Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems

In this paper, we propose a characteristic tailored finite point method (CTFPM) for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the CTFPM. Our numerical tests show for small diffusion coefficient the CTFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that CTFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.     

A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain

In this paper, we propose a tailored-finite-point method for a kind of singular perturbation problems in unbounded domains. First, we use the artificial boundary method (Han in Frontiers and Prospects of Contemporary Applied Mathematics, [2005]) to reduce the original problem to a problem on bounded computational domain. Then we propose a new approach to construct a discrete scheme for the reduced problem, where our finite point method has been tailored to some particular properties or solutions of the problem. From the numerical results, we find that our new methods can achieve very high accuracy with very coarse mesh even for very small ε. In the contrast, the traditional finite element method does not get satisfactory numerical results with the same mesh. Han was supported by the NSFC Project No. 10471073. Z. Huang was supported by the NSFC Projects No. 10301017, and 10676017, the National Basic Research Program of China under the grant 2005CB321701. R.B. Kellogg was supported by the Boole Centre for Research in Informatics at National University of Ireland, Cork and by Science Foundation Ireland under the Basic Research Grant Programme 2004 (Grants 04/BR/M0055, 04/BR/M0055s1).     

A numerical study of singularly perturbed generalized Burgers-Huxley equation using three-step Taylor-Galerkin method

In the present work, a numerical study has been carried out for the singularly perturbed generalized Burgers-Huxley equation using a three-step Taylor-Galerkin finite element method. A Burgers-Huxley equation represents the traveling wave phenomena. In singular perturbed problems, a very small positive parameter, ?, called the singular perturbation parameter is multiplied with the highest order derivative term. As this parameter tends towards zero, the problem exhibits boundary layers. The traditional methods fail to capture the boundary layers when ? becomes very small. In this paper a three-step Taylor-Galerkin finite element method is used to capture the boundary layers. The method is third-order accurate and has inbuilt upwinding. Stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers.     

Lifting solutions of quasilinear convection-dominated problems

In certain cases, quasilinear convection–diffusion–reaction equations range from parabolic to almost hyperbolic, depending on the ratio between convection and diffusion coefficients. From a numerical point of view, two main difficulties can arise related to the existence of layers and/or the non-smoothness of the coefficients of such equations. In this paper we study the steady-state solution of a convection-dominated problem. We present a new numerical method based on the idea of solving an associated modified problem, whose solution corresponds to a lifting of the solution of the initial problem. The method introduced here avoids an a priori knowledge of the layer(s) location and allows an efficient handling of the lack of smoothness of the coefficients. Numerical simulations that show the effectiveness of our approach are included.     

An efficient method for solving nonlinear singular initial value problems

In [M.M. Hosseini, Modified Adomain decomposition method for specific second order ordinary differential equations, Appl. Math. Comput. 186 (2007), pp. 117–123] an efficient modification of Adomian decomposition method has been proposed for solving some cases of ordinary differential equations. In this paper, this method is generalized to more cases. The proposed method can be applied to linear, nonlinear, singular and nonsingular problems. Here, it is focused on nonlinear singular initial value problems of ordinary differential equations. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.     

Robust H∞ Sliding Mode Control for a Class of Singular Stochastic Nonlinear Systems

This paper concerns the problem of robust H_∞ sliding mode control for a class of singular stochastic nonlinear systems. Integral sliding mode control is developed to deal with this problem. Based on the integral sliding surface of the design and linear matrix inequality, a sufficient condition which guarantees the sliding mode dynamics is asymptotically mean square admissible and has a prescribed H_∞ performance for a class of singular stochastic nonlinear systems is proposed. Furthermore, a sliding mode control law is synthesized such that the singular stochastic nonlinear system can be driven to the sliding surface in finite time. Finally, a numerical example is proposed to illustrate the effectiveness of the given theoretical results.     

Massively parallel methods for semiconductor device modelling

In this paper we describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknowns in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations. A version of Gummel's decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solves of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method. Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solves also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.     

