Efficient Interpolation Research Articles

Scalable Constant pH Molecular Dynamics in GROMACS.

Molecular dynamics (MD) computer simulations are used routinely to compute atomistic trajectories of complex systems. Systems are simulated in various ensembles, depending on the experimental conditions one aims to mimic. While constant energy, temperature, volume, and pressure are rather straightforward to model, pH, which is an equally important parameter in experiments, is more difficult to account for in simulations. Although a constant pH algorithm based on the λ-dynamics approach by Brooks and co-workers [Kong, X.; Brooks III, C. L. J. Chem. Phys.1996, 105, 2414–2423] was implemented in a fork of the GROMACS molecular dynamics program, uptake has been rather limited, presumably due to the poor scaling of that code with respect to the number of titratable sites. To overcome this limitation, we implemented an alternative scheme for interpolating the Hamiltonians of the protonation states that makes the constant pH molecular dynamics simulations almost as fast as a normal MD simulation with GROMACS. In addition, we implemented a simpler scheme, called multisite representation, for modeling side chains with multiple titratable sites, such as imidazole rings. This scheme, which is based on constraining the sum of the λ-coordinates, not only reduces the complexity associated with parametrizing the intramolecular interactions between the sites but also is easily extendable to other molecules with multiple titratable sites. With the combination of a more efficient interpolation scheme and multisite representation of titratable groups, we anticipate a rapid uptake of constant pH molecular dynamics simulations within the GROMACS user community.

Computational investigation into the effect of constitutive formulations for flow modeling of high elasticity

• Three constitutive formulations of the Leonov model are numerically investigated. • It is verified that keeping positive-definiteness of the conformation tensor is crucial for computational evolution while appropriate interpolation of exponentially convecting domain is essential for accuracy. • Swirling asymmetric corner vortices are described in 3D contraction flow modeling. For the purpose of defeating the high Deborah (or Weissenberg) number problems in viscoelastic flow modeling, the matrix-logarithmic formulation of the constitutive equations has been suggested and proven probably the most effective. This transformation of the conformation tensor endows proper resolution of the boundary layer with exponential convection as well as conservation of its positive definiteness. Herein we make an attempt to evaluate numerical consequences of these two features independently for the Leonov viscoelastic model mainly in the contraction flow. To this end, we from the viewpoint of computational stability and accuracy investigate three formulations written with variables such as the original conformation tensor c (c form), its matrix-logarithm (h form) and additive decomposition (b form). 2D planar 4:1 contraction flow in the limit of vanishing inertia in addition to its 1D toy problem is considered in the framework of finite element method. The h and b forms are designed to preserve the positive definiteness of c , and furthermore the h form grants efficient interpolation of exponentially convecting variables. The 1D analogy expresses the characteristics qualitatively equivalent to the original problem in a simpler fashion. The c form modeling of 2D flow exhibits typical nonevolutionary behavior when the Deborah number ( De ) exceeds a certain limit, and this degradation gets worse with finer mesh. While the h and b forms have always been computationally stable with proper description of steady behavior or elastic instability otherwise, the h form manifests faster mesh convergence. As a complementary study, the h form calculation of full 3D flow along an axisymmetric contraction channel has been performed, which also attests well-posedness without upper bound in De . Its steady result is corroborated with the one under axisymmetric flow approximation. Loss of convergence in the axisymmetric flow regime at high De is verified to result from the occurrence of 3D instability, which is then well portrayed in 3D modeling. In this instability, swirling of asymmetric structure for the corner vortices about the geometrical symmetry axis plausibly mimics experimental observation. Consequently, maintaining the positive definiteness in terms of h or b form is crucial for numerically stable evolution, and the h form interpolation delineating appropriately the exponential convection assures the fastest mesh convergence, hence enhancing the accuracy.

