Quality Of Approximation Research Articles

Digitization and subduction of SU(N) gauge theories

The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of SU(2) and SU(3) to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of Σ(360×3) at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of SU(3) for irreducible representations that subduce to a single Σ(360×3) irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations. Published by the American Physical Society 2024

Deep Nonnegative Matrix Factorization With Beta Divergences.

Deep nonnegative matrix factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales. However, all existing deep NMF models and algorithms have primarily centered their evaluation on the least squares error, which may not be the most appropriate metric for assessing the quality of approximations on diverse data sets. For instance, when dealing with data types such as audio signals and documents, it is widely acknowledged that ß-divergences offer a more suitable alternative. In this article, we develop new models and algorithms for deep NMF using some ß-divergences, with a focus on the Kullback-Leibler divergence. Subsequently, we apply these techniques to the extraction of facial features, the identification of topics within document collections, and the identification of materials within hyperspectral images.

Evaluation of geomorphological classification uncertainty using rough set theory: A case study of Shaanxi Province, China

AbstractGeomorphological classification is affected by classification principles, indicators, methods, and data resolution, which can lead to uncertainty in the results. Such uncertainty directly affects the quality and subsequent applications of geomorphological classification. To quantify and control the uncertainty, it is important to select an appropriate and effective method for evaluating the uncertainty of geomorphological classification. This study evaluated the uncertainty of geomorphological classification of Shaanxi Province at the ground‐feature class and image scales, which derived from rough set theory: rough entropy, approximate classification quality, and approximate classification accuracy. The three indicators helped effectively assess the uncertainty of geomorphological classification at multi‐scale and measured the degree to which different factors affected the uncertainty of geomorphological classification. The relative impacts of three factors on the uncertainty of classification decreased in the order of classification methods, data resolution, and classification indicators. This finding is helpful to objectively evaluate and control the uncertainty generated in the process and results of geomorphological classification, and can provide targeted reference and guidance for future geomorphological classification work, which is more conducive to decision‐making and application. At the same time, this study is also a beneficial supplement to the geomorphological research based on digital terrain analysis.

The Gromov–Wasserstein Distance Between Spheres

AbstractThe Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ { d GW p , q } p , q = 1 ∞ of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ d GW 4 , 2 between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

Scalable data fusion via a scale-based hierarchical framework: Adapting to multi-source and multi-scale scenarios

Multi-source information fusion addresses challenges in integrating and transforming complementary data from diverse sources to facilitate unified information representation for centralized knowledge discovery. However, traditional methods face difficulties when applied to multi-scale data, where optimal scale selection can effectively resolve these issues but typically lack the advantage of identifying the optimal and simplest data from different data source relationships. Moreover, in multi-source, multi-scale environments, heterogeneous data (where identical samples have different features and scales in different sources) is prone to occur. To address these challenges, this study proposes a novel approach in two key stages: first, aggregating heterogeneous data sources and refining datasets using information gain; second, employing a customized Scale-based Tree (SbT) structure for each attribute to help extract specific scale information value, thereby achieving effective data fusion goals. Extensive experimental evaluations cover ten different datasets, reporting detailed performance across multiple metrics including Approximation Precision (AP), Approximation Quality (AQ) values, classification accuracy, and computational efficiency. The results highlight the robustness and effectiveness of our proposed algorithm in handling complex multi-source, multi-scale data environments, demonstrating its potential and practicality in addressing real-world data fusion challenges.

A Quasi-Newton Trust-Region Method for Well Location Optimization Under Uncertainty