Observability analysis of nonlinear tubular (bio)reactor models: a case study

This paper deals with the observability analysis of nonlinear tubular bioreactor models. Due to the lack of tools for the observability analysis of nonlinear infinite-dimensional systems, the analysis is performed on a linearized version of the model around some steady-state profile, in which coefficients can be functions of the spatial variable. The study starts from an example of tubular bioreactor that will serve as a case study in the present paper. It is shown that such linear models with coefficients dependent on the spatial variable are Sturm–Liouville systems and that the associated linear infinite-dimensional system dynamics are described by a Riesz-spectral operator that generates a C₀ (strongly continuous)-semigroup. The observability analysis based on infinite-dimensional system theory shows that any finite number of dominant modes of the system can be made observable by an approximate point measurement.     

On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so-called singular value functions of a system describe the relative importance of each state component from an input–output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, distinct sets of singular value functions and balanced realizations are possible.     

Semi-global stabilization of minimum phase nonlinear systems in special normal form via linear high-and-low-gain state feedback

We provide an alternative solution to the problem of semi-global stabilization of a class of minimum phase nonlinear systems which is considered in Reference 17. Our method yields a stabilizing linear state feedback law in contrast to a nonlinear state feedback law proposed in Reference 17. We eliminate the peaking phenomenon by inducing a specific time-scale structure in the linear part of the closed-loop system. This time-scale structure consists of a very slow and a very fast time scale. The crucial component in our method is the relation between the slow and the fast time scales. Our proposed linear state feedback control law has a single tunable gain parameter that allows for local asymptotic stability and regulation to the origin for any initial condition in some a priori given (arbitrarily large) bounded set.     

Generalized criteria on delay‐dependent stability of highly nonlinear hybrid stochastic systems

Our recent paper (Fei W, etal. Delay dependent stability of highly nonlinear hybrid stochastic systems. Automatica. 2017;82:165‐170) is the first to establish delay‐dependent criteria for highly nonlinear hybrid stochastic differential delay equations (SDDEs) (by highly nonlinear, we mean that the coefficients of the SDDEs do not have to satisfy the linear growth condition). This is an important breakthrough in the stability study as all existing delay stability criteria before could only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In this continuation, we will point out one restrictive condition imposed in our earlier paper. We will then develop our ideas and methods there to remove this restrictive condition so that our improved results cover a much wider class of hybrid SDDEs.     

Singular nonlinear H∞ optimal control problem

The theory of nonlinear H_∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H_∞ optimal control of nonlinear systems using ideas from the linear H_∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H_∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H_∞ control problem by applying the certainty equivalence principle.     

Sinc-Galerkin method for solving nonlinear weakly singular two point boundary value problems

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with non-homogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with unknown coefficients. This method tested on several test examples. We compare our numerical results with several numerical results of existing methods. The demonstrated results confirm that proposed method is considerably efficient, accurate nature and rapidly converge.     

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in $H^1$-norm and for pressure in $L^2$-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.     

Locally linear metric adaptation with application to semi-supervised clustering and image retrieval

Many computer vision and pattern recognition algorithms are very sensitive to the choice of an appropriate distance metric. Some recent research sought to address a variant of the conventional clustering problem called semi-supervised clustering, which performs clustering in the presence of some background knowledge or supervisory information expressed as pairwise similarity or dissimilarity constraints. However, existing metric learning methods for semi-supervised clustering mostly perform global metric learning through a linear transformation. In this paper, we propose a new metric learning method that performs nonlinear transformation globally but linear transformation locally. In particular, we formulate the learning problem as an optimization problem and present three methods for solving it. Through some toy data sets, we show empirically that our locally linear metric adaptation (LLMA) method can handle some difficult cases that cannot be handled satisfactorily by previous methods. We also demonstrate the effectiveness of our method on some UCI data sets. Besides applying LLMA to semi-supervised clustering, we have also used it to improve the performance of content-based image retrieval systems through metric learning. Experimental results based on two real-world image databases show that LLMA significantly outperforms other methods in boosting the image retrieval performance.     

