Spatial Basis Functions Research Articles

Time/Space-Separation-Based Gaussian Process Modeling for the Cross-Coupling Effect of a 2-DOF Nanopositioning Stage

Multiple-axis nanopositioning stages are widely used in nanotechnology, which suffer not only from hysteresis nonlinearity but also from cross-coupling effect. A vast body of literature has been proposed to model the hysteresis nonlinearity. However, few efforts have been made to describe the cross-coupling effect due to its complexity, and most of the presented works considered only the effect of the motion of the other axis from the aspect of lumped system. Actually, the cross-coupling effect has a spatiotemporal behavior, which means that it depends not only on the motion of the other axis but also on the contacting position of the axes. In this article, a time/space-separation-based Gaussian process modeling method is developed to describe the spatiotemporal cross-coupling effect. The spatiotemporal output of the system is first separated into a few spatial basis functions and the corresponding temporal coefficients by using the principle component analysis. Then, the frequency-separation-based Gaussian process is introduced to model the system dynamics in a low-dimensional temporal domain. Finally, by using the time/space synthesis, the spatiotemporal cross-coupling model can be constructed. Experiments are presented to demonstrate the effectiveness of this modeling method.

Motion limiting nonlinear dynamics of initially curved beams

An initially curved beam is considered and its motion is constrained using two elastic constraints; the corresponding non-smooth nonlinear transverse dynamics is investigated for the first time. A clamped-clamped beam with one axially movable end is modelled via Bernoulli-Euler beam theory together with the inextensibility condition, giving rise to nonlinear inertial terms along with nonlinear geometric terms. Furthermore, the damping is modelled via Kelvin-Voigt internal damping model. The proposed model is verified for linear and nonlinear behaviours via comparison to a finite element model. The impact between beam and constraints is incorporated via calculating its work contribution. The nonlinear equation of motion is derived while incorporating geometric, damping, inertial, and constraints nonlinearities. A series of spatial basis functions together with corresponding vibration modes are used as the proposed solution of the transverse displacement. A modal discretisation is performed via the weighted-residual method of Galerkin and the corresponding non-smooth terms are kept intact while conducting numerical integration. A numerical continuation technique is utilised to solve the resultant equations. The non-smooth response is obtained for various cases and the effects of several parameters are studied thoroughly.

Algorithmically deconstructing shot locations as a method for shot quality in hockey

Abstract Spatial point processes have been successfully used to model the relative efficiency of shot locations for each player in professional basketball games. Those analyses were possible because each player makes enough baskets to reliably fit a point process model. Goals in hockey are rare enough that a point process cannot be fit to each player’s goal locations, so novel techniques are needed to obtain measures of shot efficiency for each player. A Log-Gaussian Cox Process (LGCP) is used to model all shot locations, including goals, of each NHL player who took at least 500 shots during the 2011–2018 seasons. Each player’s LGCP surface is treated as an image and these images are then used in an unsupervised statistical learning algorithm that decomposes the pictures into a linear combination of spatial basis functions. The coefficients of these basis functions are shown to be a very useful tool to compare players. To incorporate goals, the locations of all shots that resulted in a goal are treated as a “perfect player” and used in the same algorithm (goals are further split into perfect forwards, perfect centres and perfect defence). These perfect players are compared to other players as a measure of shot efficiency. This analysis provides a map of common shooting locations, identifies regions with the most goals relative to the number of shots and demonstrates how each player’s shot location differs from scoring locations.

Data-driven surrogate modeling of multiphase flows using machine learning techniques

This study focuses on the development of a theoretical framework and corresponding algorithms to establish spatio-temporal surrogate models for multiphase flow processes using Gaussian process (GP) based machine learning technique trained by direct numerical simulation (DNS) data. The training (and testing) datasets are obtained by solving the incompressible form of the Navier Stokes equations with surface tension in an Eulerian reference frame. The liquid-gas interfacial evolution is resolved using a volume-of-fluid (VOF) interface capturing method. The overall framework proceeds in four steps: 1) design of experiments study to identify the training and testing points and generation of corresponding datasets using DNS calculations; 2) dimensionality reduction using proper orthogonal decomposition; 3) Gaussian process regression (supervised training) over the reduced training dataset over the entire range of operating conditions under consideration; and 4) Galerkin reconstruction and error quantification by comparing the emulated flowfields (at test conditions) with the testing dataset. The machine learning framework predicts both the spatial basis-functions and the time-coefficients, thus, predicting the entire flowfield in time and space. The capabilities of the algorithm are demonstrated for two canonical flow configurations: 1) flow over a circular cylinder for a range of Reynolds numbers from 10 to 200; and 2) diesel jet injected into a quiescent nitrogen environment at chamber pressure of 30 atm and room temperature conditions, and injection velocities from 10 to 55 m/s, corresponding to a range of gas-based Weber numbers from 11.5 to 348. The emulations from the learned GP algorithm show excellent agreement with high-fidelity numerical data for test conditions; average error (in both space and time) at the testing point of Re = 185 for the flow over cylinder case is 4.4%, and for the diesel jet injection configuration at a testing point corresponding to velovity of 22.5 m/s is found to be 15.5%. The tip penetration location of the diesel jet is predicted within 2.5% of the DNS calculations. Corresponding to these two representative test points, speedup of 256 and 8000 is achieved for flow over cylinder and diesel jet atomization configurations, respectively. This paper represents the first effort of its kind on the development of a general machine learning framework to predict multiphase flows.

Dual Extreme Learning Machines-Based Spatiotemporal Modeling for Nonlinear Distributed Thermal Processes

Many industrial thermal processes belong to distributed parameter systems (DPSs), which have two coupled nonlinear blocks. Dual least square support vector machines (LS-SVM) has been proposed to model such systems. However, due to the use of two LS-SVM, this method often leads to heavy computation and long learning time, which does not suit for online application. In this study, a dual extreme learning machine (ELM)-based spatiotemporal modeling method is proposed for such two nonlinearities embedded DPSs. Firstly, the KL method is applied to reduce the dimension of the original system and obtain the spatial basis functions (BFs). Then, dual ELM is designed to match the two nonlinear structures. Finally, through the reconstruction of space–time synthesis, the approximate spatiotemporal distribution model of the original system is obtained. In addition, simulations on a curing process is studied to confirm the effectiveness of the proposed method.

Proper Orthogonal Decomposition of High-Speed Particle Image Velocimetry in an Open Cavity

Two-dimensional velocity data of flow over an open cavity acquired by pulse-burst particle image velocimetry at 37.5 kHz for freestream Mach numbers of 0.6, 0.8, and 0.94 have been analyzed using proper orthogonal decomposition (POD). The snapshot application of the POD resulted in modes consistent with previous studies on cavity flows under similar conditions. The time-resolved nature of velocity datasets also allowed for the application of the classical POD in both space and time which, for a stationary flow, is solved in the frequency domain. Eigenvalue spectra and convergence plots revealed that these POD modes are organized by capturing the maximum energy associated with dominant frequencies. The extracted eigenvectors revealed a strong dependency on frequency, with the most dominant modes corresponding to frequencies associated with Rossiter modes known to dominate open cavities. Mach number comparisons from the classical POD application reveal that their modes efficiently capture energy of dominant frequencies associated with the flow dynamics, regardless of the freestream conditions. Using a similarity parameter between spatial unitary basis functions, it was shown that the modes at different Mach numbers were quantitatively similar at the same Strouhal and POD mode numbers.

Dimension Embedded Basis Function for Spatiotemporal Modeling of Distributed Parameter System

The construction of spatial basis functions (BFs) is critical to the time/space separation of the distributed parameter system (DPS). The spatial BFs constructed by traditional Karhunen–Loeve may not work satisfactorily for two spatial-dimensional (2-D) DPS, because of a distorted mapping of the original sensor array in the row-wise vectorization process. In this article, a novel time/space separation based method is proposed to construct dimension embedded BFs (DE-BFs) for modeling 2-D DPS. The DE-BFs are first formulated according to the spatial sensor array structure, and sequentially optimized with alternating least squares by minimizing the reconstruction error. The mapping relationship between the DE-BFs and the spatial sensor array is well preserved. In addition, the coupling across temporal and different spatial dimensions is sufficiently captured. A satisfactory model accuracy can be achieved by the DE-BFs, even with limited training data. Experiments of a 2-D curing thermal process are used to verify the effectiveness of the proposed method.

