Multiwavelet Bases Research Articles

Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes

In this paper, we introduce a new global troubled-cell indicator for the discontinuous Galerkin (DG) method in one- and two-dimensions. This is done by taking advantage of the global expression of the DG method and re-expanding it in terms of a multiwavelet basis, which is a sum of the global average and finer details on different levels. Examining the higher level difference coefficients acts as a troubled-cell indicator, thus avoiding unnecessary increased computational cost of a new expansion. In two-dimensions the multiwavelet decomposition uses combinations of scaling functions and multiwavelets in the x- and y-directions for improved troubled-cell indication. By using such a troubled-cell indicator, we are able to reduce the computational cost by avoiding limiting in smooth regions. We present numerical examples in one- and two-dimensions and compare our troubled-cell indicator to the subcell resolution technique of Harten (1989) [11] and the shock detector of Krivodonova et al. (2004) [18], which were previously investigated by Qiu and Shu (2005) [23].

BIORTHOGONAL MULTIWAVELETS RELATED BY DIFFERENTIATION

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.

Maximal-overlap adaptive multiwavelet for detecting transient vibration responses from dynastic signal of rotating machineries

Vibration signal is an important prerequisite for mechanical fault detection. However, early stage defect of rotating machineries is difficult to identify because their incipient energy is interfered with background noises. Multiwavelet is a powerful tool used to conduct non-stationary fault feature extraction. However, the existing predetermined multiwavelet bases are independent of the dynamic response signals. In this paper, a constructing technique of vibration data-driven maximal-overlap adaptive multiwavelet (MOAMW) is proposed for enhancing the extracting performance of fault symptom. It is able to derive an optimal multiwavelet basis that best matches the critical non-stationary and transient fault signatures via genetic algorithm. In this technique, two-scale similarity transform (TST) and symmetric lifting (SymLift) scheme are combined to gain high designing freedom for matching the critical faulty vibration contents in vibration signals based on the maximal fitness objective. TST and SymLift can add modifications to the initial multiwavelet by changing the approximation order and vanishing moment of multiwavelet, respectively. Moreover, the beneficial feature of the MOAWM lies in that the maximal-overlap filterbank structure can enhance the periodic and transient characteristics of the sensor signals and preserve the time and frequency analyzing resolution during the decomposition process. The effectiveness of the proposed technique is validated via a numerical simulation as well as a rolling element bearing with an outer race scrape and a gearbox with rub fault.

Identification and frequency domain analysis of non-stationary and nonlinear systems using time-varying NARMAX models

This paper introduces a new approach for nonlinear and non-stationary (time-varying) system identification based on time-varying nonlinear autoregressive moving average with exogenous variable (TV-NARMAX) models. The challenging model structure selection and parameter tracking problems are solved by combining a multiwavelet basis function expansion of the time-varying parameters with an orthogonal least squares algorithm. Numerical examples demonstrate that the proposed approach can track rapid time-varying effects in nonlinear systems more accurately than the standard recursive algorithms. Based on the identified time domain model, a new frequency domain analysis approach is introduced based on a time-varying generalised frequency response function (TV-GFRF) concept, which enables the analysis of nonlinear, non-stationary systems in the frequency domain. Features in the TV-GFRFs which depend on the TV-NARMAX model structure and time-varying parameters are investigated. It is shown that the high-dimensional frequency features can be visualised in a low-dimensional time–frequency space.

A stochastic Galerkin method for the Euler equations with Roe variable transformation

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

Legendre multiwavelet collocation method for solving the linear fractional time delay systems

Abstract In this article the Legendre multiwavelet basis with aid of collocation method has been applied to give approximate solution for fractional delay systems. The properties of Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of algebraic system. Numerical results and comparison with exact solutions in the cases when we have exact solution are given in test examples in order to demonstrate the applicability and efficiency of the method.

COMPLETE SPACE QUANTUM CHEMISTRY BY MULTIRESOLUTION MULTIWAVELET BASIS SET

Quantum chemistry program based on Multiresolution Multiwavelet (MRMW) basis set is much simpler and can be more efficient than the conventional one based on Gaussian basis set in molecular geometry optimization because the Hellmann Fynman Theorem (HFT) holds for the complete space which MRMW provides. It is numerically shown that large Gaussian basis sets can provide the chemical information with chemical accuracy for very small molecular systems despite the incompleteness of their basis set. However, even for the relatively small size molecule of CH4 , the error from the basis set incompleteness results in an error larger than an acceptable chemical accuracy. The advantage of the MRMW complete basis set in vibrational analysis is also discussed.

Confinement effects of solvation on a molecule physisorbed on a polarizable continuum particle

We investigate the effects of solvation on the static properties of a molecule physisorbed on a spherical polarizable continuum particle, with a static dielectric constant. The effective polarizability of the physisorbed molecule is enhanced by a factor of 105 in vacuo and by only 102 when solvated. The electrostatic interaction between molecules and the polarizable continuum particle (PCP) results in a magnified reflected electric field with a magnitude that depends on the PCP-molecule separation. A similar and greater magnification effect in reflected field also results from the PCP-field (external electric field) interaction. The variation of the polarizability of the molecules with respect to the changes in their environment illustrates the importance of electrostatic interaction in the enhancement of the effective polarizability. In the course of the investigation, the solute and the continuous body are represented with the same adaptive multi-wavelet basis functions, thereby, within the user specified precision, eliminating basis set error.

Divergence-Free Multiwavelets on the Half Plane

We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition.