Increasing the Lateral Resolution of 3D-GPR Datasets through 2D-FFT Interpolation with Application to a Case Study of the Roman Villa of Horta da Torre (Fronteira, Portugal)

The approach presented in this work uses an interpolation methodology to densify 3D-GPR datasets to sharpen the results obtained in GPR surveys carried out in an archaeological environment. It allows the estimation of missing data from the combined use of mathematical transforms, such as the Fourier and curvelet transforms, and predictive filters. This technique makes it possible to calculate the missing signal simply by meeting two requirements: the data in the frequency domain must be limited in a range of values and must be able to be represented by a distribution of Fourier coefficients (verified conditions). The INT-FFT algorithm uses an open-access routine (Suinterp, Seismic Unix) to interpolate the GPR B-scans based on seismic trace interpolation. This process uses automatic event identification routines by calculating spatial derivatives to identify discontinuities in space by detecting very subtle changes in the signal, thus allowing for more efficient interpolation without artifacts or signal deterioration. We successfully tested the approach using GPR datasets from the Roman villa of Horta da Torre (Fronteira, Portugal). The results showed an increase in the geometric sharpness of the GPR reflectors and did not produce any numerical artifacts. The tests performed to apply the methodology to GPR-3D data allowed for assessing the interpolation efficiency, the level of estimation of missing data, and the level of information lost when we chose to increase the distance between B-scans in the acquisition stage.

Forward Warping-Based Video Frame Interpolation Using a Motion Selective Network

Recently, deep neural networks have shown surprising results in solving most of the traditional image processing problems. However, the video frame interpolation field does not show relatively good performance because the receptive field requires a vast spatio-temporal range. To reduce the computational complexity, in most frame interpolation studies, motion is first calculated with the optical flow, then interpolated frames are generated through backward warping. However, while the backward warping process is simple to implement, the interpolated image contains mixed motion and ghosting defects. Therefore, we propose a new network that does not use the backward warping method through the proposed max-min warping. Since max-min warping generates a clear warping image in advance according to the size of the motion and the network is configured to select the warping result according to the warped layer, using the proposed method, it is possible to optimize the computational complexity while selecting a contextually appropriate image. The video interpolation method using the proposed method showed 34.847 PSNR in the Vimeo90k dataset and 0.13 PSNR improvement compared to the Quadratic Video Interpolation method, showing that it is an efficient frame interpolation self-supervised learning.

Conjugate sparse plus low rank models for efficient Bayesian interpolation of large spatial data

AbstractA key challenge in spatial data science is the analysis for massive spatially‐referenced data sets. Such analyses often proceed from Gaussian process specifications that can produce rich and robust inference, but involve dense covariance matrices that lack computationally exploitable structures. Recent developments in spatial statistics offer a variety of massively scalable approaches. Bayesian inference and hierarchical models, in particular, have gained popularity due to their richness and flexibility in accommodating spatial processes. Our current contribution is to provide computationally efficient exact algorithms for spatial interpolation of massive data sets using scalable spatial processes. We combine low‐rank Gaussian processes with efficient sparse approximations. Following recent work by Zhang et al. (2019), we model the low‐rank process using a Gaussian predictive process (GPP) and the residual process as a sparsity‐inducing nearest‐neighbor Gaussian process (NNGP). A key contribution here is to implement these models using exact conjugate Bayesian modeling to avoid expensive iterative algorithms. Through the simulation studies, we evaluate performance of the proposed approach and the robustness of our models, especially for long range prediction. We implement our approaches for remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory Unit (TIU) in a remote portion of Interior Alaska.

General Data Search Algorithms for Earth Simulation Systems with Cyclic Boundaries

Grid remapping is one of the most fundamental functions in Earth simulation systems, and is essentially a kind of data interpolation. The key to an efficient interpolation method is how to quickly find the relevant grid points required for interpolation. With the rise of unstructured grid models, the demand for general and efficient interpolation search algorithms is becoming stronger and stronger. KD (K-dimensional) tree has proven to be effective in dealing with unstructured grids. However, it is unable to tackle the cyclic boundary conditions in Earth simulation systems, which restricts the application of KD tree. Taking the nearest neighbor search as an example, this paper introduces two new KD tree-based multi-dimensional data search methods, which break through the limitations of the original method with regards to the cyclic boundary. One method is based on target points duplication, and the other method is based on source points duplication. Their time complexity and space complexity are analyzed and verified by carefully designed experiments. The results show that the method based on target points duplication generally performs better than that based on source points duplication when the data are basically evenly distributed.

