized into several sub-regions, named blocks, which can be mapped into normalized square domains. The weak-form Petrov–Galerkin method and polynomial series of interpolations are employed in each block. The computational efficiency is rigorously examined against the strong-form finite block method, the finite element technique and meshless approaches. Numerical results demonstrate that the BPGM possesses the following important properties: (1) only a few blocks are required for calculating problems in unbounded regions which saves tedious work of meshing; (2) the displacements and stresses are continuous at the interfaces of neighboring blocks; (3) due to the use of weak formulation, the continuity requirements of the approximation functions are reduced and numerical results are stable; (4) because of using Lagrange polynomial interpolation, highly accurate solutions can be obtained with a small amount of nodes in each block.

Comparing methods for analysing time scale dependent correlations in irregularly sampled time series data

Time series derived from paleoclimate archives are often irregularly sampled in time and thus not analysable using standard statistical methods such as correlation analyses. Although measures for the similarity between time series have been proposed for irregular time series, they do not account for the time scale dependency of the relationship. Stochastically distributed temporal sampling irregularities act qualitatively as a low-pass filter reducing the influence of fast variations from frequencies higher than about 0.5 (Δtmax)−1, where Δtmax is the maximum time interval between observations. This may lead to overestimated correlations if the true correlation increases with time scale. Typically, correlations are underestimated due to a non-simultaneous sampling of time series.Here, we investigated different techniques to estimate time scale dependent correlations of weakly irregularly sampled time series, with a particular focus on different resampling methods and filters of varying complexity. The methods were tested on ensembles of synthetic time series that mimic the characteristics of Holocene marine sediment temperature proxy records. We found that a linear interpolation of the irregular time series onto a regular grid, followed by a simple Gaussian filter was the best approach to deal with the irregularity and account for the time scale dependence. This approach had both, minimal filter artefacts, particularly on short time scales, and a minimal loss of information due to filter length.

Two Properties of Pseudo-Polynomials over a Galois Field

Let F be the completion, with respect to the degree valuation, of the field of rational functions over , the Galois (finite) field of q elements. A function f : F → F is integer-valued if . An integer-valued function f is called a pseudo-polynomial iff(M+K)≡f(M)(modK)for all and . Based on an interpolation series introduced by Carlitzin 1935, explicit shapes of pseudo-polynomials are established. Using an asymptotic characterization of polynomials, it is also proved that the set of all pseudo-polynomials is an integral domain but not a unique factorization domain.

Calibration of Safecast dose rate measurements

A methodology is presented to calibrate contributed Safecast dose rate measurements acquired between 2011 and 2016 in the Fukushima prefecture of Japan. The Safecast data are calibrated using observations acquired by the U.S. Department of Energy at the time of the 2011 Fukushima Daiichi power plant nuclear accident.The methodology performs a series of interpolations between the U.S. government and contributed datasets at specific temporal windows and at corresponding spatial locations. The coefficients found for all the different temporal windows are aggregated and interpolated using quadratic regressions to generate a time dependent calibration function. Normal background radiation, decay rates, and missing values are taken into account during the analysis.Results show that the standard Safecast static transformation function overestimates the official measurements because it fails to capture the presence of two different Cesium isotopes and their changing magnitudes with time. A model is created to predict the ratio of the isotopes from the time of the accident through 2020. The proposed time dependent calibration takes into account this Cesium isotopes ratio, and it is shown to reduce the error between U.S. government and contributed data. The proposed calibration is needed through 2020, after which date the errors introduced by ignoring the presence of different isotopes will become negligible.

Mortality Prediction in Severe Congestive Heart Failure Patients with Multifractal Point-Process Modeling of Heartbeat Dynamics.

Multifractal analysis of human heartbeat dynamics has been demonstrated to provide promising markers of Congestive Heart Failure (CHF). Yet, it crucially builds on the interpolation of RR intervals series, which has been generically performed with limited links to CHF pathophysiology. We devise a novel methodology estimating multifractal autonomic dynamics from heartbeat-derived series defined in the continuous time. We hypothesize that markers estimated from our novel framework are also effective for mortality prediction in severe CHF. We merge multifractal analysis within a methodological framework based on inhomogeneous point process models of heartbeat dynamics. Specifically, wavelet coefficients and wavelet leaders are computed over measures extracted from instantaneous statistics of probability density functions characterizing and predicting the time until the next heartbeat event occurs. The proposed approach is tested on data from 94 CHF patients, aiming at predicting survivor and non-survivor individuals as determined after a 4 years follow up. Instantaneous markers of vagal and sympatho-vagal dynamics display power-law scaling for a large range of scales, from s to s. Using standard SVM algorithms, the proposed inhomogeneous point-process representation based multifractal analysis achieved the best CHF mortality prediction accuracy of 79.11 % (sensitivity 90.48%, specificity 67.74%). Our results suggest that heartbeat scaling and multifractal properties in CHF patients are not generated at the sinus-node level, but rather by the intrinsic action of vagal short-term control and of sympatho-vagal fluctuations associated with circadian cardiovascular control, especially within the VLF band. These markers might provide critical information in devising a clinical tool for individualized prediction of survivor and non-survivor CHF patients.