Identification of FIR Wiener systems with unknown,non-invertible,polynomial non-linearities

Wiener systems consist of a linear dynamic system whose output is measured through a static non-linearity. In this paper we study the identification of single-input single-output Wiener systems with finite impulse response dynamics and polynomial output non-linearities. Using multi-index notation, we solve a least squares problem to estimate products of the coefficients of the non-linearity and the impulse response of the linear system. We then consider four methods for extracting the coefficients of the non-linearity and impulse response: direct algebraic solution, singular value decomposition, multi-dimensional singular value decomposition and prediction error optimization.     

The calculation of singular coefficients

We consider in this paper the calculation of the ‘singular coefficients’ associated with the solution of an elliptic partial differential equation near a singular point; a re-entrant corner, or a crack tip, etc. These are the coefficients in the relevant singular expansion of the solution near the point of singularity; they often have physical relevance, and it is of interest to be able to calculate them accurately. We consider the problem in the context of the global element method; this is a variable-order finite element method designed to be capable of producing highly accurate solutions for singular problems, even in the neighbourhood of the singularity. If the values of the singular coefficients are needed, these must be extracted from the computed solution; we show in this paper that a suitably defined least-squares fitting procedure allows the calculation of values for the leading singular coefficients which are as accurate as the underlying solution.     

Structural shape sensitivity analysis for nonlinear material models with strain softening

This paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the finite element method with a nonlinear constitutive material model. Traditionally, the structural sensitivity analysis is computed using an incremental approach based on the incremental procedures for the solution of the structural equilibrium problem. In this work, a direct (nonincremental) formulation for computing these structural sensitivities, that is valid for some specific nonlinear material models, is proposed. The material models for which the presented approach is valid are characterized by the fact that the stresses at any timet can be expressed in terms of the strains at the timet and, in some cases, the strains at a specific past timet ^u (t ^u <t). This is the case of elasticity (linear as well as nonlinear), perfect plasticity and damage models. A special strategy is also proposed for material models with strain softening.For the cases where it is applicable, the sensitivity analysis proposed here allows us to compute the structural sensitivities around any structural equilibrium point after finishing the solution process and it is completely independent of the numerical scheme used to solve the structural equilibrium problem. This possibility is particularized for the case of a damage model considering a strain-softening behaviour. Finally, the quality and reliability of the proposed approach is assessed through its application to some examples.     

Singular extremals on Lie groups

We investigate the space of singular curves associated to a distribution ofk-planes, or, what is the same thing, a nonlinear deterministic control system linear in controls. A singular curve is one for which the associated linearized system is not controllable. If a quadratic positive-definite cost function is introduced, then the corresponding optimal control problem is known as the sub-Riemannian geodesic problem. The original motivation for our work was the question “Are all sub-Riemannian minimizers smooth?” which is equivalent to the question “Are singular minimizers necessarily smooth?” Our main result concerns the singular curves for a class of homogeneous systems whose state spaces are compact Lie groups. We prove that for this class each singular curve lies in a lower-dimensional subgroup within which it is regular and we use this result to prove that all sub-Riemannian minimizers are smooth. A central ingredient of our proof is a symplectic-geometric characterization of singular curves formulated by Hsu. We extend this characterization to nonsmooth singular curves. We find that the symplectic point of view clarifies the situation and simplifies calculations.     

Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems

Cui et al. [M. Cui and F. Geng, Solving singular two point boundary value problems in reproducing kernel space, J. Comput. Appl. Math. 205 (2007), pp. 6–15; H. Yao and M. Cui, A new algorithm for a class of singular boundary value problems, Appl. Math. Comput. 186 (2007), pp. 1183–1191] presents an algorithm to solve a class of singular linear boundary value problems in the reproducing kernel space. In this paper, we will present three new algorithms to solve a class of singular weakly nonlinear boundary value problems in reproducing kernel space. The algorithms are efficiently applied to solving some model problems. It is demonstrated by the numerical examples that those algorithms are highly accurate.