Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Computational algorithms implementing gradient methods based on solution of direct and adjoint problems in weak formulations are proposed for a number of complex-valued inverse problems of parameter renewal in multicomponent parabolic distributed systems. This approach makes it unnecessary to construct Lagrangian functionals explicitly and to use Green’s functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 49–73, July–August 2007.     

Solution of integrated inverse problems for multicomponent hyperbolic distributed systems

The paper presents a technique for developing computational algorithms to solve inverse problems for multicomponent hyperbolic systems with principal and natural heterogeneous conjugation conditions. Explicit expressions for the Frechet derivatives of quadratic residual functionals are obtained to generate gradient computational algorithms. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 55–80, March–April 2008.     

Radial basis functions method for parabolic inverse problem

The radial basis functions (RBFs) method is employed to handle a class of multi-dimensional parabolic inverse problems. Because they are not modelled by classical parabolic initial-boundary value problems, theoretical behaviour and numerical approximation of these problems have been active areas of research. Based on the idea of RBF approximation, a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter. Moreover, with the meshless property, it can be used to handle multi-dimensional parabolic inverse problems defined on very complicated geometries.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

退化高阶抛物型分布参数系统的迭代学习控制

研究一类高阶分布参数系统的迭代学习控制问题,该类系统由退化高阶抛物型偏微分方程构成.根据系统所满足的性质,基于P型学习算法构建得到迭代学习控制器.利用压缩映射原理,证明该算法能使得系统的输出跟踪误差于L~2空间内沿迭代轴方向收敛于零.最后,仿真算例验证了算法的有效性.     

Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition

Applications of the multidomain Local Fourier Basis method [1], for the solution of PDEs on parallel computers are described. The present approach utilizes, in an explicit way, the rapid (exponential) decay of the fundamental solutions of elliptic operators resulting from semi-implicit discretizations of parabolic time-dependent problems. As a result, the global matching relations for the elemental solutions are decoupled into local interactions between pairs of solutions in neighboring domains. Such interactions require only local communications between processors with short communication links. Thus the present algorithm overcomes the global coupling, inherent both in the use of the spectral Fourier method and implicit time discretization scheme.This research is supported partly by a grant from the French-Israeli Binational Foundation for 1991–1992.     

A class of inverse problems for discontinuous systems

We analyze a class of inverse parametric problems for dynamic processes described by systems of ordinary differential equations whose form and piecewise-constant parameters depend on what subdomain in the state space the state of the process belongs to. The study was supported by the INTAS (Project No. 06-1000017-8909). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 142–152, November–December 2008.     

An algebraic formulation of inverse problems in MP dynamics

Metabolic P (MP) grammars are a particular class of multiset rewriting grammars introduced in the MP theory for modelling metabolic processes. In this paper, a new algebraic formulation of inverse problems, based on MP grammars and Kronecker product, is given, for further motivating the correctness of the LGSS (Log-Gain Stoichiometric Stepwise) algorithm, introduced in 2010s for solving inverse problems in the MP framework. At the end of the paper, a section is included that introduces the problem of multicollinearity, which could arise during the execution of LGSS, and that defines an algorithm, based on a hierarchical clustering technique, that solves it in a suitable way.     

Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification

Two different explicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order, 5-point Forward Time Centred Space (FTCS) explicit formula, and the (1,9) FTCS explicit scheme which is generally second-order, but is fourth order when the diffusion number takes the value s = (1/6). These schemes are economical to use, are second-order and have bounded range of stability. The range of stability for the 5-point formula is less restrictive than the (1,9) FTCS explicit scheme. The results of numerical experiments are presented, and accuracy and Central Processor (CPU) times needed for each of the methods are discussed. These schemes use less central processor times than the second-order fully implicit method for two-dimensional diffusion with temperature overspecification. We also give error estimates in the maximum norm for each of these methods.     

