Multiwavelet Bases Research Articles

A fractional multi-wavelet basis in Banach space and solving fractional delay differential equations

In this article, we construct a fractional multi-wavelet basis based on Legendre polynomials to solve fractional delay linear and nonlinear differential equations. For this we introduce an orthonormal fractional basis for Banach space L2[0,1] with suitable inner product which make it effective to decrease computational operations and increase accuracy to find approximate solution of the equations. Also, solving fractional problems by orthogonal basis such as Legendre polynomials has a lower accuracy in comparison with fractional basis. Finally, some examples are solved to show the high accuracy of the presented method, and also to compare with some other works.

Benchmarking Correlation-Consistent Basis Sets for Frequency-Dependent Polarizabilities with Multiresolution Analysis.

This paper presents the first converged frequency-dependent HF polarizability results for general molecules, on a set of 89 closed-shell atoms and molecules. The solver employs multiresolution analysis (MRA) in a multiwavelet basis to compute both ground and response states to a guaranteed precision, which are validated against independent numerical grid calculations on atoms and linear molecules. The MRA ground-state energies and response properties are used to evaluate results in correlation-consistent basis sets up to 5Z augmented with either single or double diffuse functions and core-polarization functions. Systematic trends are revealed through consideration of chemical composition as well as the use of machine learning to cluster convergence trends, the latter suggesting the possibility of learning and correcting basis-set error.

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

Full Breit Hamiltonian in the Multiwavelets Framework.

New techniques in core-electron spectroscopy are necessary to resolve the structures of oxides of f-elements and other strongly correlated materials that are present only as powders and not as single crystals. Thus, accurate quantum chemical methods must be developed to calculate core spectroscopic properties in such materials. In this contribution, we present an important development in this direction, extending our fully adaptive real-space multiwavelet basis framework to tackle the four-component Dirac-Coulomb-Breit Hamiltonian. We show that multiwavelets can reproduce one-dimensional grid-based approaches. They are however a fully three-dimensional approach which can later be extended to molecules and materials. Our multiwavelet implementation attained precise results irrespective of the chosen nuclear model, provided that the error threshold is tight enough and that the chosen polynomial basis is sufficiently large. Furthermore, our results confirmed that in two-electron species, the magnetic and Gauge contributions from s-orbitals are identical in magnitude and can account for the experimental evidence from K and L edges.

A New Method for Dynamical System Identification by Optimizing the Control Parameters of Legendre Multiwavelet Neural Network

Wavelet neural networks have been widely applied to dynamical system identification fields. The most difficult issue lies in selecting the optimal control parameters (the wavelet base type and corresponding resolution level) of the network structure. This paper utilizes the advantages of Legendre multiwavelet (LW) bases to construct a Legendre multiwavelet neural network (LWNN), whose simple structure consists of an input layer, hidden layer, and output layer. It is noted that the activation functions in the hidden layer are adopted as LW bases. This selection if based on the its rich properties of LW bases, such as piecewise polynomials, orthogonality, various regularities, and more. These properties contribute to making LWNNs more effective in approximating the complex characteristics exhibited by uncertainties, step, nonlinear, and ramp in the dynamical systems compared to traditional wavelet neural networks. Then, the number of selection LW bases and the corresponding resolution level are effectively optimized by the simple Genetic Algorithm, and the improved gradient descent algorithm is implemented to learn the weight coefficients of LWNN. Finally, four nonlinear dynamical system identification problems are applied to validate the efficiency and feasibility of the proposed LWNN-GA method. The experiment results indicate that the LWNN-GA method achieves better identification accuracy with a simpler network structure than other existing methods.

