Good Approximation Properties Research Articles

A Robust Moving Total Least-Squares Fitting Method for Measurement Data

The moving least-squares (MLS) and moving total least-squares (MTLS) methods have been widely used for fitting measurement data. They can be used to achieve good approximation properties. However, these two methods are susceptible to outliers due to the way of determining local approximate coefficients, which leads to distorted estimation. To reduce the influence of outliers and random errors of all variables without adding small weights or setting the threshold subjectively, we present a robust MTLS (RMTLS) method, in which an improved least trimmed squares (ILTS) method is used for obtaining the local approximants of the influence domain. The ILTS method divides the nodes in the influence domain into a certain number of subsamples, achieves the local approximants by the total least-squares (TLS) method with compact support weight function, and trims the node with the largest orthogonal residual from each subsample, respectively. The remaining nodes from the subsamples are used to determine the local coefficients. The measurement experiment and numerical simulations are provided to demonstrate the robustness and accuracy of the presented method in comparison with the MLS and MTLS methods.

Non-Eulerian Newmark Methods: A Powerful Tool for Free-Boundary Continuum Mechanics Problems

In this paper we introduce a new procedure to solve free-surface problems based on applying the Newmark family of time integration schemes to non-Eulerian formulations of the problem (i.e., in non-current domains). The methods obtained within this framework present important advantages over other methods in literature, as for instance, that the computational domain is independent of the current unknowns, the convective term disappears, and modelling and tracking of the free surface is straightforward. Moreover, the Newmark algorithm is convenient to obtain accurate and stable methods for solving continuum mechanics models which can be written in terms of either displacement, velocity or acceleration. We consider a viscous Newtonian fluid in a time dependent domain which may undergo large deformations along the time but not topological changes of interfaces. A unified formulation providing the general framework of the proposed schemes is stated in this context. They are combined with finite elements methods for space discretization. In particular, the Newmark family of pure Lagrange–Galerkin methods is stated. The non-Eulerian formulations are advantageous to obtain linear methods. More precisely, linearized versions of the standard (non-linear) Newmark schemes that present good approximation properties are also proposed. Moreover, we deal with problems associated with high mesh distortion by proposing a reinitialization method to be applied in the non-Eulerian Newmark framework that preserves the order of convergence. In order to assess the performance of the proposed numerical methods, we solve different problems in two space dimensions.

Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh+ scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh+ scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh+ scheme in the analysis of thermoelastic problems.

Mathematical analysis of robustness of two-level domain decomposition methods with respect to inexact coarse solves

Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. The GenEO coarse space has been shown to lead to a fully robust two-level Schwarz preconditioner which scales well over multiple cores [27, 19] as has been proved rigorously in [27]. The robustness is due to its good approximation properties for problems with highly heterogeneous material parameters. It is available in the finite element packages FreeFem++, Feel++ and recently in Dune and is implemented as a standalone library in HPDDM But the coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large and exact solves are used. It is therefore interesting to consider the effect of approximate coarse solves. In this paper, robustness of GenEO methods is analyzed with respect to approximate coarse solves. Interestingly, the GenEO-2 method has to be modified in order to be able to prove its robustness in this context.

Parabolic Profile in Heat-Conduction Problems. 4. Semi-Bounded Space with a Convective Boundary Condition

New approximate solutions of the heat-conduction problem for a semi-bounded space with a convective boundary condition have been obtained with the use of the heat balance integral method, the refined integral method, and the combined integral method as well as with new hybrid schemes involving a new generalized integral relation, the temperature momentum integral, and the heat-flow momentum integral. The solutions obtained were represented as a parabola with a time-dependent exponent n(t), and they are entirely explicit in form. It is shown that the optimization of the exponent n(t) on the basis of minimization of the Langford norm EL provides, as a rule, the obtaining of solutions with a low approximation accuracy. The new hybrid schemes proposed are more efficient, because they make it possible to obtain solutions in the explicit form with a fairly high approximation accuracy. It is proposed to define parabolic solutions of the heat-conduction problem with a convective boundary condition with the use of new generalized variables and without recourse to the Biot number as an independent variable. A triple hybrid scheme based on three different-kind integral relations was used for the first time in solving the heat-conduction problem with a convective boundary condition. All the solutions obtained with the hybrid schemes proposed are characterized by good approximation properties. The new hybrid schemes can be used in solving different practical problems.