Image Denoising in Industry Based on Multiwavelet Riesz Bases

Damage testing of components is a key point in many industry fields. In some cases, endoscope is used to inspect the damage part, while the images are often noised. In this paper, we focus on industrial image denoising based on multiwavelet Riesz bases. Starting from compactly supported vector refinement equation, we provide a characterization to form two Riesz bases and an example is given. Based on example Riesz bases, we research industrial endoscope image denoising and get satisfying result.

Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets

This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and fractional derivatives of the dual flatlet multiwavelets by multiplying some matrices is suggested. The system is then used to solve a fractional convection–diffusion equation. The biorthogonality and high vanishing moments properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from the other bases.

Implicit solvation models in a multiresolution multiwavelet basis

We describe and present results of the implementation of the surface and volume polarization for electrostatics (SVPE) solvation model. Unlike most other implementations of the solvation model where the solute and the solvent are described with multiple numerical representations, our implementation uses a multiresolution, adaptive multiwavelet basis to describe both the solute and the solvent. This requires reformulation to use integral equations throughout as well as a conscious management of numerical properties of the basis.

Multiwavelet density estimation

Accurate density estimation methodologies play an integral role in a variety of scientific disciplines with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for density estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of wavelet bases is that they cannot be simultaneously symmetric, orthogonal, and compactly supported. Multiwavelets—more general, vector-valued constructions of wavelets—overcome this disadvantage, making them natural choices for estimating density functions, many of which exhibit local symmetries around features such as a mode. We extend the methodology of wavelet density estimation to use multiwavelet bases and illustrate several empirical examples of multiwavelet density estimation.

Basis set limit computation of dynamic polarizability at near‐resonance region

AbstractWe evaluate dynamic molecular polarizability at near‐resonance region using multiresolution multiwavelet (MRMW) basis set. The results obtained by traditional Gaussian basis sets are compared against the MRMW one. For perpendicular component, d‐aug functions seem to be necessary. Very large Gaussian basis sets beyond d‐aug‐cc‐pvqz can provide the results close to the complete space value obtained by the MRMW even for near‐resonance region. Improvement by enlargement of Gaussian functions is less systematic in the parallel component than in the perpendicular component. The competitiveness of the MRMW method against Gaussian methods in the computational efficiency for property evaluation is also discussed. © 2012 Wiley Periodicals, Inc.

A new implementation of dynamic polarizability evaluation using a multi-resolution multi-wavelet basis set

An algorithm to evaluate the response function on the space spanned by Multi-resolution Multi-wavelet (MRMW) basis functions is implemented in a new efficient quantum chemical program system. Dynamic electric dipole polarizability of a molecule is evaluated for a range of applied optical frequencies as a preliminary application of the implementation. The results obtained by the MRMW provide the one in near completeness limit. We also compare a few examples of the dynamic polarizabilities evaluated using conventional Gaussian basis set functions with the one evaluated by the MMMW basis functions. For small frequencies, there is no significant difference among them and large Gaussian results are close to the MRMW value, but they diverge from the MRMW value when the applied frequency approaches to the resonance region.

A new class of multi-wavelet bases: V-system

The V-system is a complete orthogonal system of functions defined on the interval [0, 1], generated by finite Legendre polynomials and the dilation and translation of a function generator, which consists of a finite number of continuous and discontinuous functions. The V-system has interesting properties, such as orthogonality, symmetry, completeness and short compact support. It is shown in this paper that the V-system is essentially a special multi-wavelet basis. As a result, some basic properties of the V-system are established through the well-developed theory of multi-wavelets. From this point of view, more other V-systems are constructed.

Identification of Time-Varying Systems Using Multi-Wavelet Basis Functions

This brief introduces a new parametric modelling and identification method for linear time-varying systems using a block least mean square (LMS) approach where the time-varying parameters are approximated using multi-wavelet basis functions. This approach can be applied to track rapidly or even sharply varying processes and is developed by combining wavelet approximation theory with a block LMS algorithm. Numerical examples are provided to show the effectiveness of the proposed method for dealing with severely nonstationary processes. Application of the proposed approach to a real mechanical system indicates better tracking capability of the multi-wavelet basis function algorithm compared with the normalized least squares or recursive least squares routines.

A multiwavelet Galerkin boundary element method for the stationary Stokes problem in 3D

In this paper, a multiwavelet Galerkin boundary element method is presented for the fast solution of the stationary Stokes problem in three dimensions. Piecewise linear discontinuous multiwavelet bases are constructed on each patch of piecewise smooth surface individually, which allow easy and efficient evaluation of the matrix entries. Because of the use of the multiwavelets, the system matrix can be compressed to O ( N) ( N denotes the number of unknowns) nonzero entries without compromising the order of convergence as for the conventional Galerkin boundary element method. Numerical results of two test samples are given to demonstrate the availability of the present method.

The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems

In this article the Legendre multiwavelet basis with the aid of a collocation method has been applied to give the approximate solution for the fractional optimal control problems (FOCPs). The properties of the Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of an algebraic system. Numerical results and a comparison with the exact solution in the cases when we have an exact solution are given to demonstrate the applicability and efficiency of the new method.

Adaptive discontinuous Galerkin methods in multiwavelets bases

We use a multiwavelet basis with the Discontinuous Galerkin (DG) method to produce a multi-scale DG method. We apply this Multiwavelet DG method to convection and convection–diffusion problems in multiple dimensions. Merging the DG method with multiwavelets allows the adaptivity in the DG method to be resolved through manipulation of multiwavelet coefficients rather than grid manipulation. Additionally, the Multiwavelet DG method is tested on non-linear equations in one dimension and on the cubed sphere.