Basis Functions For Representation Research Articles

Adaptive passive admittance control of a compliant collaborative robot

Compliance of the collaborative robots plays an important role in human-robot interaction. Due to the complex nonlinearities caused by the compliance, and parameter uncertainties, good performance control method for the compliant collaborative robot is always a challenging task. Therefore, to overcome the challenge of the compliance, and uncertain dynamics, this paper develops an adaptive passive admittance control (APAC) approach for a novel compliant collaborative robot, whose compliant mechanism is constituted of the inerters, springs, and dampers. First, through force-current analogy between mechanical network and electrical network, the compliance of the proposed collaborative robot with the inerters, springs, and dampers, is represented by a novel and simplified admittance matrix to facilitate dynamics model of the proposed collaborative robot. Then, for the proposed compliant collaborative robot, an APAC strategy is presented to dispose of the uncertain parameters of the dynamics. The inertial matrix, Coriolis matrix, admittance matrix, and gravity vector of the compliant collaborative robot are expressed as linear representation of orthogonal basis functions by using function approximation technique (FAT). The boundedness of the robot dynamics is proved by detailed stability analysis which uses Lyapunov function. Finally, simulation results indicate that the proposed APAC of the compliant collaborative robot is effective.

3DShape2VecSet: A 3D Shape Representation for Neural Fields and Generative Diffusion Models

We introduce 3DShape2VecSet, a novel shape representation for neural fields designed for generative diffusion models. Our shape representation can encode 3D shapes given as surface models or point clouds, and represents them as neural fields. The concept of neural fields has previously been combined with a global latent vector, a regular grid of latent vectors, or an irregular grid of latent vectors. Our new representation encodes neural fields on top of a set of vectors. We draw from multiple concepts, such as the radial basis function representation, and the cross attention and self-attention function, to design a learnable representation that is especially suitable for processing with transformers. Our results show improved performance in 3D shape encoding and 3D shape generative modeling tasks. We demonstrate a wide variety of generative applications: unconditioned generation, category-conditioned generation, text-conditioned generation, point-cloud completion, and image-conditioned generation. Code: https://1zb.github.io/3DShape2VecSet/.

Multivariant spatial and geometric interpolation of field maps through model order reduction

Complex three-dimensional field maps can be efficiently denoised and compressed in data volume by high order single value decomposition. Extension of the method to inter/extrapolation and determination of basis function representation is reported here. Inter/extrapolation of denoised and compressed maps is applied to spatial coordinates and especially to the parameters defining the geometry of devices creating such maps. It significantly increases the efficiency of the provision of compact and noise-free maps combined with high resolution. The method is illustrated by creating corresponding maps for rf cells of radio-frequency quadrupoles.

Light-front wavefunctions of mesons by design

We develop a mechanism to build the light-front wavefunctions (LFWFs) of meson bound states on a small-sized basis function representation. Unlike in a standard Hamiltonian formalism, the Hamiltonian in this method is implicit, and the information of the system is carried directly by the functional form and adjustable parameters of the LFWFs. In this work, we model the LFWFs for four charmonium states, eta _c, J/psi , psi ', and psi (3770) as superpositions of orthonormal basis functions. We choose the basis functions as eigenfunctions of an effective Hamiltonian, which has a longitudinal confining potential in addition to the transverse confining potential from light-front holographic QCD. We determine the basis function parameters and superposition coefficients by employing both guidance from the nonrelativistic description of the meson states and the experimental measurements of the meson decay widths. With the obtained wavefunctions, we study the features of those meson states, including charge radii and parton distribution functions. We use the J/psi LFWF to calculate the meson production in diffractive deep inelastic scattering and ultra-peripheral heavy-ion collisions, and the eta _c LFWF to calculate its diphoton transition form factor. Both results show good agreement with experiments. The obtained LFWFs have simple-functional forms and can be readily used to predict additional experimental observables.

Applying a Set of Orthogonal Basis Functions in Numerical Solution of Hallen's Integral Equation for Dipole Antenna of Perfectly Conducting Material

The focus of this paper is on solving Hallen's integral equation for a diploe antenna of perfectly conducting material. A special representation of orthogonal triangular basis functions is used to implement an effective numerical method for solving this equation. The Hallen's formulation is treated in detail and illustrative computations are given for current distributions and radiation patterns.

