Interpolation Error Research Articles

HR-Depth: High Resolution Self-Supervised Monocular Depth Estimation

Self-supervised learning shows great potential in monocular depth estimation, using image sequences as the only source of supervision. Although people try to use the high-resolution image for depth estimation, the accuracy of prediction has not been significantly improved. In this work, we find the core reason comes from the inaccurate depth estimation in large gradient regions, making the bilinear interpolation error gradually disappear as the resolution increases. To obtain more accurate depth estimation in large gradient regions, it is necessary to obtain high-resolution features with spatial and semantic information. Therefore, we present an improved DepthNet, HR-Depth, with two effective strategies: (1) re-design the skip-connection in DepthNet to get better high-resolution features and (2) propose feature fusion Squeeze-and-Excitation(fSE) module to fuse feature more efficiently. Using Resnet-18 as the encoder, HR-Depth surpasses all previous state-of-the-art(SoTA) methods with the least parameters at both high and low resolution. Moreover, previous SoTA methods are based on fairly complex and deep networks with a mass of parameters which limits their real applications. Thus we also construct a lightweight network which uses MobileNetV3 as encoder. Experiments show that the lightweight network can perform on par with many large models like Monodepth2 at high-resolution with only20%parameters. All codes and models will be available at https://github.com/shawLyu/HR-Depth.

Melt Pool and Heat Treatment Optimization for the Fabrication of High-Strength and High-Toughness Additively Manufactured 4340 Steel

Additively manufactured (AM) components offer superior design flexibility compared to their conventionally manufactured counterparts and require process parameter optimization to achieve high-quality depositions with desirable and predictable mechanical properties. This study was focused on 4340 steel fabricated using laser powder bed fusion (LPBF), and 42 laser power and scan speed combinations have been systematically investigated to optimize the melt pool geometry and ensure fully dense parts. The AM material was compared with a wrought 4340 equivalent and studied in both as-fabricated and heat-treated conditions. Two heat treatments were designed and used to optimize the materials for strength and toughness, respectively. Microstructure, tensile, and fractographic studies were conducted to assess build integrity and establish processing–structure–performance relationships. Tensile properties of the AM materials in all studied conditions were equivalent or better than the comparable wrought materials. The high performance of the AM materials was attributed to the absence of both manganese sulfide inclusions and rolling-induced banding, typically found in the wrought materials. The fine cellular substructure is thought to additionally contribute to the high strength of the as-fabricated AM material. Complementary to the knowledge that has emerged from the experimental investigations, the dataset was further leveraged to make recommendations for future design of experiments to optimize AM build parameters in other material systems. A statistical Monte Carlo analysis was used to predict the interpolation error produced when using reduced datasets and to enable informed processing parameters selection. These findings are discussed to make recommendations for the use of AM materials for high-integrity structural applications.

An effective interpolation strategy for mitigating the day boundary discontinuities of precise GNSS orbit products

A precise Global Navigation Satellite System (GNSS) satellite orbit is an essential prerequisite for precise point positioning (PPP). Due to the daily orbit discontinuities (DODs) between adjacent daily orbit products provided by the International GNSS Service, the interpolation accuracy at the epochs near the day boundaries cannot be as high as expected and can significantly degrade the positioning accuracy. This is especially for the emerging Beidou and Galileo satellites with larger DODs. In this study, a new interpolation strategy was proposed to mitigate the DOD effect by estimating the zero- to second-order biases between the orbits of two adjacent days. The results show that the proposed interpolation strategy outperforms the traditional interpolation strategies. The 3D orbit interpolation error for the last 10 min on each day is reduced from almost a few centimeters or even meters to millimeters, an improvement of approximately 86–99% for all satellites. PPP validation also provides concrete evidence of this improvement, as the DOD-induced satellite phase residuals are reduced significantly. Thus, the new interpolation strategy mitigates the DOD effect and helps multi-GNSS users obtain the highest interpolation accuracy at the day boundaries, which can be comparable to that at the non-day boundaries.

