Gibbs phenomena for Lq-best approximation in finite element spaces

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue{L^{p}}-space,{1<p<\infty}. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in{L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

Mitigating Gibbs Phenomena in Uncertainty Quantification With a Stochastic Spectral Method

The use of spectral projection-based methods for simulation of a stochastic system with discontinuous solution exhibits the Gibbs phenomenon, which is characterized by oscillations near discontinuities. This paper investigates a dynamic bi-orthogonality-based approach with appropriate postprocessing for mitigating the effects of the Gibbs phenomenon. The proposed approach uses spectral decomposition of the spatial and stochastic fields in appropriate orthogonal bases, while the dynamic orthogonality (DO) condition is used to derive the resultant closed-form evolution equations. The orthogonal decomposition of the spatial field is exploited to propose a Gegenbauer reprojection-based postprocessing approach, where the orthogonal bases in spatial dimension are reprojected on the Gegenbauer polynomials in the domain of analyticity. The resultant spectral expansion in Gegenbauer series is shown to mitigate the Gibbs phenomenon. Efficacy of the proposed method is demonstrated for simulation of a one-dimensional stochastic Burgers equation and stochastic quasi-one-dimensional flow through a convergent-divergent nozzle.

Wavelet Denoising of SAR Images Corrupted by Speckle Noise Using Cycle Spinning

A novel and efficient speckle noise reduction algorithm based on wavelet transform by cycle spinning for removing speckle of unknown variance and minimizing the effect of pseudo-Gibbs phenomena from Synthetic Aperture Radar (SAR) images is proposed. Therefore, we show that the sub-band decompositions of logarithmically transformed SAR images. Then, we process and reconstruct multi-resolution wavelet coefficients by wavelet-threshold using cycle spinning, a technique estimating the true images as the linear average of individual estimates derived from wavelet thresholded translated versions of the noise images. Experimental results show that the proposed de-noising algorithm is possible to achieve an excellent balance between suppresses speckle effectively and weaken as many image Gibbs phenomena as possible. Quantitative and qualitative comparisons of the results obtained by the new method with the results achieved from the other speckle noise reduction techniques demonstrate its higher performance for speckle reduction in SAR images.

Image Fusion Method for Strip Steel Surface Detect Based on Bandelet-PCNN

To solve the problem of the pseudo-Gibbs phenomena around singularities when we implement image fusion with images of strip surface detects obtained from different angles, a novel image fusion method based on Bandelet-PCNN(Pulse coupled neural networks) is proposed. Low-pass sub-band coefficient of source image by Bandelet is inputted into PCNN. And the coefficient is selected by ignition frequency by the neuron iteration. At last the fused image can be got through inverse Bandelet using the coefficient and Geometric flow parameters. Experimental results demonstrate that for the scrip surface detects of scratches, abrasions and pit, fused image effectively combines defect information of multiple image sources. Contrast to the classical wavelet transform and Bandelet transform the method reserves more detailed and comprehensive detect information. Consequently the method proposed in this paper is more effective.