An Embedded Tactical Decision Aid Framework for Environmentally Adaptive Autonomous Underwater Vehicle Communication and Navigation

Motivated by a changing acoustic environment in the Arctic Beaufort Sea, in this article, we present a tactical decision aid framework for a human decision-maker collaborating with an autonomous underwater vehicle (AUV) to integrate the vertical sound-speed profile for underwater navigation and communication. In a predeployment phase, using modeled and real oceanographic data, we load basis function representations of the sound-speed perturbations onto one or more AUVs on deck, where a handful of weights can estimate a sound-speed profile. During deployment, these weights are updated on an AUV through a digital acoustic message to improve navigation and reciprocal communication throughout the duration of an under-ice mission. Field work applying this framework in the Beaufort Sea is presented, highlighting key decisions regarding predeployment oceanographic data assimilation, CTD cast collection, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> weight choice. Selected examples evaluate the framework’s ability to adapt to a depth-limited CTD cast and the appearance of an anomalous microlens feature in the profile. We show that the framework effectively balances the need to adapt in a changing acoustic environment in real time while maintaining operator trust in an AUV’s embedded intelligence.

Critical functions and inf-sup stability of Crouzeix-Raviart elements

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p≥5, p odd, are inf-sup stable for the Stokes problem on triangulations. For p≥4, p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula, the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability.The generalization of this theory to cover odd polynomial orders p≥5 is involved; one reason is that the macroelement classes, which have been used for even p, are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.

Exp: N-body integration using basis function expansions

ABSTRACT We present the N-body simulation techniques implemented in the exp code. exp uses empirically chosen basis functions to expand the potential field of an ensemble of particles. Unlike other basis function expansions, the derived basis functions are adapted to an input mass distribution, enabling accurate expansion of highly non-spherical objects, such as Galactic discs. We measure the force accuracy in three models, one based on a spherical or aspherical halo, one based on an exponential disc, and one based on a bar-based disc model. We find that exp is as accurate as a direct-summation or tree-based calculation, and in some ways is better, while being considerably less computationally intensive. We discuss optimizing the computation of the basis function representation. We also detail numerical improvements for performing orbit integrations, including time-steps.

Two-sample Tests for Functional Data Using Characteristic Functions

In this paper, we consider the two-sample problem for univariate and multivariate functional data. To solve this problem, we use tool of characteristic function and a basis function representation of functional data. We construct test statistics for conformity of distributions based on a weighted distance between characteristic functions of random vectors obtained in basis representation. Different weight functions result in different test statistics, whose distributions are approximated by permutation method. Testing procedures are implemented in the R program and the code is available. Simulation study shows good finite sample properties of proposed methods, while real data example illustrates the application of them.

System Identification of Lightly Damped Systems Using Recursive Kautz Functions-Case Study With Coriolis Mass Flow Meter

This manuscript deals with the system identification of lightly (under) damped processes to find a suitable model structure. To identify such processes, the Kautz model, which is a two-parameter representation of orthogonal basis functions (OBF) in state space format, is used. The process parameters are obtained through the subspace method by sequencing the states obtained from the Kautz model. The obtained states are transformed into convex optimization in which the eigen values poles) to be estimated are confined to lie within user-defined regions in the convex plane. To derive the optimal states, an efficient control scheme based on the state observer design is implemented. The viability of the proposed work has been verified on real-time Coriolis Mass Flow Meter (CMFM) under no flow condition and the corresponding responses are plotted using MATLAB.

Light mesons within the basis light-front quantization framework

We study the light-unflavored mesons as relativistic bound states in the nonperturbative Hamiltonian formalism of the basis light-front quantization (BLFQ) approach. The dynamics for the valence quarks of these mesons is specified by an effective Hamiltonian containing the one-gluon exchange interaction and the confining potentials both introduced in our previous work on heavy quarkonia, supplemented additionally by a pseudoscalar contact interaction. We diagonalize this Hamiltonian in our basis function representation to obtain the mass spectrum and the light-front wave functions (LFWFs). Based on these LFWFs, we then study the structure of these mesons by computing the electromagnetic form factors, the decay constants, the parton distribution amplitudes (PDAs), and the parton distribution functions (PDFs). Our results are comparable to those from experiments and other theoretical models.