水文系列插补延长均化处理验证及应用

水文系列插补延长均化处理验证及应用

Wavelet Analysis for Temporal Disaggregation

A problem often faced by economic researchers is the interpolation or distribution of economic time series observed at low frequency into compatible higher frequency data. A method based on wavelet analysis is presented to temporal disaggregate time series. A standard `plausible' method is applied, not to the original time series, but to the smooth components resulting from a discrete wavelet transformation. This first step generates a smoothed component at the desired frequency. Subsequently, a noisy component is added to the smooth series to enforce the natural constraint of the series. The method is applied to national accounts for Euro Area, to study both ow and stock variables, and it outperforms other standard methods, as Stram and Wei or low pass interpolation when the series of interest is volatile.

Conditioned empirical orthogonal functions for interpolation of runoff time series along rivers: Application to reconstruction of missing monthly records

Reconstruction of missing runoff data is of important significance to solve contradictions between the common situation of gaps and the fundamental necessity of complete time series for reliable hydrological research. The conventional empirical orthogonal functions (EOF) approach has been documented to be useful for interpolating hydrological series based upon spatiotemporal decomposition of runoff variation patterns, without additional measurements (e.g., precipitation, land cover). This study develops a new EOF-based approach (abbreviated as CEOF) that conditions EOF expansion on the oscillations at outlet (or any other reference station) of a target basin and creates a set of residual series by removing the dependence on this reference series, in order to redefine the amplitude functions (components). This development allows a transparent hydrological interpretation of the dimensionless components and thereby strengthens their capacities to explain various runoff regimes in a basin. The two approaches are demonstrated on an application of discharge observations from the Ganjiang basin, China. Two alternatives for determining amplitude functions based on centred and standardised series, respectively, are tested. The convergence in the reconstruction of observations at different sites as a function of the number of components and its relation to the characteristics of the site are analysed. Results indicate that the CEOF approach offers an efficient way to restore runoff records with only one to four components; it shows more superiority in nested large basins than at headwater sites and often performs better than the EOF approach when using standardised series, especially in improving infilling accuracy for low flows. Comparisons against other interpolation methods (i.e., nearest neighbour, linear regression, inverse distance weighting) further confirm the advantage of the EOF-based approaches in avoiding spatial and temporal inconsistencies in estimated series.

Temporal Interpolation of Satellite-Derived Leaf Area Index Time Series by Introducing Spatial-Temporal Constraints for Heterogeneous Grasslands

Continuous satellite-derived leaf area index (LAI) time series are critical for modeling land surface process. In this study, we present an interpolation algorithm to predict the missing data in LAI time series for ecosystems with high within-ecosystem heterogeneity, particularly heterogeneous grasslands. The algorithm is based on spatial-temporal constraints, i.e., the missing data in the LAI time series of a pixel are predicted by the phenological links with other pixels. To address the uncertainties in the construction and selection of reference curves in a heterogeneous landscape, the algorithm constructs a reference dataset for each missing data in the LAI time series from all pixels showing very strong linear phenological links with the target pixel within a region. We also use an iterative process to update the available spatial-temporal constraints. We tested the algorithm with an eight-day composite Moderate Resolution Imaging Spectroradiometer (MODIS) LAI product in the Songnen grasslands, Northeast China in 2010 and 2011. The validation dataset was generated based on high quality time series by artificially adding data gaps. The algorithm achieved high overall interpolation accuracies with high coefficient of determination R2 (>0.9) and low root mean square error (RMSE) (<0.2) in both dry (2010) and wet (2011) years. The algorithm showed advantages in predicting missing data for different seasons and proportions of missing data versus the algorithm that uses regional average LAI curve as a reference. These results suggest that the proposed algorithm could more effectively characterize spatial-temporal constraint information in heterogeneous grasslands for temporal interpolation.

Evaluation of stress intensity factors and T-stress by finite block method: Static and dynamic

In this paper, the Finite Block Method (FBM) for computing the Stress Intensity Factors (SIFs) and the T-stress under static load and transient dynamic load is presented. Based on the one-dimensional first order partial differential matrices derived from the Lagrange series interpolation, the higher order partial differential matrices can be determined directly. By using the mapping technique, a block with arbitrary shape of the boundary is transformed in the polar coordinate (r,θ) into the normalised coordinate (ξ,η)|ξ|⩽1,|η|⩽1. In order to capture the stress intensity factor and the T-stress, the Williams' series of stress function is introduced in the circular core, centred at the crack tip, with the consideration of traction and displacement continuities along the circumference of the circular core. Time dependent problems are analysed by using the Laplace transformation method and the Durbin’s inversion method is used to determine all the physical time dependent variables. However, in the Laplace domain, the Deng's series of stress and displacement has to be used. The accuracy and the convergence of the FBM are demonstrated by four examples. Comparisons have been made with the solutions obtained by the Boundary Collocation Method (BCM) and the Finite Element Method (FEM).

Method to Increase the Speed of EIS Series Measurements on Battery Stacks during Cycling While Avoiding Dramatic Loss of Measurement Accuracy

The amount of time expended while running experiments is a crucial factor to consider when screening new materials as candidates for improving battery performance. Electrochemical impedance spectroscopy (EIS) is among the most appropriate evaluation tools for collecting in-situ information about electrochemical processes occurring inside battery systems. However, during charging and discharging cycles, the electrodes, electrolytes, and related materials comprising battery systems are continuously undergoing changes in state over time, even at timescales shorter than the amount of time required to run a single charge-discharge experiment. For instance, voltage is changing during constant current load. Here, the application of EIS can become restricted because the prerequisite of steady state as a foundational principle to interpret the resulting EIS spectra is violated due to changes of state in between the time it takes to sweep through frequencies of interest. Trying to yield significant results accounting for drift effects requires implementing particular modeling measures. This abstract presents an ensemble of solutions to combat this issue. Parallel EIS, automatic drift correction of the measured time dependent signals, time interpolation by series measurements, and the application of the Z-HIT algorithm as post-processing of the data are the tools of choice. Simultaneous EIS is an effective time saving procedure. Otherwise, sequential EIS evaluation of n single cells in a stack increases time consumption by the factor n. Also, comparing the results for each individual cell to other cells is no longer directly comparable to an exact degree because the measurement drifts due to time effects are difficult to correct for. Second, precautions must be provided during the data acquisition phase in order to suppress the erroneous assignment of drift signals to the system response [1]. Furthermore, “non causal” spectra in the meaning of the Kramers-Kronig relations can be converted into correct spectra by interpolation of spectra series vs. time and, by removal of residual non-causalities with the help of the Z-HIT transform, which has the capability to reconstruct drift-affected data [2]. In many cases, the user would favor the comfortable possibility to significantly reduce the time for recording the EIS spectra by the application of multi-sine excitation. However, conventional multi-sine excitation normally is accompanied by significant losses of accuracy. An advanced technique provides the features of automatic drift correction and error detection known from single-sine excitation [3]. As a result, substantial loss in accuracy can be avoided. Experimental data obtained from a four-cell Li-FePO4 stack (13.2 V / 40 Ah) using this new multi-sine algorithm will be presented upon approval of this abstract for presentation. Keywords: Multi-sine, Battery stacks, EIS, True parallel measurements, Z-HIT

RESEARCH IN THE INTERPOLATION REPRESENTATIONS OF STOCHASTIC PROCESSES IN THE TWO TYPES OF INTERPOLATION KNOTS

The article deals with some interpolation representations of stochastic processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first and second orders at some types of knots and observations of the process and its derivatives of the first orders at other types of knots. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimate of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row that the interpolation row converges to the stochastic process uniformly over in any bounded area of changing of parameter is obtained. The main purpose of this article is the obtained convergence with probability 1 of the corresponding interpolation series to a stochastic process in any bounded domain of changes of parameter. Obtained results may be applied in the modern theory of information transmission.

$p$-Adic Interpolation of Automorphic Periods for GL$_2$

We give a new and representation theoretic construction of p -adic interpolation series for central values of self-dual Rankin-Selberg L -functions for GL _2 in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of GL _2 over a totally real field corresponding to a \mathfrak{p} -ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying GL _2 -representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a p -adic interpolation series for central derivative values when the root number is generically equal to -1 , in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.

Implicit Volterra series interpolation for model reduction of bilinear systems

We propose a new interpolatory framework for model reduction of large-scale bilinear systems. The input–output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions, each representing a subsystem of the bilinear system. If a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions, we say that the reduced system satisfies a Volterra series interpolation (Flagg and Gugercin, 2015). These interpolatory conditions can also ensure the necessary conditions for H2-optimal model reduction (Flagg and Gugercin, 2015; Benner and Breiten, 2012). We observe that, by carefully selecting the weights of the series, the Volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence. Such linear parametric systems can then be reduced by some method for parametric model order reduction.Linear systems where the affine parameter dependence is given as low-rank variation in the state matrix can be mapped into a non-parameterized multi-input multi-output linear system. This allows us to utilize the standard (non-parametric) linear IRKA (Gugercin et al., 2008) for the problem of parameterized/bilinear interpolation. Numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational Krylov algorithm (Benner and Breiten, 2012). The proposed approach, however, has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only.

Post buckling analysis for composite plate by finite block Petrov-Galerkin method

Abstract Based on the Lagrange series interpolation, the Finite Block Petrov-Galerkin Method (FBPGM) has been developed to the non-linear analysis of Reissner plate theory with Functionally Graded Materials (FGMs). By introducing the mapping technique, a quadratic type block is mapped into a normalized domain with 8 seeds. Then the domain nodes distribution can be determined by using shape functions. The load incremental technique and displacement control scheme are adopted to solve nonlinear problems including large deformation and post buckling of plates under pressure load and in-plane compressive forces. Numerical examples are given to demonstrate the efficiency and accuracy of proposed method in comparing with the Boundary Element Method (BEM), point collocation method (PCM) and Finite Element Method (FEM).

Additive self-similar approximants

A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory, allowing for accurate extrapolation and interpolation of asymptotic series. The method is illustrated by the examples possessing the structure typical of many nonlinear problems in mathematical chemistry. Good numerical convergence is demonstrated for the cases that can be compared with exact solutions, when these are available. The method is shown to be not less and as a rule essentially more accurate than that of Pade approximants. Comparison with other approximation methods is also given.

An effective toolkit for the interpolation and gross error detection of GPS time series

As data gaps and gross errors inevitably exist in GPS time series, robust detection and interpolation procedures are needed to obtain a uniform time series for various geospatial studies and applications. The use of traditional methods for this purpose is usually based on some improper assumptions, which are not derived based on the real properties of the data. Moreover, different interpolation methods may need to be investigated for interpolation for various gaps with different types and amount of missing data. These make the interpolation for missing data not easy. To address the issue of mentioned above, in this study, a data-analysis method named singular spectrum analysis (SSA) for missing data is assessed for reconstructing a reliable model from unevenly sampled time series without the need for any a priori knowledge of the time series data. In this method, the interpolation and detection of gross errors are carried out in one go along with the reliable reconstructed model. Both simulation data and real GPS data testing results showed that this was an efficient method for interpolation and gross error detection.

Finite block method for interface cracks

The finite block method in both the Cartesian coordinate and polar coordinate systems is developed to evaluate the stress intensity factors and the T-stress for interface cracks in bi-materials. The first order partial differential matrices can be constructed straightaway based on the Lagrange series interpolation. The nodal values of displacement can be obtained from a set of linear algebraic equations in strong form from both the governing equation and the boundary conditions. In order to capture the stress intensity factors and the T-stress at the crack tip accurately, the asymptotic expansions of the stress and displacement around the crack tip are introduced with a singular core technique. For elastodynamic fracture problems, the Laplace transform method and the Durbin’s inverse techniques are utilised. The accuracy and the convergence of the finite element method are demonstrated in three examples. Comparisons have been made with numerical solutions by using the boundary collocation method and the finite element method. Satisfactory numerical solutions are obtained with very few blocks in each example.

The Effects of Increasing the Minimum Wage on Prices: Analyzing the Incidence of Policy Design and Context

We analyze the price pass-through effect of the minimum wage and use the results to provide insight into the competitive structure of low-wage labor markets. Using monthly price series, we find that the pass-through effect is entirely concentrated on the month that the minimum wage change goes into effect, and is much smaller than what the canonical literature has found. We then discuss why our results differ from that literature, noting the impact of series interpolation in generating most of the previous results. We then use the variation in the size of the minimum wage change to evaluate the competitive nature of low-wage labor markets. Finally, we exploit the rich variation in minimum wage policy of the last 10–15 years — including the rise of state and city-level minimum wage changes and the increased use of indexation — to investigate how the extent of price pass-through varies by policy context. This paper contributes to the literature by clarifying our understanding of the dynamics and magnitude of the pass-through effect and enriching the discussion of how different policies may shape the effect that minimum wage hikes have on prices.