A nonconforming Trefftz virtual element method for the Helmholtz problem

We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e. functions belonging to the kernel of the Helmholtz operator. The global trial and test spaces are not fully discontinuous, but rather interelement continuity is imposed in a nonconforming fashion. Although their functions are only implicitly defined, as typical of the VEM framework, they contain discontinuous subspaces made of functions known in closed form and with good approximation properties (plane-waves, in our case). We carry out an abstract error analysis of the method, and derive [Formula: see text]-version error estimates. Moreover, we initiate its numerical investigation by presenting a first test, which demonstrates the theoretical convergence rates.

Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media

Our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media. We consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures. To construct our upscaled model, we will apply the nonlocal multicontinua (NLMC) upscaling technique. The upscaled coefficients are obtained by using some multiscale basis functions, which are solutions of local problems defined on oversampled regions. For each continuum within a target coarse element, we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers, with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise. The resulting multiscale basis functions have been shown to have good approximation properties. To illustrate the idea of our approach, we will consider a dual continua background model consisting of discrete fractures in two space dimensions, that is, we consider a system with three continua. We will present several numerical examples, and they show that our method is able to capture the interaction between matrix continua and discrete fractures on the coarse grid efficiently.

Computing the density of states for optical spectra of molecules by low-rank and QTT tensor approximation

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm–Dancoff approximation (TDA) of the Bethe–Salpeter equation (BSE). The presented approach for approximating the DOS is based on two main techniques. First, we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block-diagonal plus low-rank matrix structure described in [6,3] for the BSE/TDA problem. This allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size N can be accurately represented in a low rank quantized tensor train (QTT) format that can be determined through a least squares fitting procedure. The QTT tensor provides good approximation properties for strictly oscillating DOS functions with multiple gaps, in contrast to interpolation by problem independent functions like polynomials, trigonometric functions, etc. Moreover, the QTT approximant requires asymptotically much fewer (e.g., O(log⁡N)) functional calls compared with the full grid size N. Numerical tests indicate that the QTT approach yields accurate recovery of DOS associated with problems that contain relatively large spectral gaps. The QTT tensor rank only weakly depends on the size of a molecular system that paves the way for treatment of large-scale spectral problems.

DISCRETE ATOMIC COMPRESSION OF DIGITAL IMAGES

The subject matter of this paper is the discrete atomic compression (DAC) of digital images, which is a lossy compression process based on the discrete atomic transform (DAT). The goal is to investigate the efficiency of the DAC algorithm. We solve the following tasks: to develop a general compression scheme using discrete atomic transform and to compare the results of DAC and JPEG algorithms. In this article, we use the methods of digital image processing, atomic function theory, and approximation theory. To compare the efficiency of DAC with the JPEG compression algorithm we use the sets of the classic test images and the classic aerial images. We analyze compression ratio (CR) and loss of quality, using uniform (U), root mean square (RMS) and peak signal to noise ratio (PSNR) metrics. DAC is an algorithm with flexible parameters. In this paper, we use “Optimal” and “Allowable” modes of this algorithm and compare them with the corresponding modes of JPEG. We obtain the following results: 1) DAC is much better than JPEG by the U-criterion of quality loss; 2) there are no significant differences between DAC and JPEG by RMS and PSNR criterions; 3) the compression ratio of DAC is much higher than the compression ratio of JPEG. In other words, the DAC algorithm saves more memory than the JPEG compression algorithm with not worse quality results. These results are due to the fundamental properties of atomic functions such as good approximation properties, the high order of smoothness and existence of locally supported basis in the spaces of atomic functions. Since generalized Fup-functions have the same convenient properties, it is clear that such compression results can be achieved by application of a generalized discrete atomic transform, which is based on these functions. We also discuss the obtained results in the terms of approximation theory and function theory. Conclusions: 1) it is possible to achieve better results with DAC than with JPEG; 2) application of DAC to image compression is more preferable than JPEG in the case when it is planned to use recognition algorithms; 3) further development and investigation of the DAC algorithm are promising

A Genetic Algorithm Based Fuzzy Inference System for Pattern Classification and Rule Extraction

