On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method

This paper is devoted to the construction and analysis of an adapted and nonlinear multiresolution algorithm designed for interpolation or approximation of discontinuous univariate functions. The adaption attained allows to avoid numerical artifacts that appear when using linear algorithms and, at the same time, to obtain a high order of accuracy close to the singularities. It is known that linear algorithms are stable and convergent for smooth functions, but diffusion and Gibbs effect appear if the functions are piecewise continuous. Our aim is to develop an algorithm for function approximation with full accuracy that is capable to adapt to corners (kinks) and jump discontinuities, that uses a centered stencil and that does not use extrapolation. In order to reach this goal, we will need some information about the jumps in the function that we want to approximate and its derivatives. If this information is available, the algorithm is the most compact possible in the sense that the stencil is fixed and we do not need a stencil selection procedure as other algorithms do, such as ENO subcell resolution (ENO-SR). If the information about the jumps is not available, we will show a technique to approximate it. The algorithm is based on linear interpolation plus correction terms that provide the desired accuracy close to corners or jump discontinuities.     

The application of gradient-only optimization methods for problems discretized using non-constant methods

We study the minimization of objective functions containing non-physical jump discontinuities. These discontinuities arise when (partial) differential equations are discretized using non-constant methods and the resulting numerical solutions are used in computing the objective function. Although the functions may become discontinuous, gradient information may be computed at every point. Gradient information is computable everywhere since every point has an associated discretization for which (semi) analytical sensitivities can be calculated. Rather than the construction of global approximations using only function value information to overcome the discontinuities, we propose to use only the gradient information. We elaborate on the modifications of classical gradient based optimization algorithms for use in gradient-only approaches, and we then present gradient-only optimization strategies using both BFGS and a new spherical quadratic approximation for sequential approximate optimization (SAO). We then use the BFGS and SAO algorithms to solve three problems of practical interest, both unconstrained and constrained.     

Locating Discontinuities of a Bounded Function by the Partial Sums of Its Fourier Series

A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on asymptotic expansion formulae obtained in our earlier work. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function given its truncated Fourier series. We investigate the stability of the method and study its complexity. Finally, we consider several numerical examples in order to emphasize strong and weak points of the algorithm.     

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.     

Discontinuous perturbation analysis of discrete-event dynamicsystems

The discontinuous perturbation analysis (DPA) method is presented to deal with discontinuous sample performance functions. This method is applicable to derivative estimation with respect to most threshold-type problems in discrete-event dynamic systems (DEDS). Through modeling of discontinuities by step function, we are able to provide a unified framework for constructing the derivative estimation of a DEDS. As a result, we offer an alternate approach to derive derivative estimation in DEDS's. Finally, some relationships with other PA techniques are discussed, and a numerical example is presented     

Numerical implementation of a discontinuous finite element algorithm for phase-change problems

This paper is devoted to the numerical solution of phase-change problems in two dimensions. The technique of finite elements is employed. The discretization is carried out using linear isoparametric elements and special attention is given to the accurate integration of functions presenting discontinuities at arbitrarily curved interfaces. This type of function arises in a natural way when dealing with phase-change problems because the enthalpy attains a discontinuity at the phase change temperature. To integrate the discontinuous functions in the phase-changing elements a second mapping is performed from the master element onto a new one for which the interface iis a straight line. The integrals are calculated using the Gaussian technique applied to each part of the divided element, which may be triangular or quadrilateral. The discontinuous integration technique improves the behaviour of the numerical method avoiding any possible loss of latent heat due to an inaccurate evaluation of the residual vector. Some important aspects of the solution of the nonlinear system of equations are discussed and several numerical examples are presented together with the details of the computational implementation of the algorithm.     

