A Hybridizable Discontinuous Galerkin Method for the Navier–Stokes Equations with Pointwise Divergence-Free Velocity Field

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells (SIAM J Sci Comput 34(2):A889–A913, 2012). We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.     

Pointwise control of a counterflow process

A linear dynamics-quadratic cost control problem is considered for a counterflow process with pointwise inflows, The process is modelled by a hyperbolic system in which the inflows act both as distributed nod as boundary controls. Riccati equation characterizing the optimal control is reduced to a system of X—Y equations.     

Power Control Algorithm for Wireless Cellular System

A novel distributed power control algorithm based on interference estimation is presented for wireless cellular system. A classical result of stochastic approximation is applied in this scheme. The power control algorithm is converted to seeking for the zero point problem of a certain function. In this distributed power algorithm, each user iteratively updates its power level by estimating the interference. It does not require any knowledge of the channel gains or state information of other users. Hence, the proposed algorithm is robust. It is proved that the algorithm converges to the optimal solution by stochastic approximation approach.     

无线蜂窝系统功率控制算法

A novel distributed power control algorithm based on interference estimation is presented for wireless cellular system. A classical result of stochastic approximation is applied in this scheme. The power control algorithm is converted to seeking for the zero point problem of a certain function. In this distributed power algorithm, each user iteratively updates its power level by estimating the interference. It does not require any knowledge of the channel gains or state information of other users. Hence, the proposed algorithm is robust. It is proved that the algorithm converges to the optimal solution by stochastic approximation approach.     

Optimal segmented approximations

Segmented approximations with free knots for a given continuous function are discussed in a general form. This form includes any kind of continuous approximating segments, and a great variety of error criteria, such as anyL _s -norm, or the maximum norm of the pointwise error. It is shown that a solution to the problem exists under very general conditions, and that one solution at least must have continuous pointwise error modulus. Finally, algorithms for the iterative reduction of the approximation error are proposed, which lead to good, but not necessarily best approximations.     

A library for computing the filtered and non-filtered 3D Green's tensor associated with infinite homogeneous space and surfaces

We describe a library to compute various types of Green's tensor for three-dimensional electromagnetic scattering calculations. This library includes the retarded and non-retarded (quasi-static) Green's tensors for infinite homogeneous space and the non-retarded Green's tensor associated with a surface. Both standard and filtered Green's tensor can be computed. Filtered Green's tensor can be used to accurately investigate high permittivity scatterers with the coupled-dipole approximation.     

Recursive pointwise design for nonlinear systems

This paper presents a recursive pointwise design (RPD) method for a class of nonlinear systems represented by x(t)=f(x(t))+g(x(t))u(t). A main feature of the RPD method is to recursively design a stable controller by using pointwise information of a system. The design philosophy is that f(x(t)) and g(x(t)) can be approximated as constant vectors in very small local state spaces. Based on the design philosophy, we numerically determine constant control inputs in very small local state spaces by solving linear matrix inequalities (LMIs) derived in this paper. The designed controller switches to another constant control input when the states move to another local state space. Although the design philosophy is simple and natural, the controller does not always guarantee the stability of the original nonlinear system x(t)=f(x(t))+g(x(t))u(t). Therefore, this paper gives ideas of compensating the approximation caused by the design philosophy. After addressing outline of the pointwise design, we provide design conditions that exactly guarantee the stability of the original system. The controller design conditions require to give the maximum and minimum values of elements in the functions f(x(t)) and g(x(t)) in each local state - space. Therefore, we also present design conditions for unknown cases of the maximum and minimum values. Furthermore, we construct an effective design procedure using the pointwise design. A feature of the design procedure is to subdivide only infeasible regions and to solve LMIs again only for the subdivided infeasible regions. The recursive procedure saves effort to design a controller. A design example demonstrates the utility of the RPD method.     

Spectrum assignability of parabolic distributed parameter systems

The pole assignment problem of a parabolic distributed parameter system with boundary or pointwise inputs is studied. The problem is solved under a weak restriction on the separation of the open-loop poles and the closed-loop ones. A characteristic equation whose roots are the poles of the closed-loop system is first given and it is shown that the infinite number of open-loop poles can be uniformly moved by means of a suitable linear feedback.     

Asynchronous self-organizing maps

A recently defined energy function which leads to a self-organizing map is used as a foundation for an asynchronous neural-network algorithm. We generalize the existing stochastic gradient approach to an asynchronous parallel stochastic gradient method for generating a topological map on a distributed computer system (MIMD). A convergence proof is presented and simulation results on a set of problems are included. A practical problem using the energy function approach is that a summation over the entire network is required during the computation of updates. Using simulations we demonstrate effective algorithms that use efficient sampling for the approximation of these sums.     

$$\mathbf {L^\infty }$$-error estimates for the obstacle problem revisited

In this paper, we present an alternative approach to a priori \(L^\infty \)-error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all.     

A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion

We develop a spectral element method to price single factor European options with and without jump diffusion. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element, which allows an exact representation of the non-smooth payoff function. The convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The method is spectrally accurate (exponentially convergent) in space for the solution and Greeks, and second-order accurate in time. The spectral element solution to the Black-Scholes equation is ten to one hundred times faster than commonly used second order finite difference methods.     