A preliminary isotope‐based evapotranspiration partitioning approach for tropical Costa Rica

AbstractSpatially and seasonally distributed information on transpiration (T) fluxes is limited in the tropics. Here, we applied a coupled isotope mass balance model to separate water fluxes of T and evapotranspiration (ET) from precipitation (P). The mean annual T was estimated at a resolution of 100 m for Costa Rica (51,100 km2) and a partitioning of monthly T and evaporation (E) for the 2370‐km2 San Carlos catchment. The dominant flux in the forest ecosystems was T with a mean annual T of 1086 mm that ranged from 700 mm in Tropical Montane Very Humid Forest to 1400 mm in Subtropical and Tropical Low Montane Rainforests. We estimated an average 85% of ET was T, which is concurrent with expectations for forested tropical regions, but varied according to model parameterization and data sources. A model comparison exercise showed a range of mean annual T estimates from 988 to 1465 mm and a range of T/P from 0.35 to 0.5 with temperature and relative humidity exhibiting the highest impact on the model results. Across Costa Rica, we estimated an average loss of precipitation by T of 38% (1085 mm), whereas interception (I) constitutes 10% (230 mm) and direct evaporation (E) only 7% (192 mm). Similarly, the results at the catchment scale indicated that monthly T contributes 76% (85‐mm monthly average) to total ET and E corresponds to 24% (24‐mm monthly average). The T rates exhibited an opposite seasonality to rainfall with highest T over the drier months from December to April with a peak in March (101–144 mm) and the minimum T in September (53–71 mm). Around 17% (79–130 mm) of precipitation over the catchment area is lost to T, both E (10–35 mm) and I (15–38 mm) correspond to 5%. Despite the inherent uncertainties of the data assumptions and simplifications, including data interpolation errors, the coupled isotope mass balance model showed in comparison to other global products reasonable water partitioning for different ecosystems in Costa Rica and the San Carlos catchment area. These results can help to evaluate the impact of land cover conversion on the hydrological cycle in Costa Rica, and the simple isotope‐based model could be transferred to different biomes of the tropics.

Feature-based and goal-oriented anisotropic mesh adaptation for RANS applications in aeronautics and aerospace

The scope of this paper is to demonstrate the viability of unstructured anisotropic mesh adaptation for commercial aircraft drag and high-lift prediction studies. The main achievement of this work is to demonstrate that mesh-independent certified numerical solutions can be achievable thanks to anisotropic mesh adaptation and that it is possible to run high-fidelity CFD on unstructured adapted meshes composed only of tetrahedra which is fundamental to design robust meshing process for complex geometries. It also points out the early capturing property of the solution-adaptive process in the sense that accurate output functional values are obtained on relatively coarse adapted meshes. On a more practical point of view, this paper demonstrates how mesh adaptation, thanks to its automation, is able to generate meshes that are extremely difficult to envision and almost impossible to generate manually, leading to highly accurate numerical solutions. Moreover, as the process can start from any coarse initial mesh, it greatly simplifies the overall meshing process. This study also analyze the influence of different strategies in the mesh adaptation algorithm and in the error analysis which are key components of the process. Several error estimates are considered: feature-based ones which are based on the standard multiscale Lp interpolation error estimate and goal-oriented ones to control the error on output functionals which rely on an accurate computation of the adjoint state. The adjoint problem proves to be a stiff problem for RANS equations, failing to converge the adjoint state to machine zero may impact negatively the adaptive process. The maturity of the solution-adaptive process is demonstrated on numerous drag and high-lift prediction cases. It has also excelled in sonic boom and turbomachine applications.