Impulse controls and uncertainty in economics: Method and application

We develop a stochastic optimal control framework to address an important class of economic problems where there are discontinuities and a decision maker is able to undertake impulse controls in response to unexpected disturbances. Our contribution is two fold: (1) to develop a linear programming algorithm that produces a consistent approximation of the maximum value and optimal policy functions in the context of stochastic impulse controls; and (2) to illustrate the economic benefits of impulse controls optimized, using our framework, and calibrated to the population dynamics of a marine fishery. We contend that the framework has wide applicability and offers the possibility of higher economic pay-off for a wide-range of policy problems in the presence of discontinuities and adverse shocks.     

An improved finite-element contact model for anatomical simulations

This work focuses on the simulation of mechanical contact between nonlinearly elastic objects, such as the components of the human body. In traditional methods, contact forces are often defined as discontinuous functions of deformations, which leads to poor convergence characteristics and high-frequency noises. We introduce a novel penalty method for finite-element simulation based on the concept of material depth, which is the distance between a particle inside an object and the objects boundary. By linearly interpolating precomputed material depths at node points, contact forces can be analytically integrated over contact surfaces without raising the computational cost. The continuity achieved by this formulation reduces oscillation and artificial acceleration, resulting in a more reliable simulation algorithm.     

Bernoulli functions and periodic B-splines

Bernoulli polynomials and the related Bernoulli functions are of basic importance in theoretical numerical analysis. It was shown by Golomb and others that the periodic Bernoulli functions serve to construct periodic polynomial splines on uniform meshes. In an unknown paper Wegener investigated remainder formulas for polynomial Lagrange interpolation via Bernoulli functions. We will use Wegener's kernel function to construct periodicB-splines. For uniform meshes we will show that Locher's method of interpolation by translation is applicable to periodicB-splines. This yields an easy and stable algorithm for computing periodic polynomial interpolating splines of arbitrary degree on uniform meshes via discrete Fourier transform.     

Adaptive grid generation technique of sub-5?nm flying height air bearing slider with clearance discontinuities

An adaptive grid generation technique including three kinds of weight functions related to the maximum pressure gradient is proposed to simulate the pressure distribution of a sub-5?nm flying height air bearing slider with clearance discontinuities in the interface of the head and the disk in hard disk drives. Considering the clearance discontinuities of a slider with complex geometrical shape, we have defined a discontinuous factor to describe the mass flux crossing the discontinuous boundaries. The effect of different parameters in the weight functions on the node distribution of a typical slider is investigated. The pressure profile of a slider with sub-5?nm flying height is obtained based on the grid distribution calculated from the weight functions. The computational efficiency for simulating the pressure distributions is compared for different kinds of weight functions.     

基于多阶误差曲面的纹理合成

针对Image Quilting纹理合成算法拼贴块时会出现的边界不连续现象,提出一种基于多阶误差曲面的纹理合成改进算法。该算法通过多阶误差曲面来计算最小误差边界分割,从而得到更精确的最佳切割路径;此外在按照最佳切割路径拼贴块后,利用泊松混合（Poisson Blending）来修复边界不连续区域,使得不连续的边界区域能够变得平滑,纹理合成效果更符合视觉要求,并且将改进后的算法扩展到纹理传输的实现。实验结果表明,改进后的算法可以较好地克服Image Quilting算法存在的不足,得到良好的合成结果。     

Global minimization of constrained problems with discontinuous penalty functions

With the integral approach to global optimization, a class of discontinuous penalty functions is proposed to solve constrained minimization problems. Optimality conditions of a penalized minimization problem are generalized to a discontinuous case; necessary and sufficient conditions for an exact penalty function are examined; a nonsequential algorithm is proposed. Numerical examples are given to illustrate the effectiveness of the algorithm.     

Optimality of Euler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients

We consider the pointwise approximation of solutions of scalar stochastic differential equations with discontinuous coefficients. We assume the singularities of coefficients to be unknown. We show that any algorithm which does not locate the discontinuities of a diffusion coefficient has the error at least Ω(n^{?min{1/2, ?}}), where ?∈(0, 1] is the Hölder exponent of the coefficient. In order to obtain better results, we consider algorithms that adaptively locate the unknown singularities. In the additive noise case, for a single discontinuity of a diffusion coefficient, we define an Euler-type algorithm based on adaptive mesh which obtains an error of order n^??. That is, this algorithm preserves the optimal error known from the Hölder continuous case. In the case of multiple discontinuities we show, both for the additive and the multiplicative noise case, that the optimal error is Θ(n^{?min{1/2, ?}}), even for the algorithms locating unknown singularities.     

