The cascadic multigrid method for elliptic problems

On the convergence of an extrapolation cascadic multigrid method for elliptic problems

An optimal cascadic multigrid method for the radiative transfer equation

Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.

: To illustrate Brandt's multigrid method this report considers the Helmholtz equation in cases where the frequency is close to an eigenfrequency. In a second part the eigenvalue problem is discussed. The basic idea for the treatment of the inhomogeneous problem is contained in Brandt's publications. Difficulties are encountered because the coarse grid eigenvalues differ from those of the finer grid. As a consequence, approximations to a solution in the coarse grid may differ widely from the corresponding approximation in a finer grid, if the frequency defined by the inhomogeneous problem is close to one of the eigenfrequencies. This may be detrimental to the aim of the coarse grid step; namely to remove from an approximation in a fine grid long waves which cause divergence of fine grid iterations. This is remedied by eliminating long wave contributions to the residual directly in the next finer grid. The success of this procedure still depends upon the choice of the subspace from which the long wave corrections in the finer grid are taken. Even this modified method fails if the frequency pertaining to the inhomogeneous term is too close to an eigenvalue, unless the subspace contains the pertinent eigenfunction.

In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.

Flow fields in a centrifugal fan for an indoor unit of an air-conditioner were calculated with finite element method-based large eddy simulation (LES) with the aim of predicting fan performance and aerodynamic noise in this study. The numerical simulation code employed throughout the LES was called FrontFlow/blue (FFB). We compared 10M grid [coarse grid] and 60M grid [fine grid] calculation results for investigation of influence of grid resolution. In the fine grid, the number of grid elements in blade-to-blade direction, and of region between the shroud and the bell mouth increased in particular. By calculating with the fine grid, calculated distributions of absolute velocities at blade exit reasonably agreed with experimental results. Because of this, maximum absolute velocity by fine grid near hub decreased as compared to those by coarse grid. Calculated sound pressure level by fine grid was therefore smaller than that by coarse grid, and the overestimation of sound pressure was suppressed by calculating with fine grid. This decrease of the absolute velocity was a first factor for the improvement of calculation accuracy. Moreover, number of captured streaks on the blade, hub, and shroud surfaces by fine grid increased as compared to those by coarse grid. As a result, size of streak by fine grid became smaller than that by coarse grid. Static pressure fluctuations by fine grid on the blade, hub, and shroud surfaces therefore reduced as compared to those by coarse grid. Aerodynamic noise was related to static pressure fluctuations according to Curle’s equation. This reduction of static pressure fluctuations was therefore a second factor for improvement of calculation accuracy.

Human activity recognition (HAR) is one of the leading research fields in ubiquitous computing working to integrate seamless technologies in our daily lives. The researches in this field focus on the technological advancement of fine activity recognition through using minimal technological deployments along with consideration of human factors. The domain specific knowledge of human activity recognition has shed significant importance on the activity data processing, due to the innate random nature of the activity data. The necessity for improved prediction accuracy of activities within limited infrastructural support and cost is necessary for the research works to be conducted at large. The data set utilized in this study is modified through tree-based feature-engineering method, which has significantly impacted the intrinsic time-series pattern associated with the original data set acquired from UCI machine learning repository. In this paper, the primary focus lies on improving classification accuracy of the human activities included in the data set through utilizing deep learning classifier models. The particular modified pattern of the data devoid of time-series pattern to major extent has resulted in lower classification accuracy by benchmark time-series classification models namely Keras-LSTM and RNN-LSTM. CNN based networks have demonstrated consisted improved training performance on the data set. The state-of-the-art classifier ResNet introduced by Microsoft Research in 2015 has achieved 99.9% classification accuracy on the data set of this study. Another important aim of this study is to infer the risk attached with adversarial attack on machine learning models. The high vulnerability of machine learning models in face of adversarial attacks has emerged as security concern of the deployment pipeline of machine learning models. To the author’s best knowledge, there has been no prior study to find out the high insecurity associated with the human activity recognition data set and concerned benchmark models. In this study, two prevalent adversarial approaches namely Fast Gradient Sign Method (FGSM) and Basic Iterative Method (BIM) have been employed to carry out non-targeted attack on the pre-trained ResNet model. The accuracy loss by the adversarial attack over a small range of added perturbation is tremendous, nearly 96% accuracy is diminished in the process. Finally, we have demonstrated the comparative performance by FGSM and BIM on reducing the classification accuracy of pre-trained model. The findings shed light on the nature of the data influencing the classification accuracy and the resiliency of state-of-the-art classifier model on facing adversarial attacks.

Efficient forward modeling of electromagnetic (EM) fields is the basis of interpretation and inversion of the EM data, which plays an important role in practical geophysical exploration. A novel extrapolation cascadic multigrid (EXCMG) method is developed to solve large linear systems encountered in geophysical EM modeling. The original curl-curl equation with arbitrary anisotropic conductivity is regularized by including the gradient of a scaled divergence correction term, and a linear edge element method is used to discretize the equation. We develop a novel approach to address the issue of edge unknowns in 3D edge element discretizations on nonuniform rectilinear grids. Inspired by the original EXCMG method for nodal elements, we introduce a new prolongation operator that treats edge unknowns as defined on the midpoints of edges. This operator aims to provide an accurate approximation of the finite-element solution on the refined grid. Using a good initial guess significantly reduces the number of iterations required by the preconditioned biconjugate gradient stabilized solver, which is used as a smoother for the EXCMG algorithm. Numerical experiments are carried out to validate the accuracy and efficiency of our EXCMG method, including models from magnetotellurics and controlled-source EM modeling. The testing results indicate that EXCMG is more efficient than traditional Krylov-subspace iterative solvers, the algebraic multigrid solver, and those depending on the auxiliary-space Maxwell solver, especially for large-scale problems where the number of unknowns exceeds 10 million. The EXCMG method can also be applied to solve other large-scale forward modeling problems encountered in geophysics.