Developing an entropy and copula-based approach for precipitation monitoring network expansion

Extreme precipitation events have been increasing in recent decades and have been causing impacts to human life and property, which requires expansion of existing networks to provide essential information for hydrometeorological analysis. This study developed an entropy and copula-based approach for precipitation monitoring network expansion, aiming at adding stations in ungauged areas with high value of monitoring (VOM), which was estimated through information content and redundancy. The approach was applied to a network in the Taihu Lake basin, China. We grouped the adjacent stations based on Thiessen polygons, computed VOM for each group and identified candidate locations for additional stations. The entropy measures, e.g., joint entropy and total correlation, required to compute VOM were estimated through the principle of maximum entropy (POME) and copula-based joint distribution fitting. We evaluated the performance of five copula families and adopted the Student t-copula as the optimal copula model. Results showed that orographic effect and network density were the main factors that influence VOM variation, i.e., high VOM appeared in the southwest mountain area and the east area with sparsely distributed stations. We also compared the results with those derived from three data quantization methods, among which the rounding technique produced consistent results. The final network schemes were generated by adding stations following a strategy that supports decision preference on the positioning scheme and station number, and were evaluated through the Kriging interpolation error. The proposed approach provides an efficient and a practical way for strengthening a precipitation monitoring system.

Interpolation with Specified Error of a Point Series Belonging to a Monotone Curve.

The paper addresses the problem of modeling a smooth contour interpolating a point series belonging to a curve containing no special points, which represents the original curve with specified accuracy. The contour is formed within the area of possible location of the parts of the interpolated curve along which the curvature values are monotonously increased or decreased. The absolute interpolation error of the point series is estimated by the width of the area of possible location of the curve. As a result of assigning each intermediate point, the location of two new sections of the curve that lie within the area of the corresponding output section is obtained. When the interpolation error becomes less than the given value, the area of location of the curve is considered to be formed, and the resulting point series is interpolated by a contour that lies within the area. The possibility to shape the contours with arcs of circles specified by characteristics is investigated.

Asymptotics of Polynomial Interpolation and the Bernstein Constants

It is well known that the interpolation error for left| xright| ^{alpha },alpha >0 in L_{infty }left[ -1,1right] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when alpha tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to left| xright| ^{alpha },alpha >0 in the L_{infty } norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

Nielson interpolation operators on an arbitrary triangle with one curved side

We extend some Nielson type interpolation operators to the cases of standard and arbitrary triangles with one curved side. The correspondence between the operators defined on standard triangles and arbitrary triangles is made using barycentric coordinates. We study the interpolation properties of the obtained operators and the interpolation errors. For illustration, we give some numerical examples.

Directed graph mapping exceeds phase mapping in discriminating true and false rotors detected with a basket catheter in a complex in-silico excitation pattern

Atrial fibrillation (AF) is the most frequently encountered arrhythmia in clinical practise. One of the major problems in the management of AF is the difficulty in identifying the arrhythmia sources from clinical recordings. That difficulty occurs because it is currently impossible to verify algorithms which determine these sources in clinical data, as high resolution true excitation patterns cannot be recorded in patients. Therefore, alternative approaches, like computer modelling are of great interest. In a recent published study such an approach was applied for the verification of one of the most commonly used algorithms, phase mapping (PM). A meandering rotor was simulated in the right atrium and a basket catheter was placed at 3 different locations: at the Superior Vena Cava (SVC), the Crista Terminalis (CT) and at the Coronary Sinus (CS). It was shown that although PM can identify the true source, it also finds several false sources due to the far-field effects and interpolation errors in all three positions. In addition, the detection efficiency strongly depended on the basket location.Recently, a novel tool was developed to analyse any arrhythmia called Directed Graph Mapping (DGM). DGM is based on network theory and creates a directed graph of the excitation pattern, from which the location and the source of the arrhythmia can be detected. Therefore, the objective of the current study was to compare the efficiency of DGM with PM on the basket dataset of this meandering rotor. The DGM-tool was applied for a wide variety of conduction velocities (minimal and maximal), which are input parameters of DGM.Overall we found that DGM was able to distinguish between the true rotor and false rotors for both the SVC and CT basket positions. For example, for the SVC position with a CVmin=0.01cmms, DGM detected the true core with a prevalence of 82% versus 94% for PM. Three false rotors where detected for 39.16% (DGM) versus 100% (PM); 22.64% (DGM) versus 100% (PM); and 0% (DGM) versus 57% (PM). Increasing CVmin to 0.02cmms had a stronger effect on the false rotors than on the true rotor. This led to a detection rate of 56.6% for the true rotor, while all the other false rotors disappeared. A similar trend was observed for the CT position. For the CS position, DGM already had a low performance for the true rotor for CVmin=0.01cmms (14.7%). For CVmin=0.02cmms the false and the true rotors could therefore not be distinguished.We can conclude that DGM can overcome some of the limitations of PM by varying one of its input parameters (CVmin). The true rotor is less dependent on this parameter than the false rotors, which disappear at a CVmin=0.02cmms. In order to increase to detection rate of the true rotor, one can decrease CVmin and discard the new rotors which also appear at lower values of CVmin.

