L2-norm Term Research Articles

Strongly damped wave equations with mass-like terms of the logarithmic-Laplacian

We consider strongly damped wave equations with logarithmic mass-like terms with a parameter θ∈(0,1]. This research is a part of a series of wave equations that was initiated by Charão-Ikehata [6], Charão-D'Abbicco-Ikehata considered in [5] depending on a parameter θ∈(1/2,1) and Piske-Charão-Ikehata [26] for small parameter θ∈(0,1/2). We derive a leading term as t→∞ of the solution, and by using it, a growth and a decay property of the solution itself can be precisely studied in terms of L2-norm. An interesting aspect appears in the case of n=1, roughly speaking, a small θ produces a diffusive property, and a large θ gives a kind of singularity, expressed by growth rates.

Sparse adversarial image generation using dictionary learning

Adversarial examples are used to evaluate the robustness of convolutional neural networks (CNNs) to input perturbations. Researchers have proposed different types of adversarial examples that attack CNNs to fool them. These attacks pose a serious threat to applications that use deep neural networks. Existing methods for adversarial image generation struggle in maintaining a balance between attack success rate and imperceptibility (measured in terms of ℓ2-norm) of the generated adversarial examples. Recent sparse methods for this problem focus on limiting the number of pixels in an image but do not cater to the overall imperceptibility of the adversarial images. To address these problems, we introduce adversarial attacks based on K-singular value decomposition sparse dictionary learning. The dictionary is learned using feature maps of the targeted images from the first layer of CNN. The proposed method is evaluated in terms of attack success rate and ℓ2-norm. The extensive experimentation shows our attack achieves a high success rate while maintaining a low imperceptibility score compared to state-of-the-art methods.

A General Single-Node Second-Order Dirichlet Boundary Condition for the Convection–Diffusion Equation Based on the Lattice Boltzmann Method

A general single-node second-order Dirichlet boundary condition for curved boundaries for the convection–diffusion equation based on the lattice Boltzmann method has been developed. The boundary condition simply utilizes the bounce back rule for the temperature distribution and linear interpolation. The developed boundary condition is quite general and stable. The asymptotic analysis indicates that the proposed boundary scheme is of second-order accuracy. The comparison in terms of L2-norm between the simulation results and the analytical solutions confirms that it is indeed second-order accurate for 2D and 3D symmetric flows comprising straight and curved boundaries. Moreover, the Nusselt numbers for a spherical particle in uniform flow obtained by current simulations agree very well with the predictions of empirical correlations and the data of previous direct numerical simulations with a maximal relative deviation within 8%.

Resilient Adaptive Neural Control for Uncertain Nonlinear Systems With Infinite Number of Time-Varying Actuator Failures.

Existing studies on adaptive fault-tolerant control for uncertain nonlinear systems with actuator failures are restricted to a common result that only system stability is established. Such a result of not being asymptotically stable is a tradeoff paid for reducing the number of online learning parameters. In this article, we aim to obviate such restrictions and improve the bounded error control to asymptotic control. Toward this end, a resilient adaptive neural control scheme is newly proposed based on a new design of the Lyapunov function candidates, a projection-associated tuning functions method, and an alternative class of smooth functions. It is proved that the system stability is guaranteed for the case of an infinite number of failures and when the number of failures is finite, asymptotic tracking performance can be automatically recovered, and besides, an explicit bound for the tracking error in terms of L2 norm is established. Illustrative examples demonstrate the methods developed.

Pareto frontier analysis of spatio-temporal total-variation based four-dimensional cone-beam CT

Compressed sensing (CS) based four dimensional (4D) cone-beam (CB) computed tomography (CT) reconstruction intrinsically is a multi-objective optimization problem. In practice, however, often a heuristically chosen single objective cost function is minimized. In this paper Pareto-frontier analysis was utilized to explore the image quality of the multi-objective solution space. The 4D-CBCT CS reconstruction was solved as an unconstrained optimization problem with the objective function consisting of data fidelity, spatial sparsity term (with regularization parameter λ) and temporal sparsity term (with regularization parameter β). Data fidelity was defined by an l2-norm term and the sparsity was extracted by the total-variation (TV) norm. The Pareto front was explored for a large range of λ and β parameter combinations optimized using Neterov’s descent method. Image quality was evaluated using correlation ratio (CR), mutual information (MI), structural similarity index (SSI), and contrast-to-noise ratio (CNR) on both phantom and clinical data. Higher image quality metrics were obtained with the CS optimized 4D-CBCT compared to Feldkamp (FDK) reconstructed 4D-CBCT. The optimal parameter values, however, were distinctly different for the different image quality metrics, scans and regions-of-interest. For the phantom data, the optimized CBCT have similar numerical results by CR and MI, but the results differ for SSI. For clinical data, it was found that kept the CS CBCT with 15% difference to the CBCT of optimal CNR. The optimal regularization parameter values in CS optimized 4D-CBCT were found to be patient and image quality metric dependent. The regularization should therefore be tailored to specific clinical applications.

Recovery of quantile and quantile density function using the frequency moments

The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.

