Approximation Space Research Articles

A priori error analysis of new semidiscrete, Hamiltonian HDG methods for the time-dependent Maxwell’s equations

We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell’s equations introduced in Sánchez et al. [Comput. Methods Appl. Mech. Eng. 396 (2022) 114969]. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell’s equations. This is why they are called ``Hamiltonian’’ HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell’s equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis.

Observer-Based Approximate Affine Nonlinear Model Predictive Controller for Hydraulic Robotic Excavators with Constraints

Given the highly nonlinear and strongly constrained nature of the electro-hydraulic system, we proposed an observer-based approximate nonlinear model predictive controller (ANMPC) for the trajectory tracking control of robotic excavators. A nonlinear non-affine state space equation with identified parameters is employed to describe the dynamics of the electro-hydraulic system. Then, to mitigate the plant-model mismatch caused by the first-order linearization, an approximate affine nonlinear state space model is utilized to represent the explicit relationship between the output and input and an ANMPC is designed based on the approximate nonlinear model. Meanwhile, the Extended Kalman Filter was introduced for state observation to deal with the unmeasurable velocity information and heavy measurement noises. Comparative experiments are conducted on a 1.7-ton hydraulic robotic excavator, where ANMPC and linear model predictive control are used to track a typical excavation trajectory. The experimental results provide evidence of convincing trajectory tracking performance.

Rough Groups over Rough Sets

In this study the rough algebraic structure especially on rough group is focused. The author make over the inadequate definition of rough group of an approximation space. Here, the rough groups over the rough sets is defined, in the sense of Pawlak's rough sets. Later, the structure of rough groups is given.

Wavenumber Explicit Convergence of a Multiscale Generalized Finite Element Method for Heterogeneous Helmholtz Problems

In this paper, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, (i) wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains and (ii) a quasi-optimal convergence of the method is established by assuming that the size of subdomains is ( is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as ( denoting the mesh size) toward the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [I. Babuška and J. M. Melenk, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727–758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient noniterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.

THE IMPLEMENTATION OF A ROUGH SET OF PROJECTIVE MODULE

In ring and module theory, one concept is the projective module. A module is said to be projective if it is a direct sum of independent modules. (U, R) is an approximation space with non-empty set and equivalence relation If X subset U, we can form upper approximation and lower approximation. X is rough set if upper Apr(X) is not equal to under Apr(X). The rough set theory applies to algebraic structures, including groups, rings, modules, and module homomorphisms. In this study, we will investigate the properties of the rough projective module.

Fuzzy rough dimensionality reduction: A feature set partition-based approach

Dimensionality reduction is considered in many learning methods using discriminative features to obtain optimal performance. In general, feature extraction and feature selection are two independent methods that cherry-pick the informative features in the mapping data space or original feature space. Fuzzy rough set theory is widely used in feature selection, which well keeps the interpretability of features. How to comprehensively consider the intrinsic information contained in features and combine the feature extraction with feature selection is a challenging and meaningful task. Most existing fuzzy rough-based feature selection methods select features from the whole feature set. However, some nonsignificant features may lead to a decline in useful information. Meanwhile, these existing methods directly remove redundant and irrelevant features. This study scrutinizes features to distinguish the significant degree of features and deal with them in different ways. The feature selection and feature extraction methods are combined to obtain compound features for the learning method. A feature set partition-based approach to fuzzy rough dimensionality reduction (FSPFRdr) by simultaneously considering feature selection and fuzzy similarity relation-based supervised locally linear embedding (FRSLLE) is proposed. First, the ϑ-fuzzy rough set model is defined to avoid the affection of noise, and approximation space is constructed based on fuzzy approximations and granular structure. The normalized independent classification information (NICI) is defined in the approximation space. According to NICI, the original feature set is divided into a nonsignificant feature set, a weak significant feature set, and a significant feature set. Then, the nonsignificant feature set is deleted before dimensionality reduction. Next, the proposed FRSLLE is applied to a weak significant feature set, and the embedded feature set is obtained. Furthermore, a feature evaluation function is proposed to select features from the significant and embedded feature sets. Finally, the performance of FSPFRdr is compared with thirteen dimensionality reduction algorithms on twenty-three public datasets. Experimental results on four different classifiers show that FSPFRdr has higher classification performance.