Preasymptotic Error Analysis of the HDG Method for Helmholtz Equation with Large Wave Number

This paper addresses several aspects of the linear hybridizable discontinuous Galerkin method (HDG) for the Helmholtz equation with impedance boundary condition at high frequency. First, error estimates with explicit dependence on the wave number k for the HDG approximations to the exact solution u and its negative gradient $${\mathbf {q}}=-\nabla u$$ are derived. It is shown that $$k\Vert u-u_h\Vert _{L^2(\varOmega )}+ \Vert {\mathbf {q}}-{\mathbf {q}}_h\Vert _{L^2(\varOmega )}=O(k^2h^2+k^4h^3)$$ under the conditions that $$k^3h^2$$ is sufficiently small and that the penalty parameter $$\tau \eqsim k$$ , where h is the mesh size. Note that the convergence order in $${\mathbf {q}}_h$$ is full and the pollution error is $$O(k^4h^3)$$ , which improve the existent results. Secondly, by using a standard postprocessing procedure from the HDG method for elliptic problems, a piecewise quadratic function $$u_h^*$$ is obtained so that $$k\Vert u-u_h^*\Vert _{L^2(\varOmega )}=O(k^3h^3+k^4h^3)$$ . Note that the postprocessing procedure improves only the interpolation error (from $$O(k^2h^2)$$ to $$O(k^3h^3)$$ ) but leaves the pollution error $$O(k^4h^3)$$ unchanged. Thirdly, a dispersion analysis and extensive numerical tests show that the pollution effect can be reduced greatly in 2D case by selecting appropriate penalty parameters.

Error model-based iterative compensation method for improving the estimation of frequency response function

For the estimation of frequency response function (FRF) using the window-based spectral estimation method, the error models of the FRF estimation are introduced, including window interpolation error and transient error models, and an error model-based iterative compensation (EMIC) method is then proposed to improve the FRF estimation accuracy. The specific iteration steps of the proposed EMIC method are presented, fast computations are adopted in the steps as far as possible; and that, iteration convergence and noise uncertainty are analyzed. The analyses reveal that, the proposed EMIC method is almost always convergent as long as the error of the FRF pre-estimation is small, intercepting enough data and using a window which has smaller and fewer side DFT bins as well as smaller front-end value for FRF estimation can result in better iteration convergence; whereas, the proposed EMIC method will slightly increase the noise uncertainty of FRF estimation compared with that without iterative compensation because it removes the window interpolation effects on noise just like removing the error. To verify the effectiveness and practicability of the proposed EMIC method, model simulation and actual sensor data verification are implemented, and multi existing methods are employed for comparison. The results show that, the proposed EMIC method can largely decrease the error of the FRF estimated by the window-based spectral estimation method, it has the best maximum error restraining capabilities especially in the resonance frequency band and relatively better comprehensive error restraining capabilities especially when the data is not long enough compared with the mentioned existing methods, also, the time consumption of the proposed EMIC method is acceptable and is usually seconds or less; and that, the theoretical analyses, model simulation results and actual data verification results are consistent with each other. Consequently, the proposed EMIC method is proven to be effective and practical for improving the FRF estimation of a stable LTI system from its step or impulse response data.