Eccentric self-forced inspirals into a rotating black hole

We develop the first model for extreme mass-ratio inspirals (EMRIs) into a rotating massive black hole driven by the gravitational self-force (GSF). Our model is based on an action angle formulation of the method of osculating geodesics for eccentric, equatorial (i.e., spin-aligned) motion in Kerr spacetime. The forcing terms are provided by an efficient spectral interpolation of the first-order GSF in the outgoing radiation gauge. We apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion, while maintaining all secular effects of the first-order GSF at post-adiabatic order. This implies that the model can be evolved without having to resolve all orbit cycles of an EMRI, yielding an inspiral model that can be evaluated in less than a second for any mass-ratio. In the case of a non-rotating central black hole, we compare inspirals evolved using self-force data computed in the Lorenz and radiation gauges. We find that the two gauges generally produce differing inspirals with a deviation of comparable magnitude to the conservative self-force correction. This emphasizes the need for including the (currently unknown) dissipative second order self-force to obtain gauge independent, post-adiabatic waveforms.

Recursive analytic spherical harmonics gradient for spherical lights

AbstractWhen rendering images using Spherical Harmonics (SH), the projection of a spherical function on the SH basis remains a computational challenge both for high‐frequency functions and for emission functions from complex light sources. Recent works investigate efficient SH projection of the light field coming from polygonal and spherical lights. To further reduce the rendering time, instead of computing the SH coefficients at each vertex of a mesh or at each fragment on an image, it has been shown, for polygonal area light, that computing both the SH coefficients and their spatial gradients on a grid covering the scene allows the efficient and accurate interpolation of these coefficients at each shaded point. In this paper, we develop analytical recursive formulae to compute the spatial gradients of SH coefficients for spherical light. This requires the efficient computation of the spatial gradients of the SH basis function that we also derive. Compared to existing method for polygonal light, our method is faster, requires less memory and scales better with respect to the SH band limit. We also show how to approximate polygonal lights using spherical lights to benefit from our derivations. To demonstrate the effectiveness of our proposal, we integrate our algorithm in a shading system able to render fully dynamic scenes with several hundreds of spherical lights in real time.

Efficient Vector Interpolation Method Based on Lebedev Quadrature for MLFMA

In this letter, an efficient vector interpolation method based on Lebedev quadrature to accelerate the matrix-vector multiplications in far-field regions for the multilevel fast multipole algorithm (MLFMA) is presented. This approach avoids the singularity on sphere poles in the traditional interpolation method based on Gaussian quadrature and maintains the same accuracy without increasing computational costs. As a consequence, both memory consumption and processing time can be reduced considerably, providing a viable alternative to solve large-scale electromagnetic scattering problems using MLFMA.

Video frame interpolation via down–up scale generative adversarial networks

Frame interpolation finds many applications in video applications, including frame rate up-conversion and video compression. Deep learning-based methods have been proposed for frame interpolation, but a long runtime is typically required to achieve good visual quality. In this paper, we introduce an efficient frame interpolation method based on a modified generative adversarial network. The proposed framework consists of a generator with a pair of down–up scale modules, where the down-scaled-input module attempts to capture the overall structure of the scene while the original-scale-input module aims to restore finer textures. Skip connections and an input processing block are further incorporated into the minimal two-scale generator design to expedite processing without losing image features. The difference between the synthesized frame and the ground truth is measured by a combined loss function, including one adversarial loss and three reconstruction losses. Compared to the state-of-the-art motion compensation and deep-learning based frame interpolation approaches, the proposed framework achieves the most satisfactory trade-off between the synthesis quality and runtime.