Numerical solution of three-dimensional boundary-value problems by generalized approximate inverse matrix techniques

Generalized Approximate Inverse Matrix (GAIM) techniques based on the concept of LU-type sparse factorization procedures are introduced for calculating explicitly approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the factors L and U. Explicit first and second-order iterative methods in conjunction with modified forms of the GAIM techniques are presented for solving numerically three-dimensional initial/boundary-value problems on multiprocessor systems. Applications of the new methods on a 3D boundary-value problem is discussed and numerical results are given.     

On the optimal tuning of a robust controller for parabolic distributed parameter systems

Optimal tuning of robust multivariable controllers for stable parabolic distributed parameter systems is discussed. In order to evaluate the behaviour of the closed loop system, a quadratic criterion is selected. Then it is proved that the system may be decomposed into slow and fast subsystems, and that the value of the criterion will mainly depend on the slow subsystem. Tuning of the controller, for this subsystem, turns out to be a standard finite-dimensional quadratic regulator problem. Two examples are given to show the improvement due to the new selection of the control parameters.     

新的鲁棒粒子群算法在电磁场逆问题中应用研究

提出了一种搜索鲁棒优化解的粒子群算法。为解决期望适值函数计算需要大量新采样点而导致的计算效率过低问题,提出了一种期望适值赋值的新机制。该机制只对每一代粒子中的个体最优解和整体最优解分配期望适值。此外,为便于算法搜索鲁棒优化解,重新定义了粒子的邻域关系。最后,通过两个实例计算证明了新算法求解电磁场逆问题鲁棒优化解的可行性和优点。     

具连续分布时滞的抛物型系统的变结构控制

基于比较原理,利用推广的向量Hanalay微分不等式,Dini导数,结合Green公式及不等式分析技术,研究一类具分布时滞的抛物型控制系统的变结构控制问题．首先对所导出的滑动模运动方程,在仅要求系数矩阵是个M-矩阵的条件下,获得了滑动模运动方程全局指数稳定性的充分条件,建立了滑动模运动方程全局指数稳定性定理．其次,设计了仅由状态函数描述的变结构控制器,给出了运动轨线到达滑动模态区的时间的估计．     

信赖域共轭梯度法求解二次规划逆问题

为了有效地求解二次规划逆问题,提出了一种求解其对偶问题的子问题的光滑化信赖域共轭梯度法。该方法采用增广拉格朗日法求解其对偶问题,引入光滑函数将对偶问题的子问题转换成连续的无约束优化问题,将信赖域法与共轭梯度法结合,设计出求解二次规划逆问题的算法流程。数值实验结果表明,该方法可行且有效,与牛顿法相比,更适合求解大规模问题。     

全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

一般非线性系统的构造性逆系统方法

针对由于逆系统方法之求逆算法中求解隐函数方程的困难, 通过消元法避开对隐函数方法的求解, 提出了一般非线性多变量系统的构造性求逆算法. 在此基础上讨论对逆系统方法的相应改进, 通过动态补偿进一步提出了构造性逆系统方法. 这种改进使得逆系统方法原则上可以构造性地适用于任意足够光滑的非线性系统.     

Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems

This paper studies the parameter identification problem of nonlinear abstract parabolic distributed parameter systems via variational method ^[1]. Based on the fundamental optimal control theory and the transposition method studied in ^[2], the existence of optimal parameter is proved, and the necessary condition for the optimal parameter is established.     

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.     

A Guass–Newton-like method for inverse eigenvalue problems

In this paper, we propose a Guass–Newton-like method for finding least-square solutions to inverse eigenvalue problems. We show that the proposed method converges under some mild conditions. In particular, if the method converges to the exact solution, the convergence rate is at least quadratic in the root sense. Numerical examples are given to justify the theoretical result.     

Sensor and controller location problems for distributed parameter systems

Optimal location of point sensors and controllers in stochastic regulator problems for evolution equations is considered using a semigroup approach and the separation principle. The existence of an optimal location is established under very general hypotheses. The theory is illustrated by specializing to a diffusion equation and some computational results are presented.