Running a fine grid model with 107 - 109 of cells is possible using a supercomputer with 103 - 106 of CPUs but may not be always cost-effective. The most cost-effective way is to use a coarse grid model that is much smaller but with static/dynamic profiles very close to the fine grid model. This paper proposes a new layer optimization and upscaling method with the aim for creating a consistent coarse grid model. Unlike the industry's existing layer optimization and upscaling methods, the proposed method performs layer optimization and upscaling fully integrated with the Lorenz coefficient and curves (LCC). Coarse grid layers and their permeabilities are created by minimizing the difference between fine and coarse grid LCCs. The process consists of static and dynamic optimizations. The former is measured by LCC while the latter by pressure, GOR, and water-cut. A new LCC-based permeability upscaling method is developed to preserve the fine grid multiphase flow behaviors. A satisfactory coarse grid model is achieved when both static and dynamic criteria are met. The proposed method has been successfully applied to a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within and between the matrix and fracture/karst features. The reservoir drive mechanisms are fluid expansion, miscible gas injection and aquifer drive. The reservoir is undersaturated and has an abnormally high initial reservoir pressure. The fine-grid static model contains 104 million cells (370×225×625×2) and the optimized upscaled coarse-grid dynamic model has 8.3 million cells (370×225×50×2). The upscaled model can be run efficiently on the company's existing HPC infrastructure with a maximum of 64 CPUs. Excellent matches of the Lorenz coefficient maps for reservoir total/zones and Lorenz curves at all wells between the fine and coarse grid models have been achieved. Matches on the dynamic variables, e.g., pressure, gas breakthrough time, and GOR growth, in all producers are within the defined acceptable tolerances. The high quality of the static and dynamic matches between the coarse- and fine-grid models confirms that the reservoir properties of the coarse-grid model is very close to the fine-grid model and can be used a base model for history matching and uncertainty analysis.

Very often in real applications, one has to solve some complicated equations, where a formula for the result is either impossible to obtain, or it is too complicated for any practical use. In this case one has to rely on a wide range of numerical methods, which very often represent approximations to the real results. These methods are very important in practice and usually offer both an algorithm generating increasingly exact approximations and an approximation for the error. In this chapter we shall discuss the Lagrange polynomial for interpolation data given by a table of values, basic iterative methods used for the numerical integration of real functions and some iterative methods for root finding.

Although deep neural networks (DNNs) have achieved high performance across various applications, they are often deceived by adversarial examples generated by adding small perturbations. To combat adversarial attacks, many detection methods have been proposed, including feature squeezing and trapdoor. However, these methods rely on the output of DNNs or involve training a separate network to detect adversarial examples, which leads to high computational costs and low efficiency. In this study, we propose a simple and effective approach called the entropy-based detector (EBD) to protect DNNs from various adversarial attacks. EBD detects adversarial examples by comparing the difference in entropy between the input sample before and after bit depth reduction. We show that EBD can detect over 98% of the adversarial examples generated by attacks using fast-gradient sign method, basic iterative method, momentum iterative method, DeepFool and CW attacks when the false positive rate is 2.5% for CIFAR-10 and ImageNet datasets.

The design of a security scheme for beamforming prediction is critical for next-generation wireless networks (5G, 6G, and beyond). However, there is no consensus about protecting beamforming prediction using deep learning algorithms in these networks. This paper presents the security vulnerabilities in deep learning for beamforming prediction using deep neural networks in 6G wireless networks, which treats the beamforming prediction as a multi-output regression problem. It is indicated that the initial DNN model is vulnerable to adversarial attacks, such as Fast Gradient Sign Method , Basic Iterative Method , Projected Gradient Descent , and Momentum Iterative Method , because the initial DNN model is sensitive to the perturbations of the adversarial samples of the training data. This study offers two mitigation methods, such as adversarial training and defensive distillation, for adversarial attacks against artificial intelligence-based models used in the millimeter-wave (mmWave) beamforming prediction. Furthermore, the proposed scheme can be used in situations where the data are corrupted due to the adversarial examples in the training data. Experimental results show that the proposed methods defend the DNN models against adversarial attacks in next-generation wireless networks.

We propose to enhance the stability of IMEX schemes applied to the numerical solution of nonlinear parabolic equations by using proper smoothing procedures on bands of frequency components of the solution. The separation of those different components is realized by considering a multigrid (MG) approach. The time scheme on the coarsest grid is implicit and energy diminishing, while time marching is done by a weighted damped IMEX scheme on the finer grids. This approach allows to concentrate most computational efforts on a small set of components (the coarsest grid) and to fine-tune the damping parameters so that there is limited consistency perturbation while enhancing stability. We provide stability results and numerical illustrations on pattern dynamics, Swift-Hohenberg, and crystal-phase fields equations, in both 2D and 3D cases. The results obtained with MG are comparable with those given by a one-grid simulation.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation