Function Approximation Scheme Research Articles

Quasi-interpolation for multivariate density estimation on bounded domain

The paper proposes a new nonparametric scheme for multivariate density estimation under the framework of quasi-interpolation, a classical function approximation scheme in approximation theory. Given samples of a random variable obeyed by an unknown density function with compact support, we first partition the support into several bins and compute frequency of samples falling into each bin. Then, by viewing these frequencies as (approximate) integral functionals of density function over corresponding bins, we construct a quasi-interpolation scheme for approximating the density function. Maximal mean squared errors of the scheme demonstrates that our scheme keeps the same optimal convergence rate as classical nonparametric density estimations. In addition, the scheme includes classical boundary kernel density estimation as a special case when the number of bins equals to the number of samples. Moreover, it can dynamically allocate different amounts of smoothing via selecting the bin widths and shape parameters of kernels with the prior knowledge. Numerical simulations provide evidence that the proposed nonparametric scheme is robust and is capable of producing high-performance estimation of density function.

High-Order Close Evaluation of Laplace Layer Potentials: A Differential Geometric Approach

This paper presents a new approach for solving the close evaluation problem in three dimensions, commonly encountered while solving linear elliptic partial differential equations via potential theory. The goal is to evaluate layer potentials close to the boundary over which they are defined. The approach introduced here converts these nearly-singular integrals on a patch of the boundary to a set of non-singular line integrals on the patch boundary using the Stokes theorem on manifolds. A function approximation scheme based on harmonic polynomials is designed to express the integrand in a form that is suitable for applying the Stokes theorem. As long as the data -- the boundary and the density function -- is given in a high-order format, the double-layer potential and its derivatives can be evaluated with high-order accuracy using this scheme both on and off the boundary. In particular, we present numerical results demonstrating seventh-order convergence on a smooth, warped torus example achieving 10-digit accuracy in evaluating double layer potential at targets that are arbitrarily close to the boundary.

End-Effector Position Estimation and Control of a Flexible Interconnected Industrial Manipulator Using Machine Learning

The control of flexible robot manipulators is a challenging task, especially when one considers parallel and interconnected manipulators under flexibility considerations. This paper proposes a method to estimate the position of the end-effector of a flexible interconnected manipulator based on a virtual sensor principle and function approximation schemes. By using SolidWorks/MSC ADAMS software, we developed a virtual prototype of a flexible interconnected manipulator, and rigorously evaluated the feasibility of using function approximation schemes such as Neural Networks (NN), Support Vector Machines (SVM), and Gaussian Process (GP) in estimating the deflection error arising due to the flexibility of the robot structure. Our rigorous computational experiments have shown that: (1) the NN, SVM, and GP models were are able to attain the promising and reasonable prediction accuracy, (2) a feedforward NN with 535 neurons and an Ascending distribution of its nodes achieves the best prediction and generalization to unseen environments (the upper bound of the error was <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.15\times 10^{-3}$ </tex-math></inline-formula> m); implying the robust estimation of the position of the end-effector under flexibility considerations, and (3) the control based on the inverse Jacobian and a NN-based estimator was able to follow a sinusoidal trajectory with reasonable tracking and error performance in MSC ADAMS & MATLAB/Simulink co-simulation. Our results show the feasibility and effectiveness of the nonlinear relationships learned by NN, SVM, and GP in aiding estimation and control of the position of the end-effector of the flexible manipulator with a promising/desirable capability.

Analysis of Tensor Approximation Schemes for Continuous Functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

