Adaptive Wavelet-based Method Research Articles

Wavelet-based adaptive wall-modeled large eddy simulation method for compressible turbulent flows

We develop a wavelet-based adaptive wall-modeled large eddy simulation (WA-WMLES) method to overcome the stringent restriction on step size for time integration, caused by mesh resolution requirement to resolve inner viscous sublayer, which becomes computationally expensive as the Reynolds number increases. This approach uses a wavelet-based adaptive large eddy simulation, incorporated into the anisotropic-adaptive wavelet collocation framework, to resolve the outer region of turbulent boundary layer, while the inner part is approximated by the wall-shear-stress model, and extends the application of the wavelet-based adaptive methods to a realistic wall-bounded turbulent flow configuration at a relatively high (order of a million) Reynolds number.

Wavelet-based adaptive large-eddy simulation of supersonic channel flow

Abstract

An Adaptive Recursive Wavelet Based Algorithm for Real-Time Measurement of Power System Variables During Off-Nominal Frequency Conditions

An adaptive wavelet-based method of phasor and frequency estimations and its application for online monitoring, control, and protective equipment are presented in this paper. The method is recursive with a low computational burden, suited for real-time implementations, and does not need any preprocessing stages. It computes the frequency of the analyzing signal accurately during nominal and off-nominal frequency conditions. To reduce computational errors of the phase angle and amplitude estimations that occur during off-nominal frequency conditions, the proposed method adapts itself according to the frequency drift. The proposed recursive adaptive method along with three other approaches based on recursive wavelet transform, discrete Fourier transform (DFT), and recursive DFT is implemented and compared using MATLAB, considering dynamic and static conditions. The methods are also employed for real-time computation of the phasor and frequency of the recorded data of the South West Interconnected System (SWIS) in Western Australia, and their efficiencies are assessed and compared. In addition, a real 33.3 MVA, 132/23.3 kV transformer located in SWIS is modeled in the electromagnetic transients program (EMTP) software, and the recorded signals have been employed for performance evaluation of the above-mentioned methods. The method is also implemented on a computer-based system in the laboratory and its performance is appraised accordingly. Extensive MATLAB simulations, laboratory implementation, field data, and EMTP results show that the proposed adaptive method has better static and dynamic performances in comparison to the other approaches.

Adaptive wavelet collocation methods for image segmentation using TV–Allen–Cahn type models

An adaptive wavelet-based method is proposed for solving TV(total variation)---Allen---Cahn type models for multi-phase image segmentation. The adaptive algorithm integrates (i) grid adaptation based on a threshold of the sparse wavelet representation of the locally-structured solution; and (ii) effective finite difference on irregular stencils. The compactly supported interpolating-type wavelets enjoy very fast wavelet transforms, and act as a piecewise constant function filter. These lead to fairly sparse computational grids, and relax the stiffness of the nonlinear PDEs. Equipped with this algorithm, the proposed sharp interface model becomes very effective for multi-phase image segmentation. This method is also applied to image restoration and similar advantages are observed.

A wavelet-based adaptive method for determining eigenstates of electronic systems

The possibilities for reducing the necessary computation power in wavelet-based electronic structure calculations are studied. The expansion of the expectation values of energy operators, the integrals of basis functions are mostly system-independent, consequently it is not necessary to compute them in each calculations. Fixed building blocks, such as a parameterized expansion of the nuclear and electron–electron cusp can reduce the amount of necessary calculation. An algorithm for local expansion refinement is also given. It is possible to determine the significant expansion coefficients of a high resolution level without solving the Schrodinger equation using only lower resolution results.

Spatially adaptive thresholding in wavelet domain for despeckling of ultrasound images

Ultrasound imaging is widely used for diagnostic purposes among the clinicians. A major problem concerning the ultrasound images is their inherent corruption by the multiplicative speckle noise that hampers the quality of the diagnosis, and reduces the efficiency of the algorithms for automatic image processing. In this paper, we propose a new spatially adaptive wavelet-based method in order to reduce the speckle noise from ultrasound images. A spatially adaptive threshold is introduced for denoising the coefficients of log-transformed ultrasound images. The threshold is obtained from a Bayesian maximum a posteriori estimator that is developed using a symmetric normal inverse Gaussian probability density function (PDF) as a prior for modelling the coefficients of the log-transformed reflectivity. A simple and fast method is provided to estimate the parameters of the prior PDF from the neighbouring coefficients. Extensive simulations are carried out using synthetically speckled and ultrasound images. It is shown that the proposed method outperforms several existing techniques in terms of the signal-to-noise ratio, edge preservation index and structural similarity index and visual quality, and in addition, is able to maintain the diagnostically significant details of ultrasound images.

Wavelet-based image denoising with the normal inverse Gaussian prior and linear MMSE estimator

A new spatially adaptive wavelet-based method is introduced for reducing noise in images corrupted by additive white Gaussian noise. It is shown that a symmetric normal inverse Gaussian distribution is highly suitable for modelling the wavelet coefficients. In order to estimate the parameters of the distribution, a maximum- likelihood-based technique is proposed, wherein the Gauss-Hermite quadrature approximation is exploited to perform the maximisation in a computationally efficient way. A Bayesian minimum mean-squared error (MMSE) estimator is developed utilising the proposed distribution. The variances corresponding to the noise- free coefficients are obtained from the Bayesian estimates using a local neighbourhood. A modified linear MMSE estimator that incorporates both intra-scale and inter-scale dependencies is proposed. The performance of the proposed method is studied using typical noise-free images corrupted with simulated noise and compared with that of the other state-of-the-art methods. It is shown that the proposed method gives higher values of the peak signal-to-noise ratio compared with most of the other denoising techniques and provides images of good visual quality. Also, the performance of the proposed method is quite close to that of the state-of-the-art Gaussian scale mixture (GSM) method, but with much less complexity.