LW-BPNN: A Novel Feature Extraction Method for Rolling Bearing Fault Diagnosis

Efficiently diagnosing bearing faults is of paramount importance to enhance safety and reduce maintenance costs for rotating machinery. This paper introduces a novel bearing fault diagnosis method (LW-BPNN), which combines the rich properties of Legendre multiwavelet bases with the robust learning capabilities of a BP neural network (BPNN). The proposed method not only addresses the limitations of traditional deep networks, which rely on manual feature extraction and expert experience but also eliminates the complexity associated with designing and training deep network architectures. To be specific, only two statistical parameters, root mean square (RMS) and standard deviation (SD), are calculated on different Legendre multiwavelet decomposition levels to thoroughly represent more salient and comprehensive fault characteristics by using several scale and wavelet bases with various regularities. Then, the mapping relation between the extracted features and the health conditions of the bearing is automatically learned by the simpler BPNN classifier rather than the complex deep network structure. Finally, a few experiments on a popular bearing dataset are implemented to verify the effectiveness and robustness of the presented method. The experimental findings illustrate that the proposed method exhibits a high degree of precision in diagnosing various fault patterns. It outperforms other methods in terms of diagnostic accuracy, making it a viable and promising solution for real-world industrial applications in the field of rotating machinery.

ECG Sparsity Evaluation on a Multiwavelet Basis

Abstract In this paper, an evaluation of the multiwavelet basis’ capability to represent the ECG signal sparsely was performed. The paper includes the mathematical formulation of sparsity, a brief introduction to the multiwavelet transform, as well as details about the simulation setup used for evaluation. Throughout the paper, various multiwavelets were investigated. The reported results show that the BAT and DB multiwavelets performed well, thus they could be used in the ECG signal sparsification. The investigation also focused on the ECG signals displaying deformations associated with illnesses. Preliminary results suggest that multiwavelets may prove beneficial for diverse processing of ECG signals.

A multiwavelet-based sparse time-varying autoregressive modeling for motor imagery EEG classification

Brain-computer Interface (BCI) system based on motor imagery (MI) heavily relies on electroencephalography (EEG) recognition with high accuracy. However, modeling and classification of MI EEG signals remains a challenging task due to the non-linear and non-stationary characteristics of the signals. In this paper, a new time-varying modeling framework combining multiwavelet basis functions and regularized orthogonal forward regression (ROFR) algorithm is proposed for the characterization and classification of MI EEG signals. Firstly, the time-varying coefficients of the time-varying autoregressive (TVAR) model are precisely approximated with the multiwavelet basis functions. Then a powerful ROFR algorithm is employed to dramatically alleviate the redundant model structure and accurately recover the relevant time-varying model parameters to obtain high resolution power spectral density (PSD) features. Finally, the features are sent to different classifiers for the classification task. To effectively improve the accuracy of classification, a principal component analysis (PCA) algorithm is utilized to determine the best feature subset and Bayesian optimization algorithm is performed to obtain the optimal parameters of the classifier. The proposed method achieves satisfactory classification accuracy on the public BCI Competition II Dataset III, which proves that this method potentially improves the recognition accuracy of MI EEG signals, and has great significance for the construction of BCI system based on MI.

On the Multiwavelets Galerkin Solution of the Volterra–Fredholm Integral Equations by an Efficient Algorithm

We develop the multiwavelet Galerkin method to solve the Volterra–Fredholm integral equations. To this end, we represent the Volterra and Fredholm operators in multiwavelet bases. Then, we reduce the problem to a linear or nonlinear system of algebraic equations. The interesting results arise in the linear type where thresholding is employed to decrease the nonzero entries of the coefficient matrix, and thus, this leads to reduction in computational efforts. The convergence analysis is investigated, and numerical experiments guarantee it. To show the applicability of the method, we compare it with other methods and it can be shown that our results are better than others.

A novel time-varying modeling and signal processing approach for epileptic seizure detection and classification

Electroencephalogram (EEG) signal analysis plays an essential role in detecting and understanding epileptic seizures. It is known that seizure processes are nonlinear and non-stationary, discriminating between rhythmic discharges and dynamic change is a challenging task in EEG-based seizure detection. In this paper, a new time-varying (TV) modeling framework, based on an autoregressive (AR) model structure, is proposed to characterize and analyze EEG signals. The TV parameters of the AR model are approximated through a multi-wavelet basis function expansion (MWBF) approach. An effective ultra-regularized orthogonal forward regression (UROFR) algorithm is employed to significantly reduce and refine the resulting expanded model. Given a time-varying process, the proposed TVAR–MWBF–UROFR method can generate a parsimonious TVAR model, based on which a high-resolution power spectrum density (PSD) estimation can be obtained. Informative features are then defined and extracted from the PSD estimation. The TVAR–MWBF–UROFR method is applied to a number of real EEG datasets; features obtained from these datasets are then used for seizure detection and classification. To make the results more accurate and reliable, a PCA algorithm is adopted to select the optimal feature subset, and a Bayesian optimization technique based on the Gaussian process is performed to determine the coefficients associated with each of the classifiers. The performance of the proposed method is tested on two benchmark datasets, and the experimental results indicate that TVAR–MWBF–UROFR outperforms the compared state-of-the-art classifiers in terms of accuracy, specificity, sensitivity and robustness.