Analysis, Design, and Generalization of Electrochemical Impedance Spectroscopy (EIS) Inversion Algorithms

We introduce a framework for analyzing and designing EIS inversion algorithms. Our framework stems from the observation of four features common to well-defined EIS inversion algorithms, namely (1) the representation of unknown distributions, (2) the minimization of a metric of error to estimate parameters arising from the chosen representation, subject to constraints on (3) the complexity control parameters, and (4) a means for choosing optimal control parameter values. These features must be present to overcome the ill-posed nature of EIS inversion problems. We review three established EIS inversion algorithms to illustrate the pervasiveness of these features, and show the utility of the framework by resolving ambiguities concerning three more algorithms. Our framework is then used to design the generalized EIS inversion (gEISi) algorithm, which uses Gaussian basis function representation, modality control parameter, and cross-validation for choosing the optimal control parameter value. The gEISi algorithm is applicable to the generalized EIS inversion problem, which allows for a wider range of underlying models. We also considered the construction of credible intervals for distributions arising from the algorithm. The algorithm is able to accurately reproduce distributions which have been difficult to obtain using existing algorithms. It is provided gratis on the repository https://github.com/suryaeff/gEISi.git.

Assessing Social Interest in Burnout Using Google Trends Data

Burnout is a serious problem in modern society and early detection methods are needed to successfully handled its multiple effects. The latter refer to working well-being, as well as to the affective, psychological, physiological, and behavioral well-being of workers. However, in many countries official statistics on this topic are not available. For this reason, we propose to use Google Trends data as proxies for the interest in burnout and to analyze them through the functional data analysis approach. The latter allows to address the so-called ‘curse of dimensionality’ of big data, enabling an effective statistical analysis when the number of variables exceeds the number of observations. Under this framework, the functional analysis of variance (FANOVA) model is used for testing a macro geographic area effect on search queries for the keyword “burnout” in Italy. The estimation of the FANOVA model is proposed in a finite dimensional space generated by a basis function representation. Thus, the functional model is reduced to a MANOVA model on the basis coefficients.

Spatial Statistical Downscaling for Constructing High-Resolution Nature Runs in Global Observing System Simulation Experiments

ABSTRACTObserving system simulation experiments (OSSEs) have been widely used as a rigorous and cost-effective way to guide development of new observing systems, and to evaluate the performance of new data assimilation algorithms. Nature runs (NRs), which are output from deterministic models, play an essential role in building OSSE systems for global atmospheric processes because they are used both to create synthetic observations at high spatial resolution, and to represent the “true” atmosphere against which the forecasts are verified. However, most NRs are generated at resolutions coarser than actual observations from satellite instruments or predictions from data assimilation algorithms. Our goal is to develop a principled statistical downscaling framework to construct high-resolution NRs via conditional simulation from coarse-resolution numerical model output. We use nonstationary spatial covariance function models that have basis function representations to capture spatial variability. This approach not only explicitly addresses the change-of-support problem, but also allows fast computation with large volumes of numerical model output. We also propose a data-driven algorithm to select the required basis functions adaptively, in order to increase the flexibility of our nonstationary covariance function models. In this article we demonstrate these techniques by downscaling a coarse-resolution physical numerical model output at a native resolution of of global surface concentrations to 655,362 equal-area hexagons.

Creating smooth SI. B-spline basis function representations of insulin sensitivity

Abstract In the intensive care unit (ICU), stress-induced insulin resistance leading to hyperglycemia is commonplace. If safe and effective glycemic control (GC) can be provided, a significant reduction in the negative effects of dysglycemia can be achieved. The Intensive Control Insulin-Nutrition-Glucose (ICING) model has worked particularly well in guiding patient-specific GC. The current method to identify patient- and time- specific insulin sensitivity (SI) with this model employs a discrete stepwise hourly function, which is effective, but not physiologically representative at hourly transitions. 2nd Order B-spline basis functions (BF) are investigated as a more continuous and physiologically relevant alternative for two different cohorts of patient data (Benchmark, 20 patients; Stochastic TARgeted (STAR) sub-cohort, 72 patients). Various knot-widths (KW) of 2nd Order B-spline BFs were investigated and compared to the currently used step function BF in terms of physiological relevance, identifiability, robustness to false measurements, and susceptibility to noise to ensure the most physiologically realistic representation of SI. The 180 min KW 2nd Order B-spline BFs provided the most physiologically realistic fit to the blood glucose (BG) measurements in both cohorts (Standard Deviation of relative fitting error: Benchmark 8.7% vs. 9.35% BG meter, STAR sub-cohort 6.0% vs. 6.0% BG meter), while also having significantly less susceptibility to false BG measurements compared to the current BF. The 180 min 2nd Order B-spline BF provides a smooth second order continuous, more physiologically representative, and identifiable model of the dynamics of SI, and a more realistic resulting BG solution. The outcome provides a promising new addition to the ICING model and STAR GC protocol, while also opening up several new potential continuous time analyses of SI dynamics, not previously possible.