Diamonds in the Rough: Leveraging Click Data to Spotlight Underrated Products

Problem Definition. Inspired by a real data set from the Chinese retailer JD.com, we study the click and purchase behavior of customers in an online retail setting by employing a structural estimation approach. In particular, we aim to understand what characteristics of products drive the click versus purchase decisions of customers.With the boom in e-commerce, which has been further fueled due to the current COVID-19 pandemic, an ever increasing number of customers are shopping online. Such a rapid growth in online shopping provides retailers with abundance of data to study customer shopping behavior. For example, online retailers can benefit from a better understanding of the customer search (click) and purchase behavior to improve their operational decisions such as assortment planning. From an academic standpoint, even though there is an extensive body of work on consumer search models, most prior work do not incorporate search sequences and purchase decisions of customers jointly in their analysis. We, however, combine the click and order data to estimate both the click and purchase behavior of customers.Methodology/Results. Using a large data set from JD.com, we propose a structural model to estimate the click and purchase behavior of customers according to a dynamic discrete choice model. In particular, we assume that the customer's utility from each product has an observed and (pre-click) unobserved part (in addition to a random shock). The observed part of the utility is known to the customer prior to the click; however, the unobserved part of the utility can only be learned after the customer clicks on the product. We assume that the customer solves a dynamic program in order to find the optimal search strategy. Due to the curse of dimensionality, we propose a novel value function approximation scheme inspired by the Conditional Choice Probability approach. This, in turn, reduces the estimation into a computationally tractable two-stage process. By combining the click and order data, our proposed structural framework allows us to disentangle and estimate the observed and unobserved parts of product utilities. Our estimation results show that the value of click for customers can be quite significant. This is evidenced by the fact that the unobserved utilities of products vary significantly across products. Most importantly, we are able to identify underrated products which we call diamonds in the rough: these are products with low “face” values (i.e., low observed utilities), but high total utilities due to their high unobserved utilities. Thus, even though such products have a low chance of being clicked (due to their low observed utilities), they have a high chance of being purchased, if clicked.Managerial Implications. Our structural framework provides an online retailer with new tools and insights to better manage the product assortment based on customer click and purchase behavior. In particular, our structural model allows the retailer to disentangle the observed and unobserved parts of product utilities and identify underrated diamond-in-the-rough products. The online retailer can increase her revenue by bringing such products into the spotlight by promoting them on the search page or using tags such as “spotlight product” (similar to “Amazon's Choice” tags on Amazon.com) to entice customers to click on them.

Radial basis functions and improved hyperparameter optimisation for gaussian process strain estimation

Over the past decade, a number of algorithms for full-field elastic strain estimation from neutron and X-ray measurements have been published. Many of the recently published algorithms rely on modelling the unknown strain field as a Gaussian Process (GP) – a probabilistic machine-learning technique. Thus far, GP-based algorithms have assumed a high degree of smoothness and continuity in the unknown strain field. In this paper, we propose three modifications to the GP approach to improve performance, primarily when this is not the case (e.g. for high-gradient or discontinuous fields); hyperparameter optimisation using k-fold cross-validation, a radial basis function approximation scheme, and gradient-based placement of these functions.

Approximate dynamic programming for the military aeromedical evacuation dispatching, preemption-rerouting, and redeployment problem

Military medical planners must consider how aeromedical evacuation (MEDEVAC) assets will be utilized when preparing for and supporting combat operations. This research examines the MEDEVAC dispatching, preemption-rerouting, and redeployment (DPR) problem. The intent of this research is to determine high-quality DPR policies that improve the performance of United States Army MEDEVAC systems and ultimately increase the combat casualty survivability rate. A discounted, infinite-horizon Markov decision process (MDP) model of the MEDEVAC DPR problem is formulated and solved via an approximate dynamic programming (ADP) strategy that utilizes a support vector regression value function approximation scheme within an approximate policy iteration algorithmic framework. The objective is to maximize the expected total discounted reward attained by the system. The applicability of the MDP model is examined via a notional, representative planning scenario based on high-intensity combat operations to defend Azerbaijan against a notional aggressor. Computational experimentation is performed to determine how selected problem features and algorithmic features affect the quality of solutions attained by the ADP-generated DPR policies and to assess the efficacy of the proposed solution methodology. The results from the computational experiments indicate the ADP-generated policies significantly outperform the two benchmark policies considered. Moreover, the results reveal that the average service time of high-precedence, time-sensitive requests decreases when an ADP policy is adopted during high-intensity conflicts. As the rate at which requests enter the MEDEVAC system increases, the performance gap between the ADP policy and the first benchmark policy (i.e., the currently practiced, closest-available dispatching policy) increases substantially. Conversely, as the rate at which requests enter the system decreases, the ADP performance improvement over both benchmark policies decreases, indicating the ADP policy provides little-to-no benefit over a myopic approach (e.g., as utilized in the benchmark policies) when the intensity of a conflict is low. Ultimately, this research informs the development and implementation of future tactics, techniques, and procedures for military MEDEVAC operations.