Summary Subsurface development involves well-placement decisions considering the highly uncertain understanding of the reservoir in the subsurface. The simultaneous optimization of a large number of well locations is a challenging problem. Traditional gradient-based methods can be adapted for well location optimization (WLO) when these problems are converted into real-valued representations and equipped with protocols to handle noisy objective functions. However, their application to large-scale scenarios often remains impractical. This impracticality arises because computing gradients of the objective function can be prohibitively expensive in realistic settings, particularly without using the adjoint method. In this paper, we explore the application of a novel quasi-Newton trust-region (TR) method that employs the stochastic simplex approximate gradient (StoSAG). We have implemented the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton updating algorithm together with line-search (LS) and TR optimization strategies. The StoSAG-based optimization methods have been applied to a realistic synthetic reservoir featuring 26 wells considering two distinct cases: Each includes 20 realizations of porosity and permeability. The first case exhibits mild heterogeneity, while the second exhibits significant heterogeneity with a large correlation length. We have conducted a series of runs to evaluate the performance of these algorithms in addition to comparisons to the finite-difference (FD) and particle-swarm-optimization (PSO) algorithms. We introduce a novel approach to enhance the accuracy of StoSAG gradients by proposing modified StoSAG formulations. These formulations are tailored to exploit the structure of the objective function and to capture the relationships between its components and the individual optimization parameters. This approach involves using a correction matrix W informed by problem-specific knowledge. The entries of W vary from 0 to 1 and are proportional to the interference effects the neighboring wells have on each other. We determine those entries (or weights) based on the radii of investigation around the wells and the distance between the well pairs. Results indicate that the steepest-descent (SD) algorithm coupled with StoSAG has superior performance to PSO and FD. Although the objective function is prone to numerical noise and not continuously differentiable with respect to well locations, StoSAG overcomes this challenge because it acts as a smooth approximation. Comparative tests further confirm that the TR-BFGS is more effective than the LS-BFGS. Moreover, we show that using the proposed gradient correction procedure results in a significant acceleration in convergence, indicating an enhancement in the StoSAG gradient approximation quality. This enhancement allows the TR-BFGS algorithm to achieve considerably higher performance than SD, illustrating that the accuracy of the BFGS approximation benefits from improved gradient quality.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

The kinetics of hydrogen evolution reaction accompanied by hydrogen absorption reaction with consideration of subsurface hydrogen as an adsorbed species: Faradaic impedance

An equation has been obtained for the Faradaic impedance of the hydrogen evolution reaction (HER), parallel to which the hydrogen absorption reaction (HAR) occurs, taking into account subsurface hydrogen (hydrogen atoms between the first and second layers of metal atoms) as a separate state. A new equivalent electrical circuit for the HER + HAR process has been proposed. It has been shown that the improvement in the quality of approximation of the impedance spectra of TiFe and TiFe2 cathodes in 0.5 M H2SO4 when using this equivalent circuit is statistically significant. Based on the impedance data, a HER + HAR mechanism is suggested for the TiFe electrode. The interactions in the layer of adsorbed hydrogen atoms and in the subsurface layer are discussed qualitatively.

Data-distribution-informed Nyström approximation for structured data using vector quantization-based landmark determination

We present an effective method for supervised landmark selection in sparse Nyström approximations of kernel matrices for structured data. Our approach transforms structured non-vectorial input data, like graphs or text, into a dissimilarity representation, facilitating the identification of data-distribution-informed landmarks through prototype-based learning. Experimental results indicate competitive approximation quality when compared to existing strategies, showcasing the advantageous impact of incorporating more information into the Nyström landmark selection process. This positions our method as an efficient and versatile solution for large-scale kernel learning.

A DEIM-CUR factorization with iterative SVDs

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of one-round sampling and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify A after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

We introduce a progressive and iterative method for B-spline curve and surface approximation, incorporating parameter correction based on the Newton iterative method. While parameter corrections have been used in existing Geometric Approximation (GA) methods to enhance approximation quality, they suffer from low computational efficiency. Our approach unifies control point updates and parameter corrections in a progressive and iterative procedure, employing a one-step strategy for parameter correction. We provide a theoretical proof of convergence for the algorithm, demonstrating its superior computational efficiency compared to current GA methods. Furthermore, the provided convergence proof offers a methodology for proving the convergence of existing GA methods with location parameter correction.

New methods for quasi-interpolation approximations: Resolution of odd-degree singularities

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function’s Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and – with a particular focus – polynomial reproduction and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.

Approximating the Shapley Value without Marginal Contributions

The Shapley value, which is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, has recently been used intensively in explainable artificial intelligence. Its meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley value, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contribution. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods.