Incremental Spatiotemporal Learning for Online Modeling of Distributed Parameter Systems

An incremental spatiotemporal learning scheme is proposed for online modeling of distributed parameter systems (DPSs). A novel incremental learning method is developed to recursively update the spatial basis functions and the corresponding temporal model based on the Karhunen–Loeve decomposition for time-space separation. The time-space synthesis continually evolves by adding new increment data with more updated information and revising the existing parameters of the dynamic system. In this way, the spatiotemporal structure is inherited and updated efficiently as output data increases over time. The adaptive nature of this evolving structure makes it promising for online modeling of DPSs under streaming data environment. The proposed incremental modeling scheme is evaluated on the classical benchmark of a catalytic rod problem. The simulation results demonstrate the viability and efficiency of the proposed method for online modeling of DPSs.

A spatial multivariable SVR method for spatiotemporal fuzzy modeling with applications to rapid thermal processing

Many industrial processes have significant spatiotemporal dynamics and they are usually called distributed parameter systems (DPSs). Modeling such system is challenging due to its nonlinearity, time-varying dynamics, and spatiotemporal coupling. Using model reduction techniques, traditional DPS modeling methods usually reduce an infinite-dimensional system to a finite-dimensional system, which leads to unknown nonlinearity and unmodeled dynamics. The modeling method and the established model are hard to understand. Here, we propose a spatial multivariable support vector regression (SVR) based three-domain (3-D) fuzzy modeling method for complex nonlinear DPSs. The proposed 3-D modeling method integrates the time-space separation and time-space synthesis into a 3-D fuzzy model. Therefore, it does not require model reduction and owns the capability of linguistic interpretability. A spatial multivariable SVR with spatial kernel functions is proposed to deal with spatiotemporal data. The spatial fuzzy basis functions from a 3-D fuzzy model are spatial kernel functions for a spatial multivariable SVR, which satisfy Mercy theorem. Hence, the spatial multivariable SVR can be directly employed to build up a complete 3-D fuzzy rule-base of the 3-D fuzzy model. The proposed modeling method integrates the merits of learning ability from a spatial multivariable SVR and fuzzy space processing and fuzzy linguistic expression from a 3-D fuzzy model. The proposed 3-D fuzzy modeling method is successful applied to a simulated rapid thermal processing system. In comparison with several newly developed modeling methods for DPSs, the simulation results validate the superiority of the proposed modeling method.

Explicit Time Marching Schemes for Solving the Magnetic Field Volume Integral Equation

A method for constructing explicit marching-on-in-time (MOT) schemes to solve the time-domain magnetic field volume integral equation (TD-MFVIE) on inhomogeneous dielectric scatterers is proposed. The TD-MFVIE is cast in the form of an ordinary differential equation (ODE) and the unknown magnetic field is expanded using curl conforming spatial basis functions. Inserting this expansion into the TD-MFVIE and spatially testing the resulting equation yield an ODE system with a Gram matrix. This system is integrated in time for the unknown time-dependent expansion coefficients using a linear multistep method. The Gram matrix is sparse and well conditioned for Galerkin testing and consists of only four diagonal blocks for point testing. The resulting explicit MOT schemes, which call for the solution of this matrix system at every time step, are more efficient than their implicit counterparts, which call for inversion of a fuller matrix system at lower frequencies. Numerical results compare the efficiency, accuracy, and stability of the explicit MOT schemes and their implicit counterparts for low-frequency excitations. The results show that the explicit MOT scheme with point testing is significantly faster than the other three solvers without sacrificing from accuracy.