Attentive multi-view reinforcement learning

The reinforcement learning process usually takes millions of steps from scratch, due to the limited observation experience. More precisely, the representation approximated by a single deep network is usually limited for reinforcement learning agents. In this paper, we propose a novel multi-view deep attention network (MvDAN), which introduces multi-view representation learning into the reinforcement learning framework for the first time. Based on the multi-view scheme of function approximation, the proposed model approximates multiple view-specific policy or value functions in parallel by estimating the middle-level representation and integrates these functions based on attention mechanisms to generate a comprehensive strategy. Furthermore, we develop the multi-view generalized policy improvement to jointly optimize all policies instead of a single one. Compared with the single-view function approximation scheme in reinforcement learning methods, experimental results on eight Atari benchmarks show that MvDAN outperforms the state-of-the-art methods and has faster convergence and training stability.

A Learning-Based Power Management Method for Networked Microgrids Under Incomplete Information

This paper presents an approximate Reinforcement Learning (RL) methodology for bi-level power management of networked Microgrids (MG) in electric distribution systems. In practice, the cooperative agent can have limited or no knowledge of the MG asset behavior and detailed models behind the Point of Common Coupling (PCC). This makes the distribution systems unobservable and impedes conventional optimization solutions for the constrained MG power management problem. To tackle this challenge, we have proposed a bi-level RL framework in a price-based environment. At the higher level, a cooperative agent performs function approximation to predict the behavior of entities under incomplete information of MG parametric models; while at the lower level, each MG provides power-flow-constrained optimal response to price signals. The function approximation scheme is then used within an adaptive RL framework to optimize the price signal as the system load and solar generation change over time. Numerical experiments have verified that, compared to previous works in the literature, the proposed privacy-preserving learning model has better adaptability and enhanced computational speed.

Single-Agent Indirect Herding of Multiple Targets With Uncertain Dynamics

In this paper, an indirect herding problem is considered for a single herder that is outnumbered by multiple target agents. Indirect herding is a problem that involves a network of one or more controllable herding agents and indirectly controlled target agents (i.e., the target agents can only be controlled through the herder by exploiting herder–target interactions), where the goal is to achieve a network-wide objective. This paper investigates the unique problem where a single herder is required to regulate all the target agents to some desired formation. The problem is further complicated by the fact that the target agents have uncertain nonlinear dynamics, including uncertainties in the herder–target interaction. Neural network function approximation methods are used along with switched systems methods to ensure uniformly ultimately bounded convergence of the agent trajectories provided the developed sufficient dwell-time conditions are satisfied. Simulation and experimental results that involve one herding agent and multiple target agents demonstrate the validity of the designed controller and function approximation scheme for multiple herding objectives.

A novel FABP quantization scheme for improving performance in low-bit IMDD-FBMC system

In this paper, a back propagation neural network-based function approximation scheme is proposed to provide the capacity of the nonuniform quantization (FABP) in intensity modulated direct detection-based optical filter bank multicarrier (IMDD-FBMC) system. In the means of leveraging this scheme, the quantization noise can be reduced to further improve the ratio of Signal-to-Quantization noise, namely SQNR. Further, with the help of the nonlinear programming, the optimal quantization levels are accurately calculated. And, to validate the feasibility of our method, the simulation and experiment investigation are conducted on the 10-Gb/s transmission system with optical order offset quadrature amplitude modulation based optical orthogonal frequency division multiplexing (OQAM-OFDM) signals through 80-km fiber. Results verified that, our scheme can achieve the error vector magnitude (EVM) improvement of ∼2.0% and received sensitivity improvement of at least 2 dB in the same quantization bit case compared with uniform quantization scheme.

Approximate dynamic programming for the aeromedical evacuation dispatching problem: Value function approximation utilizing multiple level aggregation

Sequential resource allocation decision-making for the military medical evacuation of wartime casualties consists of identifying which available aeromedical evacuation (MEDEVAC) assets to dispatch in response to each casualty event. These sequential decisions are complicated due to uncertainty in casualty demand (i.e., severity, number, and location) and service times. In this research, we present a Markov decision process model solved using a hierarchical aggregation value function approximation scheme within an approximate policy iteration algorithmic framework. The model seeks to optimize this sequential resource allocation decision under uncertainty of how to best dispatch MEDEVAC assets to calls for service. The policies determined via our approximate dynamic programming (ADP) approach are compared to optimal military MEDEVAC dispatching policies for two small-scale problem instances and are compared to a closest-available MEDEVAC dispatching policy that is typically implemented in practice for a large-scale problem instance. Results indicate that our proposed approximation scheme provides high-quality, scalable dispatching policies that are more easily employed by military medical planners in the field. The identified ADP policies attain 99.8% and 99.5% optimal for the 6- and 12-zone problem instances investigated, as well as 9.6%, 9.2%, and 12.4% improvement over the closest-MEDEVAC policy for the 6-, 12-, and 34-zone problem instances investigated.