RITA: Group Attention is All You Need for Timeseries Analytics

Timeseries analytics is important in many real-world applications. Recently, the Transformer model, popular in natural language processing, has been leveraged to learn high quality feature embeddings from timeseries: embeddings are key to the performance of various timeseries analytics tasks such as similarity-based timeseries queries within vector databases. However, quadratic time and space complexities limit Transformers' scalability, especially for long timeseries. To address these issues, we develop a timeseries analytics tool, RITA, which uses a novel attention mechanism, named group attention, to address this scalability issue. Group attention dynamically clusters the objects based on their similarity into a small number of groups and approximately computes the attention at the coarse group granularity. It thus significantly reduces the time and space complexity, yet provides a theoretical guarantee on the quality of the computed attention. The dynamic scheduler of RITA continuously adapts the number of groups and the batch size in the training process, ensuring group attention always uses the fewest groups needed to meet the approximation quality requirement. Extensive experiments on various timeseries datasets and analytics tasks demonstrate that RITA outperforms the state-of-the-art in accuracy and is significantly faster --- with speedups of up to 63X.

A Class of Block Multi-derivative Numerical Techniques for Singular Advection Equations

The integration of some differential equations is hard to acquire because of the presence of singular point(s) in these equations. These equations are best solved by some unique technique. Multi-derivative techniques have a long history of powerful integration of such equations yet till date, a couple of class of this technique has been explored for integrating partial differential equations. This work centers around the development, analysis, and implementation of a class of multi-derivative technique on partial differential equations. The approaches were effectively analyzed and were turned out to be consistent, stable and convergent. Numerical outcomes got likewise demonstrated the approximation quality of the technique over existing techniques in the literature.

Stable Spectral Methods for Time-Dependent Problems and the Preservation of Structure

AbstractThis paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any $$\text {C}^1(a,b)$$ C 1 ( a , b ) weight function such that $$w(a)=w(b)=0$$ w ( a ) = w ( b ) = 0 , we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $$a=-\infty $$ a = - ∞ , $$b=+\infty $$ b = + ∞ , only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function $$x^\alpha \textrm{e}^{-x}$$ x α e - x for $$x>0$$ x > 0 and $$\alpha >0$$ α > 0 and the ultraspherical weight function $$(1-x^2)^\alpha $$ ( 1 - x 2 ) α , $$x\in (-1,1)$$ x ∈ ( - 1 , 1 ) , $$\alpha >0$$ α > 0 , and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of $$\alpha $$ α , and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.

USE OF ARTIFICAL INTELLIGENCE MODELS IN ACCEPTANCE TESTS

This paper describes the application of fuzzy perсeptron in acceptance testing problems. A study on the use of various fuzzy functions in solving the problem of approximation of probability densities was carried out. Experiments were conducted on time series with preset distribution laws, evaluating approximation quality using standard deviation in the series of 29 tests. As a result of the research, using Gaussian fuzzy function, the mean value of standard deviation equal to 0.00145 determined which confirms good approximative ability of the fuzzy perсeptron. The system was tested on the acceptance test data of the gas compressor engine. In this case, approximation quality was assessed using the Kolmogorov-Smirnov agreement criterion. Analysis results confirm the high approximative ability of the fuzzy perсeptron, emphasizing its applicability in real acceptance tests.

Estimation of scalar field distribution in the Fourier domain

In this paper we consider the problem of estimation of scalar field distribution (e.g. pollutant, moisture, temperature) from noisy measurements collected by unmanned autonomous vehicles such as UAVs. The field is modelled as a sum of Fourier components/modes, where the number of modes retained and estimated determines in a natural way the approximation quality. An algorithm for estimating the modes using an online optimisation approach is presented, under the assumption that the noisy measurements are quantized. The algorithm can also be used to estimate time-varying fields. Simulation studies demonstrate the effectiveness of the proposed approach.

Improving the approximation quality of tensor product B-spline surfaces by local parameterization

Abstract Freeform surfaces like tensor product B-spline surfaces have been proven to be a suitable tool to model laser scanner point clouds, especially those representing artificial objects. However, when it comes to the modelling of point clouds representing natural surfaces with a lot of local structures, tensor product B-spline surfaces reach their limits. Refinement strategies are usually used as an alternative, but their functional description is no longer nearly as compact as that of classical tensor product B-spline surfaces, making subsequent analysis steps considerably more cumbersome. In this publication, the approximation quality of classical tensor product B-spline surfaces is improved by means of local parameterization. By using base surfaces with a local character, relevant information about local structures of the surface to be estimated are stored in the surface parameters during the parameterization step. As a consequence, the resulting tensor product B-spline surface is able to represent these structures even with only a small number of control points. The developed locally parameterized B-spline surfaces are used to model four data sets with different characteristics. The results reveal a clear improvement compared to the classical tensor product B-spline surfaces in terms of correctness, goodness-of-fit and stability.