Global smoothing for five-axis linear paths based on an adaptive NURBS interpolation algorithm

The five-axis tool path generated by CAM software usually consists of a series of linear paths. The tangent direction at the corner of the adjacent line segment will suddenly change, and the curvature is also discontinuous, which will cause vibration and shock during the machining process. Thus, a global corner smoothing algorithm based on cubic NURBS interpolation is proposed to smooth the linear paths in this paper, so as to achieve G2 continuous for five-axis linear paths. The algorithm proposed does not require matrix operations to solve the control points, and it can also reduce the number of control points while satisfying the interpolation error. The algorithm is then used to generate smooth NURBS path for ceramic core burrs and blockage repair. The simulation and experiment show that the algorithm proposed can satisfy the error constraints, reduce the vibration of the motion axis, and improve the surface quality of laser cutting.

A finite element method for Allen–Cahn equation on deforming surface

The paper studies an Allen–Cahn-type equation defined on a time-dependent surface as a model of phase separation with order–disorder transition in a thin material layer. By a formal inner–outer expansion, it is shown that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates. A geometrically unfitted finite element method, known as a trace FEM, is considered for the numerical solution of the equation. The paper provides full stability analysis and convergence analysis that accounts for interpolation errors and an approximate recovery of the geometry.

Experimental function estimation from quantum phase measurements

Characterizing a system often demands learning its response function to an applied field. Such knowledge is rooted on the experimental evaluation of punctual fiducial response and interpolation to access prediction at arbitrary values. Quantum metrological resources are known to provide enhancement in assessing these fiducial points, but the implications for improved function estimation have only recently been explored, and have not been yet demonstrated. Here we show an experimental realization of function estimation based on a photonic achitecture. The phase response of a liquid-crystal to a voltage has been reconstructed by means of quantum and classical phase estimation, providing evidence of the superiority of the former and highlighting the interplay between punctual statistical error and interpolation error. Our results show how quantum resources should successfully be employed to access the rich information contained in continuous signals.

New Developments in the Comprehensive Inner Magnetosphere‐Ionosphere Model

AbstractA Comprehensive Inner Magnetosphere‐Ionosphere (CIMI) model was developed to study the dynamic variations of the radiation belts, ring current, and plasmasphere as well as the couplings between these populations and with the ionosphere. The CIMI model has been used in multiple studies to simulate a number of magnetic storms and reproduce many observable features that involve cross‐region and cross‐energy processes. This model has been continuously validated and improved. In this study, we report on two major recent advancements in CIMI. The first improvement is transforming coordinates in both configuration and velocity space into uniform grids. Numerical schemes, especially those of high order, are generally more stable on such uniform grids. The second advancement is to solve both advection and diffusion in the CIMI equation in a single coordinate system. Previously, the phase space distribution was expressed in terms of the first two adiabatic invariants when solving for particle drift, whereas particle distributions were transformed into energy and pitch angle for calculating wave diffusion. In the new approach in CIMI, particles and distribution functions are identified by their momentum. This change avoids interpolation errors inherent in the mapping of the phase space distribution when switching back and forth between advective and diffusive transport. We describe the detailed implementation of these recent advancements in the CIMI model, and demonstrate how the newly improved model provides better predictions of the ring current and radiation belt dynamics.