Physics-based adaptivity of a spectral method for the Vlasov–Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space

We propose a spectral method for the 1D-1V Vlasov–Poisson system where the discretization in velocity space is based on asymmetrically-weighted Hermite functions, dynamically adapted via a scaling α and shifting u of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of α and u based on the numerical solution of the discrete Vlasov–Poisson system obtained at that time step. Once the new values of the Hermite parameters α and u are fixed, the Hermite expansion is updated and the discrete system is further evolved for the next time step. The procedure is applied iteratively over the desired temporal interval. The key aspects of the adaptive algorithm are: the map between approximation spaces associated with different values of the Hermite parameters that preserves total mass, momentum and energy; and the adaptivity criterion to update α and u based on physics considerations relating the Hermite parameters to the average velocity and temperature of each plasma species. For the discretization of the spatial coordinate, we rely on Fourier functions and use the implicit midpoint rule for time stepping. The resulting numerical method possesses intrinsically the property of fluid-kinetic coupling, where the low-order terms of the expansion are akin to the fluid moments of a macroscopic description of the plasma, while kinetic physics is retained by adding more spectral terms. Moreover, the scheme features conservation of total mass, momentum and energy associated in the discrete, for periodic boundary conditions. A set of numerical experiments confirms that the adaptive method outperforms the non-adaptive one in terms of accuracy and stability of the numerical solution.

Strong surface winds in Storm Eunice. Part 1: storm overview and indications of sting jet activity from observations and model data

Strong surface winds in Storm <i>Eunice</i>. Part 1: storm overview and indications of sting jet activity from observations and model data

A fast randomized algorithm for computing an approximate null space

Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall–skinny matrix. We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing right singular vectors using randomization by examining the quality of the solution using multiplicative perturbation theory. For an mtimes n (mge n) matrix, the algorithm runs with complexity O(mnlog n +n^3) which is faster than the standard O(mn^2) methods. In applications, numerical experiments show great speedups including a 30times speedup for the AAA algorithm and 10times speedup for the total least squares problem.

Randomized Quasi-Optimal Local Approximation Spaces in Time

We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/multiscale ansatz functions defined in space that can be combined with time stepping schemes within model order reduction or multiscale methods. To that end, we propose to perform several simulations of the PDE for a few time steps in parallel starting at different, randomly drawn start points, prescribing random initial conditions; applying a singular value decomposition to a subset of the so obtained snapshots yields the reduced basis/multiscale ansatz functions. This facilitates constructing the reduced basis/multiscale ansatz functions in an embarrassingly parallel manner. In detail, we suggest using a data-dependent probability distribution based on the data functions of the PDE to select the start points. Each local in time simulation of the PDE with random initial conditions approximates a local approximation space in one time point that is optimal in the sense of Kolmogorov. The derivation of these optimal local approximation spaces which are spanned by the left singular vectors of a compact transfer operator that maps arbitrary initial conditions to the solution of the PDE in a later point of time is one other main contribution of this paper. By solving the PDE locally in time with random initial conditions, we construct local ansatz spaces in time that converge provably at a quasi-optimal rate and allow for local error control. Numerical experiments demonstrate that the proposed method can outperform existing methods like the proper orthogonal decomposition even in a sequential setting and is well capable of approximating advection-dominated problems.

High-Throughput Computational Screening of Two-Dimensional Covalent Organic Frameworks (2D COFs) for Capturing Radon in Moist Air.