Physically constrained spatiotemporal modeling: generating clear-sky constructions of land surface temperature from sparse, remotely sensed satellite data

Satellite remote-sensing is used to collect important atmospheric and geophysical data at various spatial resolutions, providing insight into spatiotemporal surface and climate variability globally. These observations are often plagued with missing spatial and temporal information of Earth's surface due to (1) cloud cover at the time of a satellite passing and (2) infrequent passing of polar-orbiting satellites. While many methods are available to model missing data in space and time, in the case of land surface temperature (LST) from thermal infrared remote sensing, these approaches generally ignore the temporal pattern called the ‘diurnal cycle’ which physically constrains temperatures to peak in the early afternoon and reach a minimum at sunrise. In order to infill an LST dataset, we parameterize the diurnal cycle into a functional form with unknown spatiotemporal parameters. Using multiresolution spatial basis functions, we estimate these parameters from sparse satellite observations to reconstruct an LST field with continuous spatial and temporal distributions. These estimations may then be used to better inform scientists of spatiotemporal thermal patterns over relatively complex domains. The methodology is demonstrated using data collected by MODIS on NASA's Aqua and Terra satellites over both Houston, TX and Phoenix, AZ USA.

Spatial Correlation-Based Incremental Learning for Spatiotemporal Modeling of Battery Thermal Process

The thermal effect has a significant impact on the performance of a lithium-ion battery. Thus, modeling the thermal process, which always involves unknown boundary heat exchange, is significant to battery management. Two critical issues should be addressed for the thermal process modeling: 1) the nominal model, which is constructed offline, can be updated efficiently to compensate for any online disturbances; and 2) the influence of previous and recent spatiotemporal dynamics may be varying and should be handled properly. Bearing these in mind, in this paper, a spatial correlation-based incremental learning technique is designed for spatiotemporal modeling. First, the incremental learning technique is developed to update the dominant spatial basis functions of the nominal model, which is constructed by a time/space separation-based method. Then, a forgetting factor is incorporated into the incremental learning technique to handle time-varying dynamics. Additionally, the popular approximator, that is, the radial basis function neural network, is utilized to identify the low-dimensional temporal model. Simulations and experiments on a pouch type battery with boundary heat exchange have demonstrated the accuracy and efficiency of the proposed modeling method.

An Improved MOD Discontinuous Galerkin Method for Solving Time-Domain Electric Field Integral Equation

An improved marching-on-in-degree discontinuous Galerkin method for solving time-domain electric field integral equation is presented in this letter. By adopting half-RWGs as spatial basis functions, both conformal and nonconformal discretization scattering problems can be simulated accurately. A combination of three weighted Laguerre polynomials is utilized as new temporal basis functions to derive final equations, where some accumulation terms are eliminated. Therefore, the time spent on computing accumulation term during the iterative procedure is reduced significantly, with the total solution process accelerated. Some numerical examples are presented to demonstrate both accuracy and efficiency of our method.

Evolutionary Design of Spatio–Temporal Learning Model for Thermal Distribution in Lithium-Ion Batteries

Temperature monitoring is indispensable to the optimal and safe operation of a lithium-ion battery. In this paper, a spatio–temporal learning model designed by evolutionary algorithm is proposed to predict the thermal distribution. To formulate the multicharacteristic spatial dynamics, the chicken swarm optimization, based fusion of different dimensionality-reduction methods, is proposed for learning spatial basis functions. Through integration with the time/space separation based approach and equivalent circuit model based thermal model, the reduced-order model is derived. The related parameters of the reduced-order model are identified by integrating chicken swarm optimization with time/space separation based approach. A Bayesian-regularized neural-network based compensation model is developed to compensate for the model errors caused by the spatio–temporal coupled dynamics. Based on the Rademacher complexity, the generalization bound of the proposed model is analyzed. Simulations and comparisons demonstrate the superiority of the proposed model.

Symplectic interior penalty discontinuous Galerkin method for solving the seismic scalar wave equation

To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.

Kernel-Based Random Vector Functional-Link Network for Fast Learning of Spatiotemporal Dynamic Processes

Distributed parameter systems widely exist in many industrial thermal processes. Estimation of their temperature distribution in the entire operating area is not easy as the dynamics are time/space coupled, and there are only a few sensors available for measurement. In this paper, an effective spatiotemporal model is proposed for prediction of the temperature distribution. After the dominant spatial basis functions are obtained by the Karhunen–Loeve method under the time/space separation, a kernel-based random vector functional-link network is developed for learning unknown temporal dynamics. After time/space synthesis, the spatiotemporal model can be constructed to effectively estimate the temperature distribution in high learning speed. The generalization performance of this model is discussed using Rademacher complexity. Simulations on two typical industrial thermal processes show that the proposed method has superior model performance than neural networks and least square support vector machine.

