Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

Analysis of Numerical Differentiation Formulas in a Boundary Layer on a Shishkin Grid

A problem of numerical differentiation of functions with large gradients in a boundary layer is investigated. The problem is that for functions with large gradients and a uniform grid the relative error of the classical difference formulas for derivatives may be considerable. It is proposed to use a Shishkin grid to obtain a relative error of the formulas that is independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform with respect to the small parameter. In the case of a uniform grid, a boundary layer region is indicated outside of which the numerical differentiation formulas have an error that is uniform with respect to the small parameter. The results of numerical experiments are presented.     

An analysis of a vacuum diode model

One of the vacuum diode models is written as a singular boundary value problem for two ODEs of the second order with one parameter. The parameter is unknown and must be found along with the initial conditions at the origin to satisfy the boundary conditions at the end of the interval. The problem presents considerable difficulties in its theoretical and, especially, numerical study. Here we give a complete analytical and numerical solution of this problem.     

Numerical analysis of singularly perturbed delay differential turning point problem

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.     

Computation of an infinite integral using series acceleration

A numerical computation in crystallography involves an infinite integral depending on one parameter. In a recent article in this journal this computational problem is addressed using Romberg’s method and tools for error control. One observe numerical difficulties with the reported approach both near the parameter’s endpoints and near the parameter interval’s midpoint. In this short note we will present an alternative approach making use of a known infinite series formulation of the problem at hand and a simple and efficient series acceleration technique. If some care is taken to avoid cancellations the numerical results are excellent for all values of the parameter. AMS subject classification 65B05, 65B10, 65D30     

A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations

This study deals with the singularly perturbed initial value problem for a quasilinear first-order delay differential equation. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.     

Differential equations for the analytic singular value decomposition of a matrix

Summary This paper is concerned with finding a smooth singular value decomposition for a matrix which is smoothly dependent on a parameter. A previous approach to this problem was based on minimisation techniques, here, in contrast, a system of ordinary differential equations is derived for the decomposition. It is shown that the numerical solution of an initial value problem associated with these differential equations provides a feasible approach to the solution of this problem. Particular consideration is given to the situation which arises with equal modulus singular values which lead to indeterminacies in the evaluations needed for the numerical solution. Examples which illustrate the behaviour of the method are included.     

System Identifiability (Symbolic Computation) and Parameter Estimation (Numerical Computation)

A system identification based on physical laws often involves a parameter estimation. Before performing an estimation problem, it is necessary to investigate its identifiability. This investigation leads often to painful calculations. Generally, the numerical computation of the parameters does not use these calculus. In this contribution we propose least-squares methods to link identifiability approaches with numerical parameter estimation.     

Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.     

Optimization with Variable Sets of Constraints and an Application to Truss Design

We discuss the minimization of a continuous function on a subset of Rⁿ subject to a finite set of continuous constraints. At each point, a given set-valued map determines the subset of constraints considered at this point. Such problems arise e.g. in the design of engineering structures.After a brief discussion on the existence of solutions, the numerical treatment of the problem is considered. It is briefly motivated why standard approaches generally fail. A method is proposed approximating the original problem by a standard one depending on a parameter. It is proved that by choosing this parameter large enough, each solution to the approximating problem is a solution to the original one. In many applications, an upper bound for this parameter can be computed, thus yielding the equivalence of the original problem to a standard optimization problem.The proposed method is applied to the problem of optimally designing a loaded truss subject to local buckling conditions. To our knowledge this problem has not been solved before. A numerical example of reasonable size shows the proposed methodology to work well.     

Numerical Solution of the Inverse Nonequilibrium Sorption Problem with a Time-Dependent Boundary Condition

We consider the quasi-linear problem of nonequilibrium sorption dynamics with external-diffusion kinetics and a boundary condition that contains the time derivative of a solution component. A numerical method is proposed for describing the inverse problem to recover the nonlinear parameter of the system of differential equations—the inverse of the sorption isotherm. Convergence of the difference scheme for the direct problem is proved. Numerical solutions of both the direct and the inverse problem are obtained for various parameter values.     

一类非光滑分布参数系统的可辨识性及最优性条件

变压器温度场参数辨识问题是一种分片光滑的分布参数辨识问题,以流速为辨识参数,针对传质传热的一类分布参数系统参数辨识问题,证明了系统最优参数的存在性和控制参数为最优的必要条件,为变压器温度场的数值模拟研究提供了理论基础.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.     

Adaptive parameter choice for one-sided finite difference schemes and its application in diabetes technology

In this paper we discuss the problem of approximation of the first derivative of a function at the endpoint of its definition interval. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. A natural way to approach the problem is to use one-sided finite difference schemes for numerical differentiation, but, following this way, one should be aware that the values of the function to be differentiated are noisy and available only at given fixed points. Then (as we argue in the paper) the number of used point values is the only parameter to be employed for regularization of the above mentioned ill-posed problem of numerical differentiation. In this paper we present and theoretically justify an adaptive procedure for choosing such a parameter. We also demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data.     

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.     

一维波动方程联合反演的分步正则化法

本文只用一个纵波信息,对一维波动方程的速度和震源函数进行联合反演.并考虑到波动方程的反问题是一不适定问题,对震源函数和波速分别用正则化法分步迭代求解,大大减少了反问题的计算工作量,改善了该反问题的计算稳定性.为计算实际一维地震数据提供了一种方法.文中给出了只用一个反问题补充条件同时进行多参数反演的详细公式,并对相应的数值算例进行了分析和比较.     

正则参数求解的微分进化算法

研究了正则化方法中正则参数的求解问题,提出了利用微分进化算法获取正则参数.微分进化算法属于全局最优化算法,具有鲁棒性强、收敛速度快、计算精度高的优点.把正则参数的求解问题转化为非线性优化问题,通过保持在解空间不同区域中各个点的搜索,以最大的概率找到问题的全局最优解,同时还利用数值模拟将此方法与广义交叉原理、L-曲线准则、逆最优准则等进行了对比,数值模拟结果表明该方法具有一定的可行性和有效性.     

Regularization of backward heat conduction problem

We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.