Five-Axis Trajectory Generation Considering Synchronization and Nonlinear Interpolation Errors

Abstract This paper presents a novel real-time interpolation technique for five-axis machine tools to attain higher speed and accuracy. To realize computationally efficient real-time interpolation of 6-DOF tool motion, a joint workpiece–machine coordinate system interpolation scheme is proposed. Cartesian motion of the tool center point (TCP) is interpolated in the workpiece coordinate system (WCS), whereas tool orientation is interpolated in the machine coordinate system (MCS) based on the finite impulse response filtering. Such an approach provides several advantages: (i) it eliminates the need for complex real-time spherical interpolation techniques, (ii) facilitates efficient use of slower rotary drive kinematics to compensate for the dynamic mismatch between Cartesian and rotary axes and achieve higher tool acceleration, and (iii) mitigates feed fluctuations while interpolating near kinematic singularities. To take advantage of such benefits and realize accurate joint WCS–MCS interpolation scheme, tool orientation interpolation errors are analyzed. A novel approach is developed to adaptively discretize long linear tool moves and confine interpolation errors within user set tolerances. Synchronization errors between TCP and tool orientation are also characterized, and peak synchronization error level is determined to guide the interpolation parameter selection. Finally, blending errors during non-stop continuous interpolation of linear toolpaths are modeled and confined. Advantages of the proposed interpolation scheme are demonstrated through simulation studies and validated experimentally. Overall, proposed technique can improve cycle times up to 10% while providing smooth and accurate non-stop real-time interpolation of tool motion in five-axis machining.

First-principles ionized-impurity scattering and charge transport in doped materials

Scattering of carriers with ionized impurities governs charge transport in doped semiconductors. However, electron interactions with ionized impurities cannot be fully described with quantitative first-principles calculations, so their understanding relies primarily on simplified models. Here we show an ab initio approach to compute the interactions between electrons and ionized impurities or other charged defects. It includes the short- and long-range electron-defect ($e$-d) interactions on equal footing and allows for efficient interpolation of the $e$-d matrix elements. We combine the $e$-d and electron-phonon interactions in the Boltzmann transport equation to compute the carrier mobilities in doped silicon over a wide range of temperature and doping concentrations, seamlessly spanning the defect- and phonon-limited transport regimes. The individual contributions of the defect- and phonon-scattering mechanisms to the carrier relaxation times and mean-free paths are analyzed. Our method provides a powerful tool to study electronic interactions in doped materials. It broadens the scope of first-principles transport calculations, enabling studies of a wide range of doped semiconductors and oxides with application to electronics, energy, and quantum technologies.

Using ECG signal as an entropy source for efficient generation of long random bit sequences

Electrocardiogram (ECG) signal produced by the human heart has been investigated as a potential entropy source for cryptographic random bit generation for a long time. The throughput of existing methods remains as the bottleneck for its deployment in practical applications. To overcome this problem, we develop a Bernoulli entropy source by processing a single heartbeat ECG signal to obtain a long random bit sequence (RBS). The proposed method converts the signal into an IID (independent and identically distributed) source of entropy using efficient interpolation and optimization techniques. Several heartbeat signals, obtained from two different databases, were used to test the entropy source generating RBSs with one million bits. The entropy source was evaluated with the latest NIST recommendation for IID source validation and it passed all recommended tests and the average min-entropy obtained from several heartbeat signals of different individuals was close to the perfect entropy value of 1.0. It was also observed that the entropy increases monotonically with the increase of the length of keys. The proposed method can efficiently produce a long RBS for cryptographic applications, such as key generation for one-time pad and image encryption. The method could be further explored to generate a true random number with a personalized signature, which is crucial for information security in the future generation of computing.

Computationally efficient barycentric interpolation of large grain boundary octonion point sets

We present a method for performing efficient barycentric interpolation for large grain boundary octonion point sets which reside on the surface of a hypersphere. This method includes removal of degenerate dimensions via singular value decomposition (SVD) transformations and linear projections, determination of intersecting facets via nearest neighbor (NN) searches, and interpolation. This method is useful for hyperspherical point sets for applications such as grain boundaries structure-property models, robotics, and specialized neural networks. We provide a case study of the method applied to the 7-sphere. We provide 1-sphere and 2-sphere visualizations to illustrate important aspects of these dimension reduction and interpolation methods. A MATLAB implementation is available at github.com/sgbaird-5dof/interp.•Barycentric interpolation is combined with hypersphere facet intersections, dimensionality reduction, and linear projections to reduce computational complexity without loss of information•A max nearest neighbor threshold is used in conjunction with facet intersection determination to reduce computational runtime.

