Fast Interpolation Research Articles

Roundoff error problems in interpolation methods for time-fractional problems

Roundoff errors often disrupt interpolation methods for time-fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose δ-cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss–Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps.

The e-MANTIS emulator: fast predictions of the non-linear matter power spectrum in f(R)CDM cosmology

ABSTRACT In order to probe modifications of gravity at cosmological scales, one needs accurate theoretical predictions. N-body simulations are required to explore the non-linear regime of structure formation but are very time consuming. In this work, we release a new public emulator, dubbed e-mantis, that performs an accurate and fast interpolation between the predictions of f(R) modified gravity cosmological simulations, run with ecosmog. We sample a wide 3D parameter space given by the current background scalar field value $10^{-7}\lt \left|f_{R_0}\right|\lt 10^{-4}$, matter density 0.24 < Ωm < 0.39, and primordial power spectrum normalization 0.6 < σ8 < 1.0, with 110 points sampled from a Latin hypercube. For each model we perform pairs of f(R)CDM and ΛCDM simulations covering an effective volume of $\left(560 \, h^{-1}\, \mathrm{Mpc}\right)^3$ with a mass resolution of ∼2 × 1010h−1M⊙. We build an emulator for the matter power spectrum boost B(k) = Pf(R)(k)/PΛCDM(k) using a Gaussian process regression method. The boost is mostly independent of h, ns, and Ωb, which reduces the dimensionality of the relevant cosmological parameter space. Additionally, it is more robust against statistical and systematic errors than the raw power spectrum, thus strongly reducing our computational needs. According to our dedicated study of numerical systematics, the resulting emulator has an estimated maximum error of 3 per cent across the whole cosmological parameter space, for scales $0.03 \ h\, \mathrm{Mpc}^{-1} \lt k \lt 7 \ h\, \mathrm{Mpc}^{-1}$, and redshifts 0 < z < 2, while in most cases the accuracy is better than 1 per cent. Such an emulator could be used to constrain f(R) gravity with weak lensing analyses.

Neutrino structure functions from GeV to EeV energies

The interpretation of present and future neutrino experiments requires accurate theoretical predictions for neutrino-nucleus scattering rates. Neutrino structure functions can be reliably evaluated in the deep-inelastic scattering regime within the perturbative QCD (pQCD) framework. At low momentum transfers (Q2 ≲ few GeV2), inelastic structure functions are however affected by large uncertainties which distort event rate predictions for neutrino energies Eν up to the TeV scale. Here we present a determination of neutrino inelastic structure functions valid for the complete range of energies relevant for phenomenology, from the GeV region entering oscillation analyses to the multi-EeV region accessible at neutrino telescopes. Our NNSFν approach combines a machine-learning parametrisation of experimental data with pQCD calculations based on state-of-the-art analyses of proton and nuclear parton distributions (PDFs). We compare our determination to other calculations, in particular to the popular Bodek-Yang model. We provide updated predictions for inclusive cross sections for a range of energies and target nuclei, including those relevant for LHC far-forward neutrino experiments such as FASERν, SND@LHC, and the Forward Physics Facility. The NNSFν determination is made available as fast interpolation LHAPDF grids, and it can be accessed both through an independent driver code and directly interfaced to neutrino event generators such as GENIE.

Fast stratification of geological cross-section from CPT results with missing data using multitask and modified Bayesian compressive sensing

Since cone penetration test (CPT) is reasonably rapid, affordable, and repeatable, it has been widely used in situ for subsurface soil stratification and classification in geological and geotechnical engineering practice. When used for soil stratification across a 2D geological cross-section, however, it is often observed that some CPTs probe deeper than others, and that some CPT soundings may contain missing data due to presence of gravel-sized particles or intentional bypassing of gravelly soil layers. Arguments above and frequently encountered problem of a small number of CPT soundings in practice pose a great challenge for 2D soil stratification, especially for nonstationary CPT within multilayers. While certain methods have been proposed hoping to address these concerns, they are frequently constrained by either stationary assumption of data, autocorrelation function forms, or computational issues. This study introduces a data-driven multitask Bayesian compressive sensing (MT-BCS) method to estimate missing data for CPT sounding of interest, and then develops a modified 2D BCS method for fast interpolation for horizontal locations without CPT soundings. The proposed method is demonstrated and validated using both numerical and real-world CPT data. Results show that proposed method is both efficient and robust in terms of missing data estimation in each CPT sounding and soil stratification for a 2D geological cross-section.