Sampling and approximation of bandlimited volumetric data

We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is based on expanding the given function in the basis of generalized prolate spheroidal wavefunctions, with the expansion coefficients given by weighted dot products between the samples of the function and the samples of the basis functions. As numerical implementations require all expansions to be finite, we present a truncation rule for the expansions. Finally, we derive a bound on the overall approximation error in terms of the assumed space/frequency concentration.

A simultaneous space-time wavelet method for nonlinear initial boundary value problems

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with[0, 3 N -1] compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.

Singular value decomposition versus sparse grids: refined complexity estimates

Abstract We compare the cost complexities of two approximation schemes for functions that live on the product domain $\varOmega _1\times \varOmega _2$ of sufficiently smooth domains $\varOmega _1\subset \mathbb{R}^{n_1}$ and $\varOmega _2\subset \mathbb{R}^{n_2}$, namely the singular value / Karhunen–Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\varOmega _1$ and $\varOmega _2$, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by Griebel &amp; Harbrecht (2014, Approximation of bi-variate functions. Singular value decomposition versus sparse grids. IMA J. Numer. Anal., 34, 28–54). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.

Stochastic Enumeration with Importance Sampling

Many hard problems in the computational sciences are equivalent to counting the leaves of a decision tree, or, more generally, summing a cost function over the nodes. These problems include calculating the permanent of a matrix, finding the volume of a convex polyhedron, and counting the number of linear extensions of a partially ordered set. Many approximation algorithms exist to estimate such sums. One of the most recent is Stochastic Enumeration (SE), introduced in 2013 by Rubinstein. In 2015, Vaisman and Kroese provided a rigorous analysis of the variance of SE, and showed that SE can be extended to a fully polynomial randomized approximation scheme for certain cost functions on random trees. We present an algorithm that incorporates an importance function into SE, and provide theoretical analysis of its efficacy. We also present the results of numerical experiments to measure the variance of an application of the algorithm to the problem of counting linear extensions of a poset, and show that introducing importance sampling results in a significant reduction of variance as compared to the original version of SE.

A Flexible $C^2$ Subdivision Scheme on the Sphere: With Application to Biomembrane Modelling

We construct a $C^2$ multiscale approximation scheme for functions defined on the (Riemann) sphere. Based on a three-directional box-spline, a flexible $C^2$ scheme over a valence 3 extraordinary vertex can be constructed. Such a flexible $C^2$ subdivision scheme is known to be impossible for arbitrary valences. The subdivision scheme can be used to model spherical surfaces based on a recursively subdivided tetrahedron, with only valence 3 and 6 vertices in the resulted triangulations. This adds to the toolbox of subdivision methods a high order, high regularity scheme which can be beneficial to scientific computing applications. For instance, the scheme can be used in the numerical solution of the Canham--Helfrich--Evans models for spherical and toroidal biomembranes. Moreover, the characteristic maps of the subdivision scheme endow the underlying simplicial complex with a conformal structure. This in particular means that the special subdivision surfaces constructed here comes with a well-defined harmon...

Spectral Tensor-Train Decomposition

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the "cores") comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting \textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.

Differential approximation schemes for half-product related functions and their scheduling applications

We address the problem of minimizing a half-product function, with and without a linear knapsack constraint. We show how to convert known fully polynomial-time approximation schemes to differential approximation schemes that handle the problems with and without an additive constant and with and without a linear knapsack constraint. Thereby, we resolve the issue of differential approximation for a range of scheduling problems.

A three-point coupled compact integrated RBF scheme for second-order differential problems

In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first and second derivative values via their identity equations. Owing to its coupling of the information of the nodal first and second derivatives, the CCIRBF scheme becomes more accurate, stable and efficient than the normal compact integrated radial basis function (CIRBF) schemes proposed by [Thai-Quang, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T. (2012)]. The main features of the CCIRBF scheme include: three-point, high-order accuracy, stability, efficiency and inclusion of boundary values. Several analytic problems are considered to verify the present scheme and to compare its accuracy, stability and efficiency with those of the CIRBF, higher-order compact finite difference (HOC) and some other high-order schemes. Numerical results show that highly accurate and stable results are obtained with the proposed scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the CIRBF and HOC schemes.