Multiplicity of positive solutions for a fourth-order quasilinear singular differential equation

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation $$(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,$$ where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$

A Review of Adversarial Attacks in Computer Vision

Deep neural networks have been widely used in various downstream tasks, especially those safety-critical scenario such as autonomous driving, but deep networks are often threatened by adversarial samples. Such adversarial attacks can be invisible to human eyes, but can lead to DNN misclassification, and often exhibits transferability between deep learning and machine learning models and real-world achievability. Adversarial attacks can be divided into white-box attacks, for which the attacker knows the parameters and gradient of the model, and black-box attacks, for the latter, the attacker can only obtain the input and output of the model. In terms of the attacker's purpose, it can be divided into targeted attacks and non-targeted attacks, which means that the attacker wants the model to misclassify the original sample into the specified class, which is more practical, while the non-targeted attack just needs to make the model misclassify the sample. The black box setting is a scenario we will encounter in practice

Adversarial Training for Physics-Informed Neural Networks

Physics-informed neural networks have shown great promise in solving partial differential equations. However, due to insufficient robustness, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To address these issues, based on the projected gradient descent adversarial attack, we proposed an adversarial training strategy for PINNs termed by AT-PINNs. AT-PINNs enhance the robustness of PINNs by fine-tuning the model with adversarial samples, which can accurately identify model failure locations and drive the model to focus on those regions during training. AT-PINNs can also perform inference with temporal causality by selecting the initial collocation points around temporal initial values. We implement AT-PINNs to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, Burgers equation with sharp solutions and the Allen-Cahn equation. The results demonstrate that AT-PINNs can effectively locate and reduce failure regions. Moreover, AT-PINNs are suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution

Image Denoising via Nonlinear Hybrid Diffusion

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models

Modeled Antarctic Precipitation. Part I: Spatial and Temporal Variability*

Surface snow accumulation is the primary mass input to the Antarctic ice sheets. As the dominant term among various components of surface snow accumulation (precipitation, sublimation/deposition, and snow drift), pre-cipitation is of particular importance in helping to assess the mass balance of the Antarctic ice sheets and their contribution to global sea level change. The Polar MM5, a mesoscale atmospheric model based on the fifth-generation Pennsylvania State University– NCAR Mesoscale Model, has been run for the period of July 1996 through June 1999 to evaluate the spatial and temporal variability of Antarctic precipitation. Drift snow effects on the redistribution of surface snow over Antarctica are also assessed with surface wind fields from Polar MM5 in this study. It is found that areas with large drift snow transport convergence and divergence are located around escarpment areas where there is considerable katabatic wind acceleration. It is also found that the drift snow transport generally diverges over most areas of East and West Antarctica with relatively small values. The use of the dynamic retrieval method (DRM) to calculate precipitation has been developed and verified for the Greenland ice sheet. The DRM is also applied to retrieve the precipitation over Antarctica from 1979 to 1999 in this study. Most major features in the spatial distribution of Antarctic accumulation are well capture

Re-initialization-free Level Set Method via Molecular Beam Epitaxy Equation Regularization for Image Segmentation

Variational level set method has become a powerful tool in image segmentation due to its ability to handle complex topological changes and maintain continuity and smoothness in the process of evolution. However its evolution process can be unstable, which results in over flatted or over sharpened contours and segmentation failure. To improve the accuracy and stability of evolution, we propose a high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization. This method uses the crystal growth in the MBE process to limit the evolution of the level set function, and thus can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve. It also works for noisy images with intensity inhomogeneity, which is a challenge in image segmentation. To solve the variational model, we derive the gradient flow and design scalar auxiliary variable (SAV) scheme coupled with fast Fourier transform (FFT), which can significantly improve the computational efficiency compared with the traditional semi-implicit and semi-explicit scheme. Numerical experiments show that the proposed method can generate smooth segmentation curves, retain fine segmentation targets and obtain robust segmentation results of small objects. Compared to existing level set methods, this model is state-of-the-art in both accuracy and efficiency

The Sensitivity of Simulated River Discharge to Land Surface Representation and Meteorological Forcings

Abstract The discharge of freshwater into oceans represents a fundamental process in the global climate system, and this flux is taken into account in simulations with general circulation models (GCMs). Moreover, the availability of realistic river routing schemes is a powerful instrument to assess the validity of land surface components, which have been recognized to be crucial for the global climate simulation. In this study, surface and subsurface runoff generated by the 13 land surface schemes (LSSs) participating in the Second Global Soil Wetness Project (GSWP-2) are used as input fields for the Hydrology Discharge (HD) routing model to simulate discharge for 30 of the world's largest rivers. The simplest land surface models do not provide a good representation of runoff, and routed river flows using these inputs are affected by many biases. On the other hand, HD shows the best simulations when forced by two of the more sophisticated schemes. The multimodel ensemble GSWP-2 generates the best phasing of the annual cycle as well as a good representation of absolute values, although the ensemble mean tends to smooth the peaks. Finally, the intermodel comparison shows the limits and deficiencies of a velocity-constant routing model such as HD, particularly in the phase of mean annual discharge. The second part of the study assesses the sensitivity of river discharge to the variation of external meteorological forcing. The Center for Ocean–Land–Atmosphere Studies version of the SSiB model is constrained with different meteorological fields and the resulting runoff is used as input for HD. River flow is most sensitive to precipitation variability, but changes in radiative forcing affect discharge as well, presumably because of the interaction with evaporation. Also, this analysis provides an estimate of the sensitivity of river discharge to precipitation variations. A few areas (e.g., central and eastern Asia, the Mediterranean, and much of the United States) show a magnified response of river discharge to a given percentage change in precipitation. Hence, an amplified effect of droughts as indicated by the consensus of climate change predictions may occur in places such as the Mediterranean. Conversely, increasing summer precipitation foreseen in places like southern and eastern Asia may amplify floods in these poor and heavily populated regions. Globally, a 1% fluctuation in precipitation forcing results in an average 2.3% change in discharge. These results can be used for the definition and assessment of new strategies for land use and water management in the near future

An Alternative Variational Framework for Image Denoising

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method