Radial coordinate maps, radial vectors, and binormal vectors for 5/6, 5/5 and 5/4 edge island domains in W7-X

The edge island domain in Wendelstein 7-X consists of divertor islands whose position coincide with the location of rational values of the rotational transform and surround the main confinement volume (the ‘main plasma’), providing the magnetic topology for an island divertor. The 5/5 edge consists of five individual islands that are unconnected. In contrast, a single island connects onto itself after 6 or 4 toroidal transits in the 5/6 or 5/4 edge, respectively. Many interesting phenomena are related to these islands and diagnostic analyses require a mapping from ‘laboratory’ or real space coordinates to the island coordinate system (ICS). A procedure is described here to calculate several scalar and vector quantities for closed island structures which can be utilized in fast interpolation schemes for inverse maps. For the 5/5 edge, a fixed-boundary vacuum (zero beta) magneto-hydrodynamic solution of the 5/5 island is found with the Variational Moments Equilibrium Code, VMEC. The solution is compatible with already existing routines which determine the radial and binormal vectors of VMEC solutions at arbitrary laboratory coordinates. VMEC does not support solutions for the 5/4 and 5/6 islands, but the radial and binormal vectors are available in a local 2D Fourier ICS.

Extraction of Human Motion Information from Digital Video Based on 3D Poisson Equation

Based on the 3D Poisson equation, this paper extracts the features of the digital video human body action sequence. By solving the Poisson equation on the silhouette sequence, the time and space features, time and space structure features, shape features, and orientation features can be obtained. First, we use the silhouette structure features in three-dimensional space-time and the orientation features of the silhouette in three-dimensional space-time to represent the local features of the silhouette sequence and use the 3D Zernike moment feature to represent the overall features of the silhouette sequence. Secondly, we combine the Bayesian classifier and AdaBoost classifier to learn and classify the features of human action sequences, conduct experiments on the Weizmann video database, and conduct multiple experiments using the method of classifying samples and selecting partial combinations for training. Then, using the recognition algorithm of motion capture, after the above process, the three-dimensional model is obtained and matched with the model in the three-dimensional model database, the sequence with the smallest distance is calculated, and the corresponding skeleton is outputted as the results of action capture. During the experiment, the human motion tracking method based on the university matching kernel (EMK) image kernel descriptor was used; that is, the scale invariant operator was used to count the characteristics of multiple training images, and finally, the high-dimensional feature space was mapped into the low-dimensional to obtain the feature space approximating the Gaussian kernel. Based on the above analysis, the main user has prior knowledge of the network environment. The experimental results show that the method in this paper can effectively extract the characteristics of human body movements and has a good classification effect for bending, one-foot jumping, vertical jumping, waving, and other movements. Due to the linear separability of the data in the kernel space, fast linear interpolation regression is performed on the features in the feature space, which significantly improves the robustness and accuracy of the estimation of the human motion pose in the image sequence.

A partition-enhanced least-squares collocation approach (PE-LSC)

We present a partition-enhanced least-squares collocation (PE-LSC) which comprises several modifications to the classical LSC method. It is our goal to circumvent various problems of the practical application of LSC. While these investigations are focused on the modeling of the exterior gravity field the elaborated methods can also be used in other applications. One of the main drawbacks and current limitations of LSC is its high computational cost which grows cubically with the number of observation points. A common way to mitigate this problem is to tile the target area into sub-regions and solve each tile individually. This procedure assumes a certain locality of the LSC kernel functions which is generally not given and, therefore, results in fringe effects. To avoid this, it is proposed to localize the LSC kernels such that locality is preserved, and the estimated variances are not notably increased in comparison with the classical LSC method. Using global covariance models involves the calculation of a large number of Legendre polynomials which is usually a time-consuming task. Hence, to accelerate the creation of the covariance matrices, as an intermediate step we pre-calculate the covariance function on a two-dimensional grid of isotropic coordinates. Based on this grid, and under the assumption that the covariances are sufficiently smooth, the final covariance matrices are then obtained by a simple and fast interpolation algorithm. Applying the generalized multi-variate chain rule, also cross-covariance matrices among arbitrary linear spherical harmonic functionals can be obtained by this technique. Together with some further minor alterations these modifications are implemented in the PE-LSC method. The new PE-LSC is tested using selected data sets in Antarctica where altogether more than 800,000 observations are available for processing. In this case, PE-LSC yields a speed-up of computation time by a factor of about 55 (i.e., the computation needs only hours instead of weeks) in comparison with the classical unpartitioned LSC. Likewise, the memory requirement is reduced by a factor of about 360 (i.e., allocating memory in the order of GB instead of TB).