Nonstationary Multiwavelets and Multiwavelet Packets in Sobolev Space $$H^s({\mathbb {R}}^d)$$

Nonstationary multiresolution analysis has different scaling and wavelet functions at different scales. In Bastin and Laubin (Duke Math J 87:481–508, 1997), nonstationary wavelets in Sobolev spaces were introduced. Here we construct a nonstationary multiresolution analysis of multiwavelets for the higher-dimensional Sobolev spaces. We give the construction of nonstationary multiwavelets and derived their orthogonal properties in Sobolev space $$H^s({\mathbb {R}}^d)$$ . We perform some splitting trick over nonstationary multiwavelets and multiwavelet spaces to construct nonstationary multiwavelet packet and multiwavelet packet spaces in Sobolev space $$H^s({\mathbb {R}}^d)$$ . A nonstationary multiwavelet and multiwavelet packet expansion of Bessel potentials is given as an application of an orthonormal multiwavelet and multiwavelet packet basis in Sobolev space.

Compact operators for existence of solution and projection method with multi-wavelet bases to solve [formula omitted] and error analysis in Sobolev space

In this article, we discuss the existence of solution using compact operators for Fredholm integral equations system (F.IES). Also we use Galerkin method in a n-dimensional Hilbert space to solve this problem. To reduce the computational operations, we use orthonormal multi-wavelet bases which are constructed by Chebyshev polynomials. Since Chebyshev multi-wavelet bases functions are orthonormal, they help us decrease the operations used in discretizing the integral equations system to an algebraic equations system. The algebraic system contains sparse matrices and it leads to a numerical solution with a high accuracy. Also we consider error analysis in a Sobolev space. Finally for validity and applicability of the above proposed method we compare our results with homotopy perturbation method and Taylor-series expansion method.

A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel

A stable least residue method for solving a nonlinear fractional integro-differential equation with a weakly singular kernel in the reproducing kernel space is proposed. To solve the equation, the multiwavelets bases in the reproducing kernel space are constructed based on the cubic Legendre wavelets in L2[0,1]. The best approximate solution could be obtained by solving the normal equation. Meanwhile, we provide the adaptive convergence order and the stability proof of the scheme. It is encouraging that the new method is stable and the accuracy of the method is preserved even in the case if the solution has fast oscillations. Four examples strongly demonstrate the validity of the scheme.

Static Polarizabilities at the Basis Set Limit: A Benchmark of 124 Species.

Benchmarking molecular properties with Gaussian-type orbital (GTO) basis sets can be challenging, because one has to assume that the computed property is at the complete basis set (CBS) limit, without a robust measure of the error. Multiwavelet (MW) bases can be systematically improved with a controllable error, which eliminates the need for such assumptions. In this work, we have used MWs within Kohn–Sham density functional theory to compute static polarizabilities for a set of 92 closed-shell and 32 open-shell species. The results are compared to recent benchmark calculations employing the GTO-type aug-pc4 basis set. We observe discrepancies between GTO and MW results for several species, with open-shell systems showing the largest deviations. Based on linear response calculations, we show that these discrepancies originate from artifacts caused by the field strength and that several polarizabilies from a previous study were contaminated by higher order responses (hyperpolarizabilities). Based on our MW benchmark results, we can affirm that aug-pc4 is able to provide results close to the CBS limit, as long as finite difference effects can be controlled. However, we suggest that a better approach is to use MWs, which are able to yield precise finite difference polarizabilities even with small field strengths.