Glasdi: Parametric Physics-Informed Greedy Latent Space Dynamics Identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on-the-fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, an efficient k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 66 to 4,417x speed-up with 1 to 5% relative errors.

High-efficient analysis of metal target electromagnetics above the half-space based on mixed field integral equation

A new acceleration method is proposed for efficiently solving the problem of electromagnetic scattering from metal targets in half-space. The analysis of electromagnetic problems in any structure can be settled by the electric field integral equation. But the generated matrix condition number is large and the iterative solution has poor convergence. The number of the matrix condition generated by the magnetic field integral equation is small and iterative convergence is good. But only the closed structure problems can be worked out. The combined field integral equation is adopted because of the universality of the electric field integral equation and the convergence of the magnetic field integral equation. The gradient term of Green's function is involved in the integral equation of the mixed field. In order to further enhance the calculation efficiency, an efficient four-dimensional spatial interpolation method is introduced for half-space Green's function. Tabulation and lagrange interpolations are performed in the Sommerfeld integrals for the half-space Green's function. The improved efficiency can be 7.5 times higher than that of the traditional combined field integral equation(CFIE). Numerical results show that the computational time can be reduced significantly by the proposed method with encouraging accuracy.

An interpolation method based on adaptive smooth feedrate scheduling and parameter increment compensation for NURBS curve

Compared with the traditional line/circle interpolation, non-uniform rational B-splines (NURBS) interpolation has more efficient and high-quality advantages in free-form curve/surface modeling. In order to obtain high efficiency and stable interpolation feed motion, a NURBS interpolation method with adaptive smoothing federate scheduling and parameter increment compensation is studied. Firstly, according to the curvature constraint, the adaptive feedrate is obtained and the segmentation of the NURBS curve is completed. Then, in order to avoid the problem of feeding speed overrun, the improved S-shape acceleration/deceleration control is adopted to adjust the jerk. The adjacent segments meeting the combined planning conditions are treated comprehensively. Secondly, aiming at the problem of velocity error in interpolation point parameter calculation, an interpolation parameter calculation method based on parameter increment compensation is proposed to reduce the velocity fluctuation and improve the interpolation point parameter calculation accuracy. Finally, the effectiveness of the proposed method is verified by simulation and experiment. The results show that the algorithm is effective.

Displacement voxelization to resolve mesh-image mismatch: Application in deriving dense white matter fiber strains

Background and objectiveIt is common to combine biomechanical modeling and medical images for multimodal analyses. However, mesh-image mismatch may occur that prevents direct information exchange. To eliminate mesh-image mismatch, we develop a simple but elegant displacement voxelization technique based on image voxel corner nodes to achieve voxel-wise strain. We then apply the technique to derive dense white matter fiber strains along whole-brain tractography (∼35 k fiber tracts consisting of ∼3.3 million sampling points) resulting from head impact. MethodsDisplacements at image voxel corner nodes are first obtained from model simulation via scattered interpolation. Each voxel is then scaled linearly to form a unit hexahedral element. This allows convenient and efficient voxel-wise strain tensor calculation and displacement interpolation at arbitrary fiber sampling points via shape functions. Fiber strains from displacement interpolation are then compared with those from the commonly used strain tensor projection using either voxel- or element-wise strain tensors. ResultsBased on a synthetic displacement field, fiber strains interpolated from voxelized displacement are considerably more accurate than those from strain tensor projection relative to the prescribed ground-truth (determinant of coefficient (R2) of 1.00 and root mean squared error (RMSE) of 0.01 vs. 0.87 and 0.10, respectively). For a set of real-world reconstructed head impacts (N = 53), the strain tensor projection method performs similarly poorly (R2 of 0.80–0.90 and RMSE of 0.03–0.07), with overestimation strongly correlated with strain magnitude (Pearson correlation coefficient >0.9). Up to ∼15% of the fiber strains are overestimated by more than the lower bound of a conservative injury threshold of 0.09. The percentage increases to ∼37% when halving the threshold. Voxel interpolation is also significantly more efficient (15 s vs. 40 s for element strain tensor projection, without parallelization). ConclusionsVoxelized displacement interpolation is considerably more accurate and efficient in deriving dense white matter fiber strains than strain tensor projection. The latter generally overestimates with overestimation magnitude strongly correlating with fiber strain magnitude. Displacement voxelization is an effective technique to eliminate mesh-image mismatch and generates a convenient image representation of tissue deformation. This technique can be generalized to broadly facilitate a diverse range of image-related biomechanical problems for multimodal analyses. The convenient image format may also promote and facilitate biomechanical data sharing in the future.