Setting fuzzy rules is one of the paramount techniques in the design of a fuzzy system. For a simple system, fuzzy if-then rules are usually derived from the human experts. However, in the event of having multiple variables coupled with a few features, the classification problem will be getting more sophisticated, as a result human expert may not be able to derive proper rules. This paper presents a genetic-algorithm-based fuzzy inference system for extracting highly comprehensible fuzzy rules to be implemented in human practices without detailed computation (hereafter denoted as GA-FIS). The impetus for developing a new and efficient GA-FIS model arises from the need of constructing fuzzy rules directly from raw data sets that combines good approximation and classification properties with compactness and transparency. Therefore, our proposed GA-FIS method will first define the membership functions with logical interpretation which is amendable by domain experts to human understanding, and then genetic algorithm serves as an optimization tool to construct the best combination of rules in fuzzy inference system that can achieve higher classification accuracy and gain better interpretability. The proposed approach is applied to various benchmark and real world problems and the results show its validity.

ATOMIC FUNCTIONS AND THEIR GENERALIZATIONS IN DATA PRO-CESSING: FUNCTION THEORY APPROACH

Theory of atomic functions, which are solutions with a compact support of the linear functional differential equations with a constant coefficients and linear transforms of the argument, was created in the 70's of the 20th century because of the necessity to solve different applied problems, in particular, boundary value problems. One of the reasons for the appearance of atomic functions and some other classes of functions was the inability to use such classic approximation tools as algebraic and trigonometric polynomials. V.A. Rvachev up-function is the most famous and widely used atomic function. With the passage of time and the development of technologies, the existing problems are changing and fundamentally new problems appear. For instance, now the big data processing is one of the most important problems. It should be mentioned that the suitable mathematical tools must be used to obtain the desired result. This paper is devoted to fundamentals of applications of some atomic functions and their generalizations to data processing and lossy information compression. In this paper we consider the main properties of these functions from the function theory point of view and give their interpretation with respect to information processing. Smoothness, compact support and good approximation properties are the main advantages of atomic functions. Moreover, the spaces of atomic functions and the spaces of generalized Fup-functions, which are the natural generalization of V.A. Rvachev Fup-functions, are asymptotically extremal for approximation of periodic differentiable functions. This means that in the terms of A.N. Kolmogorov width these functions are just as effective as classic trigonometric polynomials {1, cos(nx), sin(nx)}. Hence, the replacement of discrete transforms based on trigonometric functions on similar transforms based on atomic functions and generalized Fup-functions is quite promising. For this purpose we introduce discrete atomic transform and generalized discrete atomic transform. We also discuss the dependence of data processing results on order of smoothness and size of support of the used functions. Theoretical justification of the application of some atomic functions and generalized Fup-functions to data processing and, in particular, lossy data compression is the main result of this paper

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider∂tu−divA(x,∇u)=f∈L1(ΩT), on a Lipschitz bounded domain in RN. The growth of the weakly monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M⁎ (neither Δ2, nor ∇2). The condition we impose on M is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.

Legendre orthogonal polynomials approximation of system gramians and its application to balanced truncation

ABSTRACTThe paper presents a procedure for computation of system gramians by using Legendre orthogonal polynomials approximation of the system state impulse and output responses. The proposed approach is trajectory based and relies on the system state and output trajectories snapshots selected either by experiment or computer simulation. It is defined in deterministic settings as opposed to similar approaches defined in stochastic settings by using the data covariance matrix. The advantage of using orthogonal series approximation for the gramians is to avoid solving the usual Lyapunov equations. The proposed method can be equally well applied to linear time-invariant as well as time-varying systems, and even to unstable systems, since the gramians approximation is performed on a finite interval of time. When the observation interval contains the whole energy of the system state impulse and output responses, the proposed method gives similar results as the gramians computed by solving Lyapunov equations. Several experiments are performed showing the good approximation properties of the presented method.