Thin conical shells with constant thickness and under axisymmetric load

The expressions for determining the stresses, strains and displacements of a truncated or complete thin conical shell with constant thickness and axisymmetric load distributed or concentrated along the meridian is presented. These expressions were obtained by the construction of the Green's function for the homogeneous differential equation based on the bending theory. It is shown that a complete cone with lateral load can be obtained as a particular case and in a second step a cone with load at the vertex. The solution for an axisymmetric load distributed along the meridian was obtained using superposition.     

Immunity-based hybrid learning methods for approximator structure and parameter adjustment

From the point of view of information processing the immune system is a highly parallel and distributed intelligent system which has learning, memory, and associative retrieval capabilities. In this paper we present two immunity-based hybrid learning approaches for function approximation (or regression) problems that involve adjusting the structure and parameters of spatially localized models (e.g., radial basis function networks). The number and centers of the receptive fields for local models are specified by immunity-based structure adaptation algorithms, while the parameters of the local models, which enter in a linear fashion, are tuned separately using a least-squares method. The effectiveness of the procedure is demonstrated through a nonlinear function approximation problem and a nonlinear dynamical system modeling problem.     

A predictor-corrector type algorithm for the pseudospectral abscissa computation of time-delay systems

The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important from a robust stability point of view. In this paper we present an accurate method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectrum and the level sets of an appropriately defined complex function. The computation is done in two steps. In the prediction step, an approximation of the pseudospectral is obtained based on a rational approximation of the characteristic matrix and the application of a bisection algorithm. Each step in this bisection algorithm relies on checking the presence of the imaginary axis eigenvalues of a complex matrix, similar to the delay free case. In the corrector step, the approximate pseudospectral abscissa is corrected to any given accuracy, by solving a set of nonlinear equations that characterizes the extreme points in the pseudospectrum contours.     

Local Adaptivity to Variable Smoothness for Exemplar-Based Image Regularization and Representation

A novel adaptive and exemplar-based approach is proposed for image restoration (denoising) and representation. The method is based on a pointwise selection of similar image patches of fixed size in the variable neighborhood of each pixel. The main idea is to associate with each pixel the weighted sum of data points within an adaptive neighborhood. We use small image patches (e.g. 7×7 or 9×9 patches) to compute these weights since they are able to capture local geometric patterns and texels seen in images. In this paper, we mainly focus on the problem of adaptive neighborhood selection in a manner that balances the accuracy of approximation and the stochastic error, at each spatial position. The proposed pointwise estimator is then iterative and automatically adapts to the degree of underlying smoothness with minimal a priori assumptions on the function to be recovered. The method is applied to artificially corrupted real images and the performance is very close, and in some cases even surpasses, to that of the already published denoising methods. The proposed algorithm is demonstrated on real images corrupted by non-Gaussian noise and is used for applications in bio-imaging.     

Controller design for distributed systems based on Hankel-norm approximations

A procedure for obtaining a finite-dimensional reduced-order model for a class of distributed systems is developed. It is based on a combination of a modal approximation and an optimal Hankel-norm approximation and the approximation error can be accurately predicted. The reduced-order model can be used to design a precompensator for the original system as its performance and robust stability characteristics can be deduced from the reduced-order model and finitely many modal parameters. The procedure is applied to a parabolic system.     

Extremum-seeking control of state-constrained nonlinear systems

An extremum-seeking control problem is posed for a class of nonlinear systems with unknown dynamical parameters, whose states are subject to convex, pointwise inequality constraints. Using a barrier function approach, an adaptive method is proposed for generating setpoints online which converge to the feasible minimizer of a convex objective function containing the unknown dynamic parameters. A tracking controller regulates system states to the generated setpoint via state feedback, while maintaining feasibility of the state constraints. A simulation example demonstrates application of the method.     

Complete calming and stabilization of linear autonomous systems with delay

The problems of complete controllability and stabilizability for linear autonomous systems of delayed and neutral type are considered. For a second-order system with one input, it is supposed that the complete controllability criterion holds, and a controller with concentrated delay is constructed in such a way that the closed-loop system turns out to be pointwise degenerate, and thus provide complete original system calming down (complete calming controller). When the complete calming controller with distributed delay is being chosen, the stabilization problem is being simultaneously solved.     

Multiscale Analysis of Stock Index Return Volatility

We present a study where wavelet approximation techniques and some relatedcomputational algorithms are applied to non-stationary high frequencyfinancial timesseries. Wavelets represent a novel instrument as far as concernedapplications in the finance setting, but have a great relevance in manydomains, from physics to statistics. Thus, while one goal of the paper isto compare the numerical performance of global and local function optimizers,another goal is to try to show that ad hoc wavelet-based functiondictionaries are very useful for financial modeling through signaldecomposition and approximation. Detecting the latent dependence featureswhich are typically found in high frequency financial returns is particularlyimportant for the scope of proposing models which are able to achievereliable results in parameter estimation and pointwise function prediction.We show that by pre-processing data with wavelet dictionaries we effectivelyaccount for hidden periodic components, whose discovery allows to attain andimprove the feature extraction power. We refer to sparse approximationthrough the Matching Pursuit algorithm, thus handling the negative effects ofcovariance non-stationarity at very high frequencies.