Adaptive compensation for infinite number of actuator failures based on tuning function approach

In controlling nonlinear uncertain systems, compensating for infinite number of actuator failures/faults based on the well-known tuning function approach is an important, yet challenging problem in the field of adaptive control. In fact, it has been illustrated through simulation examples that instability is observed when an existing tuning function based scheme designed for compensating finite number of actuator failures is applied to an infinite number case. So far, there is still no solution to this problem. In this paper, we address this issue by proposing a novel adaptive scheme. Technically, our scheme is developed from a new piecewise Lyapunov function analysis, the parameter projection and a modified tuning function method. It is proved that all closed-loop signals are ensured bounded by the control scheme even there is a possibility that the actuator failures take place infinitely, provided that the minimum time interval between two successive failures is bounded below by any positive scalar. Moreover, the ultimate bound of tracking error can be reduced arbitrarily small even for relatively frequent failures. In addition, a guideline for improving transient performance in terms ofL2-norm of tracking error is also established. Perfect asymptotic tracking is obtained when the total number of actuator failures becomes finite.

Adaptive asymptotic control of multivariable systems based on a one-parameter estimation approach

Multivariable adaptive control is an important and challenging research area, where it usually requires designing adaptive laws to estimate a number of unknown parameters. Hence, this process may excessively consume limitedly available computational resources and large amount of computational time, which may degrade system performances, particularly for the case of online updating of parameter estimates. Thus reducing the number of parameters to be estimated is a promising way to solve the problem. The state-of-the-art result in the area is to update just one adaptive parameter. However, the tracking errors are only ensured bounded and thus the resulting closed-loop system is not asymptotically stable. Hence, a question that arises is whether such a result of non-zero tracking errors is the price paid for reducing the number of updating parameters. Up to now, there is still no answer to this question. In this paper, we address such an issue and realize asymptotic stability with online estimation of just one parameter. To achieve such a goal, a novel dynamic loop gain function based approach is proposed to incorporate with the backstepping control design procedure, which enables us to solve the algebraic loop problem caused by all the existing traditional Nussbaum function based approaches and thus establish system stability. A bound for the tracking error is explicitly established in terms of L2 norm, which helps improve the transient performance by selecting the control parameters. Moreover, guaranteed by the Barbalat’s Lemma, the tracking error is further ensured to approach zero asymptotically. Finally, simulation examples are conducted to testify the effectiveness of the proposed approach.

Partition of unity methods for approximation of point water sources in porous media

Several partition of unity methods (PUM) are compared on the problem of steady water flow in an aquifer-well system. In order to improve the approximation of a singular behavior of the pressure near the wells, the standard finite element space is enriched with a cut-off fundamental solution to a Laplace problem with a point source on the whole R2 space. The optimal order of convergence of PUM in terms of L2 norm of the error is demonstrated. The error of adaptive integration is analysed and a new adaptive strategy is proposed. The influence of the choice of the enriched domain is investigated and its impact on the error is demonstrated numerically.

Consensus Proximal Support Vector Machine for Classification Problems with Sparse Solutions

Classification problem is the central problem in machine learning. Support vector machines (SVMs) are supervised learning models with associated learning algorithms and are used for classification in machine learning. In this paper, we establish two consensus proximal support vector machines (PSVMs) models, based on methods for binary classification. The first one is to separate the objective functions into individual convex functions by using the number of the sample points of the training set. The constraints contain two types of the equations with global variables and local variables corresponding to the consensus points and sample points, respectively. To get more sparse solutions, the second one is l 1–l 2 consensus PSVMs in which the objective function contains an l1-norm term and an l2-norm term which is responsible for the good classification performance while l1-norm term plays an important role in finding the sparse solutions. Two consensus PSVMs are solved by the alternating direction method of multipliers. Furthermore, they are implemented by the real-world data taken from the University of California, Irvine Machine Learning Repository (UCI Repository) and are compared with the existed models such as l1-PSVM, l p -PSVM, GEPSVM, PSVM, and SVM-light. Numerical results show that our models outperform others with the classification accuracy and the sparse solutions.

Development of a Variational Multiscale Large-Eddy Simulation Code ‘MISTRAL’ Using Double-Scale Finite Elements

A variational multiscale large-eddy simulation (VMS-LES) code, named MISTRAL, has been developed based upon the finite element method (FEM) for accurate and practical computation of geometrically complicated turbulent flow problems. The numerical strategy of the FEM-based VMS-LES is explained, especially focusing on the double-scale approximation for velocity and pressure in the incompressible Navier-Stokes equations, a pressure stabilization technique and a multiscale turbulence modeling. A unique technique is also employed in the time integration to realize an efficient inversion of the multiscale mass matrix and to form the multiscale pressure Poisson equation used in the approximate projection method for divergence-free constraint of velocity. As a numerical demonstration, a 2D driven cavity flow problem has been solved with the MISTRAL code in a wide range of Reynolds number (Re=1000 to 50000). The results are compared with reference data to quantitatively estimate the accuracy (magnitude of errors in terms of L2 norms) of the proposed VMS-LES method.

Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of l2-norms of wavelet expansion coefficients and the constraints are written in form of an l2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in l2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.