PNaV: A tool for generating a high-dose-rate brachytherapy treatment plan by navigating the Pareto surface guided by the visualization of multidimensional trade-offs

PurposeA Pareto Navigation and Visualization (PNaV) tool is presented for interactively constructing a high-dose-rate (HDR) brachytherapy treatment plan by navigating and visualizing the multidimensional Pareto surface. PNaV aims to improve treatment planning time and quality and is generalizable to any number of dose–volume histogram (DVH) and convex dose metrics. Methods and MaterialsPareto surface visualization and navigation were demonstrated for prostate, breast, and cervix HDR brachytherapy sites. A library of treatment plans was created to span the Pareto surfaces over a 30% range of doses in each of five DVH metrics. The PNaV method, which uses a nonnegative least-squares model to interpolate the library plans, was compared against pure optimization for 11,250 navigated plans using data envelopment analysis. The visualization of the metric trade-offs was accomplished using numerically estimated partial derivatives to plot the local curvature of the Pareto surface. PNaV enables the user to control both the magnitude and direction of the trade-off during navigation. ResultsProof of principle of PNaV was demonstrated using a graphical user interface with visualization tools to enabled rapid plan selection and a quantitative review of metric trade-offs. PNaV produced deliverable plans with DVH metrics within < 0.4%, 0.6%, and 1.1% (95% confidence interval) of the Pareto surface using plan libraries with nominal plan spacing of 10%, 15%, and 30% in each metric dimension, respectively. The interpolation used for the navigation executed in 0.1 s. The fast interpolation allows for quick and efficient exploration of trade-off options by the physician, after an initial preprocessing step to generate the library. ConclusionsGeneration, visualization, and navigation of the Pareto surface were validated for brachytherapy treatment planning. The PNaV method enables efficient and informed decision-making for radiotherapy.

An FPGA-based design for real-time super-resolution reconstruction

For several decades, the camera spatial resolution is gradually increasing with the CMOS technology evolution. The image sensors provide more and more pixels, generating new constraints for suitable optics. As an alternative, promising solutions propose super-resolution (SR) techniques to reconstruct high-resolution images or video without modifying the sensor architecture. However, most of the SR implementations are far from reaching real-time performance on a low-budget hardware platform. Moreover, convincing state-of-the-art studies reveal that artifacts can be observed in highly textured areas of the image. In this paper, we propose a local adaptive spatial super resolution (LASSR) method to fix this limitation. LASSR is a two-step SR method including a machine learning-based texture analysis and a fast interpolation method that performs a pixel-by-pixel SR. Multiple evaluations of our method are also provided using standard image metrics for quantitative evaluation and also a psycho-visual assessment for a perceptual evaluation. A first FPGA-based implementation of the proposed method is then presented. It enables high-quality 2–4 k super-resolution videos to be performed at 16 fps, using only 13% of the FPGA capacity, opening the way to reach more than 60 fps by executing several parallel instances of the LASSR code on the FPGA.

Fast interpolation method for surfaces with faults by multi-scale second-derivative optimization

We present a smooth surface interpolation method enabling to take discontinuities (e.g. faults) into account that can be applied to any dataset defined on a regular mesh. We use a second-derivative multi-scale minimization based on a conjugate gradient method. Our multi-scale approach allows the algorithm to process millions of points in a few seconds on a single-unit workstation. The interpolated surface is continuous, as well as its first derivative, except on some lines that have been specified as discontinuities. Application in geosciences are numerous, for instance when a structural model is to be built from points picked on seismic data. The resulting dip of interpolation extends the dip of the input data. The algorithm also works if faults are given by broken lines. We present results from a synthetic and real examples taking into account fault network.

A Fast 4K Video Frame Interpolation Using a Multi-Scale Optical Flow Reconstruction Network

Recently, video frame interpolation research developed with a convolutional neural network has shown remarkable results. However, these methods demand huge amounts of memory and run time for high-resolution videos, and are unable to process a 4K frame in a single pass. In this paper, we propose a fast 4K video frame interpolation method, based upon a multi-scale optical flow reconstruction scheme. The proposed method predicts low resolution bi-directional optical flow, and reconstructs it into high resolution. We also proposed consistency and multi-scale smoothness loss to enhance the quality of the predicted optical flow. Furthermore, we use adversarial loss to make the interpolated frame more seamless and natural. We demonstrated that the proposed method outperforms the existing state-of-the-art methods in quantitative evaluation, while it runs up to 4.39× faster than those methods for 4K videos.

Calculations for deep inelastic scattering using fast interpolation grid techniques at NNLO in QCD and the extraction of alpha _{mathrm {s}} from HERA data

The extension of interpolation-grid frameworks for perturbative QCD calculations at next-to-next-to-leading order (NNLO) is presented for deep inelastic scattering (DIS) processes. A fast and flexible evaluation of higher-order predictions for any a posteriori choice of parton distribution functions (PDFs) or value of the strong coupling constant is essential in iterative fitting procedures to extract PDFs and Standard Model parameters as well as for a detailed study of the scale dependence. The APPLfast project, described here, provides a generic interface between the parton-level Monte Carlo program NNLOjet and both the APPLgrid and fastNLO libraries for the production of interpolation grids at NNLO accuracy. Details of the interface for DIS processes are presented together with the required interpolation grids at NNLO, which are made available. They cover numerous inclusive jet measurements by the H1 and ZEUS experiments at HERA. An extraction of the strong coupling constant is performed as an application of the use of such grids and a best-fit value of alpha _{mathrm {s}} (M_{{mathrm {Z}}}) = 0.1170,(15)_text {exp},(25)_text {th} is obtained using the HERA inclusive jet cross section data.

MLACE-MLFMA Combined With Reduced Basis Method for Efficient Wideband Electromagnetic Scattering From Metallic Targets

In this paper, a fast interpolation algorithm [multilevel accelerated Cartesian expansion (MLACE)-multilevel fast multipole algorithm (MLFMA)-reduced basis method (RBM)] is proposed for the first time by leveraging the benefits of the MLACE, the classical MLFMA, and the RBM; furthermore, a novel nonblind sampling method (NBSM) is proposed to approximate an orthogonal subspace for the MLACE-MLFMA-RBM. By integrating the empirical interpolation method (EIM) and the error-based compensation method (EBCM) into the NBSM, we first efficiently establish a robust and precision-controllable orthogonal subspace. Next, with the accurate orthogonal subspace, the MLACE-MLFMA-RBM can simplify the iterative solution process of the MLACE-MLFMA or the MLFMA into N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> matrix-vector multiplications (MVMs) and the solution of a reduced N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> -dimensional linear equation (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> is the number of orthonormal vectors). Note that only the MVMs between the far-region impedance matrix and the orthogonal vector bases are calculated. In comparison with the classical MLACE-MLFMA and the MLACE-MLFMA-RBM based on different sampling methods for wideband electromagnetic (EM) responses, our proposed algorithms require less memory and CPU time and integrate higher flexibility. The numerical results are provided to demonstrate the accuracy, efficiency, and flexibility.

Gas Leakage Source Detection for Li-Ion Batteries by Distributed Sensor Array.

Lithium-based batteries operation is related to some safety risks of dangerous flaming, integrity destruction, or even explosion. Gas leakage is an early and reliable indicator for such irreversible malfunctioning of electrical accumulators. Often, accurate gas emission source location sensing is difficult especially in heavy operational conditions, related to temperature changes, vibrations, movements, accelerations, etc. In this paper we propose a gas detection system, with catalytic type sensor array, and a numerical reconstruction method for precise gas emission source location inside the battery pack. The detection system employs a distributed array of CO sensors. Proposed sensor array configurations significantly reduce the number of sensing nodes inside the battery pack and fewer sensors than the protected battery elements are used. This way, data acquisition process by sensor nodes is also simplified. Several array configurations are considered according to their measurement efficiency and accuracy. Reconstruction algorithm is based on fast interpolation technique very suitable for real-time data processing. Estimation of reconstruction method accuracy is made by computational model of the gas diffusion inside the pack.