Proper orthogonal decomposition based online power-distribution reconstruction method

A new online power-distribution reconstruction method based on the proper orthogonal decomposition (POD) is developed. The spatial basis functions, also called POD basis are extracted from the power-distribution samples, which are used to expand the power distribution. The corresponding reconstructed coefficients are calculated based on the detector measurements. Then an online power-distribution reconstruction code named CORN is developed based on the POD method. BEAVRS benchmark is used to verify the reconstruction capability of the POD method. Reference power distributions, virtual detector measurements and power-distribution samples for POD are obtained through core simulations based on NECP-Bamboo code. Numerical results show the POD method is effective for the online power-distribution reconstruction and has excellent results. Compared with the traditional online-reconstruction method, the POD method has obvious advantage that the similarity of core states between the reconstruction calculation and the diffusion-equation solution is not required. The real-time follow-up simulation is not necessary. The excessively dependence between the reconstruction accuracy and the diffusion-calculation accuracy is avoided.

Exploration and Inference in Spatial Extremes Using Empirical Basis Functions

Statistical methods for inference on spatial extremes of large datasets are yet to be developed. Motivated by standard dimension reduction techniques used in spatial statistics, we propose an approach based on empirical basis functions to explore and model spatial extremal dependence. Based on a low-rank max-stable model, we propose a data-driven approach to estimate meaningful basis functions using empirical pairwise extremal coefficients. These spatial empirical basis functions can be used to visualize the main trends in extremal dependence. In addition to exploratory analysis, we describe how these functions can be used in a Bayesian hierarchical model to model spatial extremes of large datasets. We illustrate our methods on extreme precipitations in eastern USA. Supplementary materials accompanying this paper appear online

Multi-Resolution Graph Based Volumetric Cortical Basis Functions From Local Anatomic Features.

Modern clinical MRI collects millimeter scale anatomic information, but scalp electroencephalography source localization is ill posed, and cannot resolve individual sources at that resolution. Dimensionality reduction in the space of cortical sources is needed to improve computational and storage complexity, yet volumetric methods still employ simplistic grid coarsening that eliminates fine scale anatomic structure. We present an approach to extend near-arbitrary spatial scaling to volumetric localization. Starting from a voxelwise brain parcellation, sub-parcels are identified from local cortical connectivity with an iterated graph cut approach. Spatial basis functions in each parcel are constructed using either a decomposition of the local leadfield matrix or spectral basis functions of local cortical connectivity graphs. We present quantitative evaluation with extensive simulations and use multiple sets of real data to highlight how parameter changes impact computed reconstructions. Our results show that volumetric basis functions can improve accuracy by as much as 30%, while reducing computational complexity by over two orders of magnitude. In real data from epilepsy surgical candidates, accurate localization of seizure onset regions is demonstrated. Spatial dimensionality reduction with volumetric basis functions improves reconstruction accuracy while reducing computational complexity. Near-arbitrary spatial dimensionality reduction will enable volumetric reconstruction with modern computationally intensive algorithms and anatomically driven multi-resolution methods.

Accurate Simulation of Shielding Effectiveness of Metallic Cabins Using an Improved Calderón Preconditioner-Based Time-Domain Integral Equation Method

An efficient simulation of shielding effectiveness of some metallic cabins or enclosures is performed using an improved Calderon preconditioner-based time-domain integral equation method. During its implementation, Rao–Wilton–Glisson and Buffa–Christiansen functions are used as the spatial basis functions for solving the Calderon preconditioned time-domain electric field integral equation, which generate a set of well-conditioned system matrices irrespective of the mesh density. Moreover, a set of approximate prolate spheroidal wave functions is chosen as the temporal basis functions; therefore, both late-time stability and accuracy are improved significantly for simulating electromagnetic pulse responses in some typical cabins. Numerical results are validated in comparison with those obtained by some commercial software. These results are useful for further evaluating electromagnetic environment effects in some complex systems.