GENERALIZED ATOMIC WAVELETS

The problem of big data sets processing is considered. Efficiency of algorithms depends mainly on the appropriate mathematical tools. Now there exists a wide variety of different constructive tools for information analysis. Atomic functions are one of them. Theory of atomic functions was developed by V. A. Rvachev and members of his scientific school. A number of results, which prove that application of atomic functions is reasonable, were obtained. In particular, atomic functions are infinitely differentiable. This property is quite useful for smooth data processing (for example, color photos). Also, these functions have a local support, which allows to decrease complexity of numerical algorithms. Besides, it was shown that spaces of atomic functions have good approximation properties, which can reduce the error of computations. Hence, application of atomic functions is perspective. There are different ways to use atomic functions and their generalizations in practice. One such approach is a construction and application of wavelet-like structures. In this paper, generalized atomic wavelets are constructed using generalized Fup-functions and formulas for their evaluation are obtained. Also, the main properties of generalized atomic wavelets are presented. In addition, it is shown that these wavelets are smooth functions with a local support and have good approximation properties. Furthermore, the set of generalized atomic wavelets is a wide class of functions with flexible parameters that can be chosen according to specific needs. This means that the constructive analysis tool, which is introduced in this paper, gives researches and developers of algorithms flexible possibilities of adapting to the specifics of various problems. In addition, the problem of representation of data using generalized atomic wavelets is considered. Generalized atomic wavelets expansion of data is introduced. Such an expansion is a sum of trend or principal value function and several functions that describe the corresponding frequencies. The remainder term, which is an error of approximation of data by generalized atomic wavelets, is small. To estimate its value the inequalities from the previous papers of V. A. Rvachev, V. O. Makarichev and I. V. Brysina can be used

Recovered finite element methods

We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has a number of attractive features over both classical finite element and discontinuous Galerkin approaches, most important of which is its potential to produce stable conforming approximations in a variety of settings. Moreover, for special choices of recovery operators, R-FEM produces the same approximate solution as the classical conforming finite element method, while, trivially, one can recast (primal formulation) discontinuous Galerkin methods. A priori error bounds are shown for linear second order boundary value problems, verifying the optimality of the proposed method. Residual-type a posteriori bounds are also derived, highlighting the potential of R-FEM in the context of adaptive computations. Numerical experiments highlight the good approximation properties of the method in practice. A discussion on the potential use of R-FEM in various settings is also included.

Time-Varying Nonlinear Causality Detection Using Regularized Orthogonal Least Squares and Multi-Wavelets With Applications to EEG

A new transient Granger causality detection method is proposed based on a time-varying parametric modeling framework, and is applied to the real EEG signals to reveal the causal information flow during motor imagery (MI) tasks. The time-varying parametric modeling approach employs a nonlinear autoregressive with external input model, whose parameters are approximated by a set of multi-wavelet basis functions. A regularized orthogonal least squares (ROLS) algorithm is then used to produce a parsimonious or sparse regression model and estimate the associated model parameters. The time-varying Granger causality between nonstationary signals can be detected accurately by making use of both the good approximation properties of multi-wavelets and the good generalization performance of the ROLS in the presence of high-level noise. Two simulation examples are presented to demonstrate the effectiveness of the proposed method for both linear and nonlinear causal detection respectively. The proposed method is then applied to real EEG signals of MI tasks. It follows that transient causal information flow over the time course between various sensorimotor related channels can be successfully revealed during the whole reaction processes. Experimental results from these case studies confirm the applicability of the proposed scheme and show its utility for the understanding of the associated neural mechanism and the potential significance for developing MI tasks based brain-computer interface systems.

Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider−divA(x,∇u)=f∈L1(Ω), on a Lipschitz bounded domain in RN. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M⁎ (neither Δ2, nor ∇2). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation

To numerically solve the Burgers’ equation, in this paper we propose a general method for constructing wavelet bases on the interval [0,1] derived from symmetric biorthogonal multiwavelets on the real line. In particular, we obtain wavelet bases with simple structures on the interval [0,1] from the Hermite cubic splines. In comparison with all other known constructed wavelets on the interval [0,1], our constructed wavelet bases on the interval [0,1] from the Hermite cubic splines not only have good approximation and symmetry properties with extremely short supports, but also employ a minimum number of boundary wavelets with a very simple structure. These desirable properties make them to be of particular interest in numerical algorithms. Our constructed wavelet bases on the interval [0,1] are then used to solve the nonlinear Burgers’ equation. Our method is based on the finite difference formula combined with the collocation method. Therefore, our proposed numerical scheme in this paper is abbreviated as MFDCM (Mixed Finite Difference and Collocation Method). Some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy.

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with 1≤q≤ź$1\leq q\leq \infty $. In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.