High-Performance Computing Probabilistic Fracture Mechanics Implementation for Gas Turbine Rotor Disks on Distributed Architectures Including Graphics Processing Units

Abstract We present an efficient Monte Carlo-based probabilistic fracture mechanics simulation implementation for heterogeneous high-performance computing (HPC) architectures including central processing units (CPUs) and graphics processing units (GPUs). The specific application focuses on large heavy-duty gas turbine rotor components for the energy sector. A reliable probabilistic risk quantification requires the simulation of millions to billions of Monte Carlo (MC) samples. We apply a modified Runge–Kutta algorithm in order to solve numerically the fatigue crack growth for this large number of cracks for varying initial crack sizes, locations, material, and service conditions. This compute intensive simulation has already been demonstrated to perform efficiently and scalable on parallel and distributed HPC architectures including hundreds of CPUs utilizing the message passing interface (MPI) paradigm. In this work, we go a step further and include GPUs in our parallelization strategy. We develop a load distribution scheme to share one or more GPUs on compute nodes distributed over a network. We detail technical challenges and solution strategies in performing the simulations on GPUs efficiently. We show that the key computation of the modified Runge–Kutta integration step speeds up over two orders of magnitude on a typical GPU compared to a single threaded CPU. This is supported by our use of GPU textures for efficient interpolation of multidimensional tables utilized in the implementation. We demonstrate weak and strong scaling of our GPU implementation, i.e., that we can efficiently utilize a large number of GPUs/CPUs in order to solve for more MC samples, or reduce the computational turn-around time, respectively. On seven different GPUs spanning four generations, the presented probabilistic fracture mechanics simulation tool ProbFM achieves a speed-up ranging from 16.4× to 47.4× compared to single threaded CPU implementation.

Data-Driven Interpolation of Sea Surface Suspended Concentrations Derived from Ocean Colour Remote Sensing Data

Due to complex natural and anthropogenic interconnected forcings, the dynamics of suspended sediments within the ocean water column remains difficult to understand and monitor. Numerical models still lack capabilities to account for the variabilities depicted by in situ and satellite-derived datasets. Besides, the irregular space-time sampling associated with satellite sensors make crucial the development of efficient interpolation methods. Optimal Interpolation (OI) remains the state-of-the-art approach for most operational products. Due to the large increase of both in situ and satellite measurements more and more available information is coming from in situ and satellite measurements, as well as from simulation models. The emergence of data-driven schemes as possibly relevant alternatives with increased capabilities to recover finer-scale processes. In this study, we investigate and benchmark three state-of-the-art data-driven schemes, namely an EOF-based technique, an analog data assimilation scheme, and a neural network approach, with an OI scheme. We rely on an Observing System Simulation Experiment based on high-resolution numerical simulations and simulated satellite observations using real satellite sampling patterns. The neural network approach, which relies on variational data assimilation formulation for the interpolation problem, clearly outperforms both the OI and the other data-driven schemes, both in terms of reconstruction performance and of a greater ability to recover high-frequency events. We further discuss how these results could transfer to real data, as well as to other problems beyond interpolation issues, especially short-term forecasting problems from partial satellite observations.