Local basis function representations of thermodynamic surfaces: Water at high pressure and temperature as an example

A flexible numerical framework using local basis function is developed in order to better represent thermodynamic properties of fluids as a function of pressure temperature and composition. A new equation of state for water to 100 GPa and 10,000 K is presented. Conventional equations of state are typically based on complex sets of global basis functions that are unique to each application. The use of series expansions in positive and negative powers of the independent variables (with additional exponential/log/trig. factors in some cases) is common and recent accurate equations of state are assembled from collections of such arbitrary terms. These individually crafted representations do not always fit data within uncertainties, can be difficult to implement, and are not easily modified to account for newer data. Multivariate (tensor) b splines overcome these shortcomings. The underlying basis functions are local, orthonormal and complete. Data can be represented to arbitrary precision and the relationships between model parameters and observational constraints are uncomplicated. The fittings of new data in more extended regimes of the thermodynamic space do not require modifying parameters in previously constrained regions. Models are transportable and easily implemented since robust and efficient standardized routines for evaluation of b splines are available in modern numerical environments. Because a local basis function representation can flexibly fit any surface, the articulation of a priori constraints (associated with expected physical and chemical behavior) is appropriately necessary and explicit during model construction. Basic tools are demonstrated in applications to thermodynamic properties of water where the modified equation of state matches a prior formulation below 1 GPa while providing an excellent fit to measurements at much higher pressures.

English

The paper focuses on a low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization techniques due the dimensionality of the problem. However, it can be effectively performed using the tensor representation of basis functions. Furthermore, this approach can take advantage of parallel computing.

Tensor Approximation of Slater-Type Orbital Basis Functions

This paper deals with a tensor representation of the Slater-type orbital basis functions. Localized basis systems are primarily used in electronic structure calculations. A choice of the system is usually limited to Gaussian-type orbitals due to the impossibility of the analytical evaluation of necessary integrals using other basis types. Unfortunately, it is not possible to use direct discretization techniques due to the dimensionality of the problem so the numerical integration is problematic. Tensor Numerical Methods overcome this problem by using special data representations, which are discussed. Finally, it is demonstrated how to use these methods to construct a tensor approximation of Slater-type orbital basis functions including an error estimate and its numerical verification.

Efficient Bayesian hierarchical functional data analysis with basis function approximations using Gaussian-Wishart processes.

Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected on discretized grids with measurement errors. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior inference through Markov chain Monte Carlo methods. Compared to the standard Bayesian inference that suffers serious computational burden and instability in analyzing high-dimensional functional data, our method greatly improves the computational scalability and stability, while inheriting the advantage of simultaneously smoothing raw observations and estimating the mean-covariance functions in a nonparametric way. In addition, our method can naturally handle functional data observed on random or uncommon grids. Simulation and real studies demonstrate that our method produces similar results to those obtainable by the standard Bayesian inference with low-dimensional common grids, while efficiently smoothing and estimating functional data with random and high-dimensional observation grids when the standard Bayesian inference fails. In conclusion, our method can efficiently smooth and estimate high-dimensional functional data, providing one way to resolve the curse of dimensionality for Bayesian functional data analysis with Gaussian-Wishart processes.

Reproducing kernel formulation of B-spline and NURBS basis functions: A meshfree local refinement strategy for isogeometric analysis

A reproducing kernel meshfree formulation is presented for the B-spline and non-uniform rational B-spline (NURBS) basis functions used in isogeometric analysis. It is shown that after properly introducing meshfree nodes, support size and consistency conditions, the reproducing kernel meshfree shape functions are capable of exactly representing the isogeometric B-spline and NURBS basis functions. Consequently, the proposed formulation successfully establishes a correspondence or close link between the meshfree methods and the isogeometric analysis. More importantly, the proposed reproducing kernel meshfree representation of isogeometric basis functions provides a reliable meshfree strategy to the local model refinement in isogeometric analysis. This strategy inherits the strength of meshfree methods and gives considerable easiness for the local refinement, i.e., the shape functions in the refined regions can be naturally constructed in a straightforward meshfree manner. Meanwhile, the consistency and independence of the shape functions required by the subsequent computational analysis are ensured by the consistency conditions of reproducing kernel meshfree formulation. A detailed illustration of the proposed approach for isogeometric local model refinement is presented. The effectiveness of the proposed meshfree local refinement strategy for isogeometric analysis is demonstrated through numerical examples.