A probability density estimation algorithm on multiwavelet for the high-resolution ADC

ADC (Analog-to-digital converter) is an important component in measuring instruments that converts the analog signals to digital form. The actual effective vertical resolution of ADC is one of the most important indicators for evaluating the ADC performance. However, it is restricted by various factors such as external disturbance, environmental interference, clock jitter, transmission delay, etc. These problems lead to a lower ADCs’ effective resolution than the ideal value. This paper proposes an improved algorithm dedicated to decrease errors in measured signals. In this algorithm, we make Multiwavelet decomposition to get the probability density function of measured signal first, then use singular value decomposition to remove the noise related frequency characteristic values in the multiwavelet basis coefficients, next create a Bayesian fusion iteration process to optimize the probability density function and build a confidence interval for elimination of the noise data, and finally the signal is reconstructed and average filtered for output. Through simulation experiments we show an improvement of the vertical resolution and SNR (signal-to-noise ratio) in comparison to the conventional averaging filter, high resolution acquisition and oversampling methods.

Dirac-Fock calculations on molecules in an adaptive multiwavelet basis.

We report the first fully numerical approach for relativistic quantum chemical calculations applicable to molecules. The approach uses an adaptive basis of multiwavelet functions to solve the full four-component Dirac-Coulomb equation to a user-specified accuracy. The accuracy of the code is demonstrated by comparison with ground state energy calculations of atoms performed in GRASP, and the applicability to molecules is shown via ground state calculations of some simple molecules, including water analogs up to H2Po. In the case of molecules, comparison is made with Gaussian basis set calculations in DIRAC.

Standard pairs and existence of symmetric multiscaling functions

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

On derivatives of smooth functions represented in multiwavelet bases

We construct high-order derivative operators for smooth functions represented via discontinuous multiwavelet bases. The need for such operators arises in order to avoid artifacts when computing functionals involving high-order derivatives of solutions of integral equations. Previously high-order derivatives had to be formed by repeated application of a first-derivative operator that, while uniquely defined, has a spectral norm that grows quadratically with polynomial order and, hence, greatly amplifies numerical noise (truncation error) in the multiwavelet computation. The new constructions proceed via least-squares projection onto smooth bases and provide substantially improved numerical properties as well as permitting direct construction of high-order derivatives. We employ either b-splines or bandlimited exponentials as the intermediate smooth basis, with the former maintaining the concept of approximation order while the latter preserves the pure imaginary spectrum of the first-derivative operator and provides more direct control over the bandlimit and accuracy of computation. We demonstrate the properties of these new operators via several numerical tests as well as application to a problem in nuclear physics.

Adaptive algorithm for electronic structure calculations using reduction of Gaussian mixtures

We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected in the process of solving equations. Using a fixed basis leads to the so-called basis error since orbitals may not lie entirely within the linear span of the basis. To avoid such an error, multiresolution bases are used in adaptive algorithms so that basis functions are selected from a fixed collection of functions, large enough to approximate solutions within any user-selected accuracy. Our new method achieves adaptivity without using a multiresolution basis. Instead, as part of an iteration to solve nonlinear equations, our algorithm selects the ‘best’ subset of linearly independent terms of a Gaussian mixture from a collection that is much larger than any possible basis since the locations and shapes of the Gaussian terms are not fixed in advance. Approximating an orbital within a given accuracy, our algorithm yields significantly fewer terms than methods using multiresolution bases. We demonstrate our approach by solving the Hartree–Fock equations for two diatomic molecules, HeH + and LiH, matching the accuracy previously obtained using multiwavelet bases.

A new fast method of solving the high dimensional elliptic eigenvalue problem

In this paper, we develop a novel method to solve the elliptic eigenvalue problem. The univariate multi-wavelet approach provides a simple diagonal preconditioner for second order elliptic problems, which gives an almost constant condition number for efficiently solving the corresponding linear system. Here, we shall consider a new fast numerical approach for approximating the smallest elliptic eigenvalue by using the multi-wavelet basis in the multi-grid discretization scheme. Moreover, we develop a new numerical scheme coupled with sparse grids method in the calculation. This new approach saves storage in degrees of freedom and thus is more efficient in the computation. Several numerical experiments are provided for validating the proposed numerical scheme, which show that our method retains the optimal convergence rate for the smallest eigenvalue approximation with much less computational cost comparing with ’eigs’ in full grids.