Fast Extraction of Local Underwater Terrain Features for Underwater Terrain-Aided Navigation

Terrain matching accuracy and real-time performance are affected by local underwater terrain features and structure of matching surface. To solve the extraction problem of local terrain features for underwater terrain-aided navigation (UTAN), real-time data model and selection method of beams are proposed. Then, an improved structure of terrain storage is constructed, and a fast interpolation strategy based on index is proposed, which can greatly improve the terrain interpolation–reconstruction speed. Finally, for the influences of tide, an elimination method of reference depth deviation is proposed, which can reduce the reference depth errors caused by tidal changes. As the simulation test shows, the proposed method can meet the requirements of real-time performance and effectiveness. Furthermore, the extraction time is considerably reduced, which makes the method suitable for the extraction of local terrain features for UTAN.

A local and fast interpolation method for mesh deformation

A local and fast interpolation method for mesh deformation

Efficient interpolated compressed sensing reconstruction scheme for 3D MRI

3D magnetic resonance imaging (3D MRI) is one of the most preferred medical imaging modalities for the analysis of anatomical structures where acquisition of multiple slices along the slice select gradient direction is very common. In 2D multi-slice acquisition, adjacent slices are highly correlated because of very narrow inter-slice gaps. Application of compressed sensing (CS) in MRI significantly reduces traditional MRI scan time due to random undersampling. The authors first propose a fast interpolation technique to estimate missing samples in the k-space of a highly undersampled slice (H-slice) from k-space (s) of neighbouring lightly undersampled slice/s (L-slice). Subsequently, an efficient multislice CS-MRI reconstruction technique based on weighted wavelet forest sparsity, and joint total variation regularisation norms is applied simultaneously on both interpolated H and non-interpolated L-slices. Simulation results show that the proposed CS reconstruction for 3D MRI is not only computationally faster but significant improvements in terms of visual quality and quantitative performance metrics are also achieved compared to the existing methods.

A fast method for Brillouin frequency shift estimation

A fast interpolation algorithm based on cross-correlation is proposed to estimate the BFS in BOTDA systems. The performances of the proposal are investigated through simulation and experiment in this paper. The simulation results demonstrate that the computational cost of the proposed algorithm is reduced to less than 1/22, 1/36, 1/2 and 1/5 of the Lorentzian fitting, upsampling cross-correlation, partial quadratic fitting and partial cross-correlation algorithm respectively. The experimental results indicate that the error of BFS extracted by our proposal is the same as that of the other schemes when no hotspot is presented and the error of BFS is smaller in cross-correlation based schemes than that in Lorentzian fitting scheme around hotspot.

A local and fast interpolation method for mesh deformation

In this study, a local interpolation approach is developed to deal with the deformation of a mesh containing structured and stretched viscous elements. A new strategy is proposed to compute the local influence distance at each movable mesh point. This strategy not only helps to exclude unnecessary donor points from the computation of the new position of the mesh point, but also gets those viscous elements stiffened appropriately such that they are no longer inverted easily. Further improvements on the efficiency of the proposed algorithm are achieved by parallelising the algorithm and developing a bounding box technique to speed up the procedure of searching donor points. Meanwhile, the robustness of the proposed algorithm in the case of large boundary movement is enhanced by incorporating a state-of-the-art multi-step technique. Various numerical experiments are conducted and the results show that the proposed algorithm could treat cases involving complex boundary movements at a low level of computing consumptions.

Magnetic resonance image reconstruction using fast interpolated compressed sensing

Application of the compressed sensing (CS) in magnetic resonance imaging (MRI) significantly reduces the scan time as it is capable of recovering images at diagnostic resolution from a few random measurements in the k-space. In 2D multi-slice MRI, a strong inter-slice correlation exists which promotes further reduction in the scan time. In this paper, first, a fast interpolation technique is proposed to estimate missing samples in the k-space of some highly undersampled slices from neighboring low undersampled slices. Next, a novel wavelet tree sparsity based CS reconstruction method is developed for the reconstruction of images from raw 2D multi-slice data. The combination of fast interpolation and CS reconstruction gives accurate reconstruction of multi-slice MR images at significantly less computational time. Simulation results show that the proposed method gives better preservation of edges and other image details with less computational time than some of the recently developed fast CS reconstruction techniques.