Accuracy of compact-stencil interpolation algorithms for unstructured mesh finite volume solver

This study considers the accuracy of cell-to-face centre interpolation of convected quantities in unstructured finite volume meshes with cell-centred storage of variables. The accuracy of the interpolation algorithms were tested in isolation using ideal data to determine their numerical accuracy on both standard and artificially distorted meshes. It was found that the formally second- and third-order accurate interpolations based on one-dimensional interpolation along the line connecting the cells to the right and left of the face under consideration only have first-order accuracy in standard unstructured mesh, and less than first-order accuracy in distorted unstructured mesh. L1 interpolation errors in the distorted unstructured mesh are greater than in standard unstructured mesh. The order of accuracy and L1 errors can be improved by applying spatial corrections. The formally second-order accurate multi-dimensional interpolations tested in this study that are not based on one-dimensional interpolation along lines connecting the neighbour cells have first-order accuracy in both standard and distorted unstructured mesh. Linear interpolation between end vertices produces greatest L1 error in standard mesh; polynomial interpolation, linear interpolation between cell centres and standard QUICK produce the greatest L1 error in distorted mesh. Spatially correct QUICK, spatially correct linear interpolation between cell centres, Laplacian interpolation to face centres, and Taylor series expansion about an upstream cell produce the smallest L1 error in both standard and distorted mesh. Based on accuracy and the simplicity of implementation, Taylor series expansion about an upstream cell is the best choice for use in unstructured mesh.

Implementation of higher-order velocity mapping between marker particles and grid in the particle-in-cell code XGC

The global total- $f$ gyrokinetic particle-in-cell code XGC, used to study transport in magnetic fusion plasmas or to couple with a core gyrokinetic code while functioning as an edge gyrokinetic code, implements a five-dimensional continuum grid to perform the dissipative operations, such as plasma collisions, or to exchange the particle distribution function information with a core code. To transfer the distribution function between marker particles and a rectangular two-dimensional velocity-space grid, XGC employs a bilinear mapping. The conservation of particle density and momentum is accurate enough in this bilinear operation, but the error in the particle energy conservation can become undesirably large and cause non-negligible numerical heating in a steep edge pedestal. In the present work we update XGC to use a novel mapping technique, based on the calculation of a pseudo-inverse, to exactly preserve moments up to the order of the discretization space. We describe the details of the implementation and we demonstrate the reduced interpolation error for a tokamak test plasma using first- and second-order elements with the pseudo-inverse method and comparing with the bilinear mapping.

Category-based fractal modelling: A novel model to integrate the geology into the data for more effective processing and interpretation

Parent lithology exerts a major influence on the associated regolith geochemistry and affects the setting of population or domain boundaries. In regional geochemical mapping and anomaly detection, a lack of samples pertaining to one or more key lithologies may results in interpolation errors. Such errors can be addressed via category-based fractal modelling to characterise different background and anomalous populations by taking both error propagation and geochemical data values into account. Such geochemical anomaly classification models have been developed using Monte Carlo simulation (MCSIM) within a given lithological domain to provide estimates of population breaks. Two types of category-based fractal models used are (i) related lithological groups representative samples where there is an adequate number of samples derived from a given lithological or geological domain, and (ii) related lithological groups whole samples and a group of Gaussian simulated samples regenerated based on the mean and standard deviation of the available samples per lithological group. Both models were applied to VMS-related element concentrations in the low-density till geochemical mapping of Sweden following centred log-ratio (clr) transformation. Compared with the concentration-area fractal model the category-based fractal models provided more reliable estimates for population thresholds and identification of anomalous classes.

Computable Interpolation Error Constants for the Geometric Simplex Finite Elements

In this paper, a novel high accuracy computation method for interpolation error constants is proposed over the geometric simplex finite elements. Firstly, the expansions of bounded linear operators are employed to derive the explicit estimate of interpolation error constants, which depend only on the shape of the geometric simplex finite elements and the definition of interpolation functions. Then, this method is applied to the linear interpolation function, and the results are consistent with our analysis. Finally, some numerical examples are given to validate our analysis. Such high accuracy computation method for interpolation error constants are beneficial attempts to accelerate the adaptive computation and verification of finite element solutions.