Global Output Convergence of a Class of Recurrent Delayed Neural Networks with Discontinuous Neuron Activations

This paper studies the global output convergence of a class of recurrent delayed neural networks with time-varying inputs. We consider non-decreasing activations which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. In particular, we drop the assumptions of Lipschitz continuity and boundedness on the activation functions, which are usually required in most of the existing works. Due to the possible discontinuities of the activations functions, we introduce a suitable notation of limit to study the convergence of the output of the recurrent delayed neural networks. Under suitable assumptions on the interconnection matrices and the time-varying inputs, we establish a sufficient condition for global output convergence of this class of neural networks. The convergence results are useful in solving some optimization problems and in the design of recurrent delayed neural networks with discontinuous neuron activations.     

Conservative interpolation between volume meshes by local Galerkin projection

The problem of interpolating between discrete fields arises frequently in computational physics. The obvious approach, consistent interpolation, has several drawbacks such as suboptimality, non-conservation, and unsuitability for use with discontinuous discretisations. An alternative, Galerkin projection, remedies these deficiencies; however, its implementation has proven very challenging. This paper presents an algorithm for the local implementation of Galerkin projection of discrete fields between meshes. This algorithm extends naturally to three dimensions and is very efficient.     

Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions

A family of simple, periodic basis functions with built-in discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representation of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.     

构造有理插值函数的一种参数法

对设定有理分式函数次数类型的有理插值问题研究,已有许多很多的结论。有理插值问题是否有解,取决于被插函数一些给定的函数值[f(xi),i=0,1,?,m+n]。指出分子和分母多项式次数之和为[N]的有理插值问题总有解,然后从设定的有理插值函数次数类型出发,引入正整参数[d],给出一种构造有理插值函数的方法。用该方法总可以构造出满足插值条件的有理分式函数,且有较大灵活性,计算量也不大。     

Numerical search of discontinuities in self-consistent potential energy surfaces

Potential energy surfaces calculated with self-consistent mean-field methods are a very powerful tool, since their solutions are, in theory, global minima of the non-constrained subspace. However, this minimization leads to an incertitude concerning the saddle points, that can sometimes be no longer saddle points in larger constrained subspaces (fake saddle points), or can be missing on a trajectory (missing saddle points). These phenomena are the consequences of discontinuities of the self-consistent potential energy surfaces (SPESs). These discontinuities may have important consequences, since they can, for example, hide the real height of an energy barrier, and avoid any use of an SPES for further dynamical calculations, barrier penetrability estimations, or trajectory predictions. Discontinuities are not related to the quality of the production of an SPES, since even a perfectly converged SPES with an ideally fine mesh can be discontinuous. In this paper we explain what the discontinuities are, their consequences, and their origins. We then propose a numerical method to detect and identify discontinuities on a given SPES, and finally we discuss what the best ways are to transform a discontinuous SPES into a continuous one.     

A State‐Space Based New Approach To The Directional Interpolation Problem

Directional interpolation plays an important role in robust control, system realization and model reduction. Several solutions to various directional interpolation problems have been proposed. In this paper, we consider the directional interpolation problem in a general setting and present a statespace based new approach to solving the problem. The solution is simple, and its exposition is as self‐contained as possible. We describe all the (strictly) bounded real rational matrix functions that satisfy the directional interpolation requirements by means of linear fractional transformation. Moreover, we give a necessary and sufficient condition for the interpolating function to be unique and show that the unique interpolating function is an inner (a co‐inner). The main procedures used to generate the interpolating function consist of standard matrix operations consisting of easy numerical computations, so the present solution is significant from the numerical viewpoint as well as the analytical viewpoint.