Wavelet Adaptivity for 3-D Device Simulation

A new technique has been implemented with the aim of providing an adaptive tool for the generation of computational meshes for 3-D semiconductor device simulation. The core of the proposed wavelet-based adaptive method (WAM) is a refinement algorithm based on the wavelet transform, which gives an estimation of solution regularity and progressively adapts the grid by increasing the resolution only in regions where the important physical phenomena take place. WAM is inserted into a validation tool that provides the interfacing filters with both the solver and a meshing engine. Additional features assure the quality of the generated meshes by increasing the selectivity properties of the refinement tool, preventing numerical instabilities or artifacts, such as abrupt variations of I-V curves between successive bias points during quasi-stationary simulations. Simulation results are provided, which show the effectiveness of the proposed approach as a means to guide the automatic refinement of the discretization grid, preserving accuracy with negligible computational overhead and no skilled control from the user.

Spatially Adaptive Wavelet-Based Method Using the Cauchy Prior for Denoising the SAR Images

The speckle noise complicates the human and automatic interpretation of synthetic aperture radar (SAR) images. Thus, the reduction of speckle is critical in various SAR image processing tasks. In this paper, we introduce a new spatially adaptive wavelet-based Bayesian method for despeckling the SAR images. The wavelet coefficients of the logarithmically transformed reflectance and speckle noise are modeled using the zero-location Cauchy and zero-mean Gaussian distributions, respectively. These prior distributions are then exploited to develop a Bayesian minimum mean absolute error estimator as well as a maximum a posteriori estimator. A new context-based technique with a reduced complexity is proposed for incorporating the spatial dependency of the wavelet coefficients with the Bayesian estimation processes. Experiments are carried out using typical noise-free images corrupted with simulated speckle noise as well as real SAR images, and the results show that the proposed method performs favorably in comparison to some of the existing methods in terms of the peak signal-to-noise ratio, speckle statistics and structural similarity index, and in its ability to suppress the speckle in the homogeneous regions

Adaptive Wavelet Method for Incompressible Flows in Complex Domains

Abstract An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes–Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. In this study, an adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles. The numerical results are compared to those obtained by other computational approaches as well as to experiments.

Simulating catalytic membrane reactors using orthogonal collocation with spatial coordinates transformation

It is presented in this study a new numerical scheme using orthogonal collocation together with an independent variable (spatial coordinate) transformation, useful for solving the model equations associated to membrane reactors with catalytic membranes. This new scheme takes advantage of a noticeable feature of the concentration profile inside a catalytic membrane: close to the membrane surfaces, this profile becomes steeper and steeper with the Thiele modulus. Using traditional numerical methods for solving the model equations of a catalytic membrane reactor, namely finite differences with equispaced intervals or orthogonal collocation, for example, may lead to imprecise results. In order to illustrate the ability of this new numerical scheme for solving such equations, it is applied to the resolution of a case where an analytical solution is available (a generic A ⇌ B reaction carried out in a catalytic membrane with perfectly mixed flow pattern in both retentate and permeate sides). Then, the same numerical scheme is used for solving a model describing the cyclohexane dehydrogenation, A ⇌ B + 3 C , carried out in a porous membrane with the same flow pattern as above—and the results are compared with the ones obtained using an adaptive wavelet-based method. For these two models, solutions were also obtained using straight orthogonal collocation and finite differences with homogeneously distributed grid points for the sake of the comparison. The results show that the new numerical approach is useful in dealing with this kind of problems, especially for high Thiele modulus values, showing high accuracy and demanding low computation time. Finally, this new scheme is applied to the resolution of a more complex model: a generic reaction 2 A ⇌ B , carried out in a dense catalytic cylindrical membrane with plug–flow pattern for both retentate and permeate sides.

New wavelet-based adaptive method for the breakage equation

A new wavelet-based adaptive collocation method is formulated for the solution of breakage equations. The technique of grid adaptation is based on sparse point representation (SPR). Without loss of generality, the method is applied and tested for the cases of uniform binary breakup and multiple breakup. Accurate and convergent solutions are obtained and found to match the analytical solutions very well.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework.

Adaptive Wavelet Methods II—Beyond the Elliptic Case

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.

MRFD method for numerical solution of wave propagationin layered media with general boundary condition

A fast adaptive wavelet-based method, called multiresolution finite difference (MRFD), is proposed to simulate the wave propagation in multilayered media with general boundary. It is a promising method for complex media because of its robustness and small computational burden. Numerical results derived from geophysics exploration show the effectiveness and potential of the method.

Wavelet Analysis of Structures: Statics, Dynamics and Damage Identification

Applications of the wavelet transform in solid mechanics are presented herein. The analysis problem is addressed first where the objective is to derive an adaptive wavelet-based method for the static and dynamic analysis of structures. This is done via a collocation scheme. The damage identification problem is investigated next making use of the space-localized properties of the wavelet transform: the regularity of the solution is detected by looking at the amplitude of the coefficients of the wavelet decomposition of the response. Open problems are finally outlined that will be the object of future work.