Radon (Rn) and its decay products are the primary sources of natural ionizing radiation exposure for the public, posing significant health risks, including being a leading cause of lung cancer. Porous material-based adsorbents offer a feasible and efficient solution for controlling Rn concentrations in various scenes to achieve safe levels. However, due to competitive adsorption between Rn and water, finding candidates with a higher affinity and capacity for capturing Rn in humid air remains a significant challenge. Here, we conducted high-throughput computational screening of 8641 two-dimensional covalent organic frameworks (2D COFs) in moist air using grand canonical Monte Carlo simulations. We identified the top five candidates and revealed the structure-performance relationship. Our findings suggest that a well-defined cavity with an approximate spherical inner space, with a diameter matching that of Rn, is the structural basis for a proper Rn capturing site. This is because the excellent steric match between the cavity and Rn maximizes their van der Waals dispersion interactions. Additionally, the significant polarization electrostatic potential surface of the cavity can regulate the adsorption energy of water and ultimately impact Rn selectivity. Our study offers a potential route for Rn management using 2D COFs in moist air and provides a scientific basis for further experimentation.

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

Approximation spaces inspired by subset rough neighborhoods with applications

Abstract In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of u u and ⟨ u ⟩ \langle u\rangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.

Augmenting Basis Sets by Normalizing Flows

AbstractApproximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well‐posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high‐dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.

High-order weak Galerkin scheme for [formula omitted]-elliptic interface problems

A fitted weak Galerkin (WG) finite element scheme has been demonstrated in the present paper for solving H(div)-elliptic equation with discontinuous coefficients and interface. In both L2 norm and H1 norm, error estimates of optimal orders are discussed for the H(div)-elliptic interface problems. High-order convergence rates have been achieved by using suitable WG approximation spaces of higher degrees. We have performed some typical numerical tests to confirm the theoretical findings of the proposed WG algorithm. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

Extraction and application of super-smooth cubic B-splines over triangulations

The space of C1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative C1 cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell–Sabin refined triangulation and present different types of C2 super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough–Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

The enriched quadrilateral overlapping finite elements for time-harmonic acoustics

The pronounced numerical dispersions and numerical anisotropy make solutions from the finite element model using low-order elements unreliable for the time-harmonic acoustic problems with fairly large wavenumber. In this work we propose a novel enriched quadrilateral overlapping elements for Helmholtz problems. In this scheme, the original overlapping elements are strengthened by the harmonic trigonometric functions stemming from the spectral techniques. Since all additional degrees of freedom are aligned on the vertex node of every overlapping element, the proposed method can be directly applied to the original finite element model without changing the mesh topology. Because of the enriched approximation space, the proposed method can significantly suppress the numerical dispersions with practically negligible numerical anisotropy, and can be more computationally efficient in providing comparable solution accuracy compared to the original scheme and the classic finite element method. Besides, the linear dependence issue is completely avoided and these enriched overlapping elements are distortion-insensitive. In this work, the original variational formulation is perturbed using the penalty method to impose the essential boundary conditions. Numerical experiments show that the developed method can reduce user interventions in mesh creation and adjustment, and is promising in practical engineering applications for time-harmonic acoustics.

Summation-by-Parts Operators for General Function Spaces

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

A Numerical Energy Reduction Approach for Semilinear Diffusion-Reaction Boundary Value Problems Based on Steady-State Iterations

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilized) linearized discrete problems, and tends to a critical point of the underlying energy functional in a stable way. Simultaneously, the finite-dimensional approximation spaces are adaptively refined. This is implemented in terms of a new mesh refinement strategy in the context of finite element discretizations, which again relies on the energy structure of the problem under consideration. In particular, in contrast to more traditional approaches, it does not involve any a posteriori error estimators, and is based on local energy reduction indicators instead. In combination, the resulting adaptive algorithm consists of an iterative linearization procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge toward a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.

Convergence and optimality of an adaptive modified weak Galerkin finite element method

AbstractAn adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.